MODELING STUDY OF NONLINEAR DYNAMICS IN THE GROWTH OF AQUATIC
PHYTOPLANKTON
by
Qing Guo

BSc., Jilin Agricultural University, 2015
MSc., Wenzhou University, 2018

DISSERTATION SUBMITTED IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
IN
NATURAL RESOURCES AND ENVIRONMENTAL STUDIES

UNIVERSITY OF NORTHERN BRITISH COLUMBIA
July 2023
© Qing Guo, 2023

ABSTRACT
Phytoplankton bloom has become a growing global concern in recent years due to the
excessive growth of algae, causing significant negative impacts on aquatic ecosystem and
threatening human health. Growing evidence suggests that algal blooms are a consequence of the
interplay of various hydrodynamical, chemical, and biological processes in aquatic systems. The
complexity and nonlinearity of aquatic ecosystems, and the complexity of climatic and
hydrographic events, make interpreting and predicting the blooms a very challenging task. In recent
years, many different strategies have been adopted to manage algal blooms. Among them,
mathematical models are advantageous because they can capture the ubiquitous stoichiometric
constraints for modeling species growth and interaction. Thus, mathematical models have been
widely used to investigate the dynamics of phytoplankton growth. In this study, five mathematical
models were developed based on population dynamics, ecological dynamics, dynamic modeling,
and probability theory. The models were investigated theoretically and numerically in terms of the
theory of partial differential equations, stochastic differential equations, impulsive differential
equations, and numerical simulation. The objective of this dissertation research was to gain insight
into plankton dynamics and explore potential management strategies for excessive algal growth in
aquatic systems. The main results are presented as follows:
Firstly, a nutrient-plankton model incorporating delay and diffusion was developed. The
theoretical analysis revealed that delay can trigger the nutrient-plankton oscillation via a Hopf bifurcation.
Especially, the stability switches for positive equilibrium occur with increasing delay, which indicates that
delay can generate and suppress the unstable coexistence of species population. Numerical results reveal
that the stability switches for positive equilibrium indeed exist in the model, and the homogeneous multiple
periodic solutions, as well as chaos, can occur with different values of delay, which implies that the model
ii

exhibits delay-induced complex dynamics.
Secondly, a stochastic Leslie-Gower phytoplankton-zooplankton model with prey refuge was
developed. The dynamical analysis revealed the sufficient conditions for the persistence and
extinction of plankton populations. The numerical simulations showed that environmental noise
and prey refuge play a crucial role in the survival of plankton species. The results further
demonstrate that prey refuge can enhance the oscillation range of phytoplankton population, but
the variance of zooplankton tends to increase and then decrease as prey refuge increased.
Thirdly, considering seasonal disturbances in aquatic ecosystem, a stochastic nutrientphytoplankton model with seasonal fluctuation was developed. The results indicate that periodic
solutions exist under certain conditions, suggesting that plankton populations are associated with
periodic oscillations. Furthermore, numerical results showed that seasonal fluctuation can trigger
periodic blooms of phytoplankton and promote the coexistence of plankton species. Specifically,
the results indicate that phytoplankton is more sensitive to nutrient than to seasonal fluctuation.
Fourthly, a stochastic nutrient-plankton model with regime-switching was developed by
considering regime-switching plankton mortality. The results showed that the model admits a
stationary distribution under certain conditions. Then the numerical analysis revealed that the
persistence and extinction of plankton populations are sensitive to variations in nutrient input. The
numerical results also indicate that regime-switching plankton mortality contributes to the survival
of plankton populations in the aquatic system.
Finally, a stochastic nutrient-plankton model with impulsive control was developed. The
theoretical analysis established sufficient conditions for the existence of periodic solutions. In
addition, the numerical analysis showed that nutrient impulse plays an important role in preventing
and controlling algal blooms, and appropriate environmental fluctuation can promote the
coexistence of phytoplankton and zooplankton populations. However, excess intensity noise can
iii

result in the collapse of the entire ecosystem.

iv

CO-AUTHORSHIP
This dissertation was primarily conducted by me, as I was responsible for developing
overarching research goals and aims, creating models, and developing software. I also wrote the
manuscripts and incorporated comments and feedback during revisions. Dr. Jianbing Li supervised
the research progress and provided valuable feedback during manuscript revisions. Dr. Min Zhao
supervised field data collection and provided computing resources and analysis tools. Dr. Chuanjun
Dai provided computer code, supporting algorithms, and computing resources, and contributed to
the manuscript revisions. Dr. He Liu also provided valuable input to improve the manuscripts. All
resulting publications include authorship from Dr. Jianbing Li, Dr. Min Zhao, Dr. Chuanjun Dai,
and Dr. He Liu. Additionally, Dr. Hengguo Yu, Dr. XiuXiu Sun, Dr. Yi Wang, Dr. Pankaj Kumar
Tiwari, Dr. Zhan Jin, and MS. Lijun Wang provided manuscript reviews and suggestions, and as a
result, they were included as co-authors on some of the publications.
Publication and authorships from this dissertation (Published or prepared for submission):
Guo, Q., Dai, C.J., Yu, H.G., Liu, H., Sun, X.X., Li, J., Zhao, M., 2020. Stability and bifurcation
analysis of a nutrient‐phytoplankton model with time delay. Mathematical Methods in the Applied
Sciences. 43, 3018-3039. DOI: 10.1002/mma.6098.
Guo, Q., Wang, L.J., Liu, H., Wang, Y., Li, J., Tiwari, P.K., Zhao, M., Dai, C.J., 2022. Stability
switches and chaos induced by delay in a nutrient-plankton model with diffusion. Submitted to
Journal of Biological Dynamics on October 9, 2022 (Revision Required on June 27, 2023).
(Chapter 3)
Guo, Q., Dai, C.J., Wang, L.J., Liu, H., Wang, Y., Li, J., Zhao, M., 2021. Dynamics of a stochastic
Leslie-Gower phytoplankton-zooplankton model with prey refuge. (Prepared), (Chapter 4)
Guo, Q., Dai, C.J., Wang, L.J., Liu, H., Wang, Y., Li, J., Zhao, M., 2022. Stochastic periodic
solution of a nutrient-plankton model with seasonal fluctuation. Journal of Biological Systems. 30,
695-720. DOI: 10.1142/S0218339022500255. (Chapter 5)
Guo, Q., Wang, Y., Dai, C.J., Wang, L.J., Liu, H., Li, J., Tiwari, P.K., Zhao, M., 2023. Dynamics
of stochastic nutrient-plankton model with regime-switching. Ecological Modelling. 477, 110249.
DOI: 10.1016/j.ecolmodel.2022.110249. (Chapter 6)
Guo, Q., Liu, H., Wang, Y., Li, J., Zhao, M., Tiwari, P.K., Jin, Z., Dai, C.J., 2023. Dynamics of a
stochastic nutrient-plankton model with impulsive control strategy. European Physical Journal
Plus. 138, 470. DOI: 10.1140/epjp/s13360-023-04111-0. (Chapter 7)
Zhao, X., Wang, L.J., Tiwari, P.K., Liu, H., Wang, Y., Li, J., Zhao, M., Dai, C.J., Guo, Q., 2023.
v

Investigation of a nutrient-plankton model with stochastic fluctuation and impulsive control.
Mathematical Biosciences and Engineering. Accepted on July 5, 2023.
Liu, H., Dai, C.J., Yu, H.G., Guo, Q., Li, J., Hao, A.M., Kikuchi, J., Zhao, M., 2021. Dynamics
induced by environmental stochasticity in a phytoplankton-zooplankton system with toxic
phytoplankton. Mathematical Biosciences and Engineering. 18, 4101-4126. DOI: 10.3934/mbe.20
21206.
Liu, H., Dai, C.J., Yu, H.G., Guo, Q., Li, J., Hao, A.M., Kikuchi, J., Zhao, M., 2022. Dynamics of
a stochastic phytoplankton–toxic phytoplankton–zooplankton system under regime switching.
Mathematical Methods in the Applied Sciences. 45, 9769-9786. DOI: 10.1002/mma.8334.
Liu, H., Dai, C.J., Yu, H.G., Guo, Q., Li, J., Hao, A.M., Kikuchi, J., Zhao, M., 2023. Dynamics of
a stochastic non-autonomous phytoplankton–zooplankton system involving toxin-producing
phytoplankton and impulsive perturbations. Mathematics and Computers in Simulation. 203, 368386. DOI: 10.1016/j.matcom.2022.06.012.

vi

TABLE OF CONTENTS
ABSTRACT................................................................................................................................. ii
CO-AUTHORSHIP ..................................................................................................................... v
TABLE OF CONTENTS........................................................................................................... vii
LIST OF TABLES ...................................................................................................................... xi
LIST OF FIGURES ................................................................................................................... xii
ACKNOWLEDGEMENT ......................................................................................................... xv
Chapter 1 GENERAL INTRODUCTION ................................................................................... 1
1.1. Background ....................................................................................................................... 1
1.2. Statement of the problem .................................................................................................. 3
1.3. Objectives of the study ...................................................................................................... 6
1.4. Organization of the dissertation ........................................................................................ 7
Chapter 2 LITERATURE REVIEW............................................................................................ 9
2.1. The causes of algal blooms ............................................................................................... 9
2.1.1. Nutrient supply of nitrogen (N) and phosphorus (P) .................................................. 9
2.1.2. Light intensity........................................................................................................... 10
2.1.3. Water temperature .................................................................................................... 10
2.1.4. Biotic factor .............................................................................................................. 11
2.1.5. Climate and hydrology ............................................................................................. 11
2.1.6. Human activities ....................................................................................................... 12
2.2. The characteristics of algal blooms ................................................................................. 13
2.3. Algal bloom impacts ....................................................................................................... 14
2.4. Bloom control.................................................................................................................. 16
2.5. Nonlinear dynamics of planktonic ecosystems ............................................................... 18
2.5.1. Delay-induced nonlinear dynamics .......................................................................... 20
2.5.2. Stochastic population dynamics ............................................................................... 24
2.5.3. The dynamics of impulsive control system under environmental fluctuation.......... 27
2.6. Summary of literature review.......................................................................................... 29
Chapter 3 STABILITY SWITCHES AND CHAOS INDUCED BY DELAY IN A NUTRIENTPLANKTON MODEL WITH DIFFUSION ............................................................................. 31
Abstract .................................................................................................................................. 31
3.1. Introduction ..................................................................................................................... 32
3.2. The mathematical model ................................................................................................. 36
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3.3. The main results .............................................................................................................. 38
3.3.1. The existence of equilibria ....................................................................................... 38
3.3.2. Stability and Hopf bifurcation .................................................................................. 39
3.4. Numerical simulation ...................................................................................................... 41
3.5. Conclusions ..................................................................................................................... 47
Chapter 4 DYNAMICS OF A STOCHASTIC LESLIE-GOWER PHYTOPLANKTONZOOPLANKTON MODEL WITH PREY REFUGE ............................................................... 50
Abstract .................................................................................................................................. 50
4.1. Introduction ..................................................................................................................... 51
4.2. The methods and results .................................................................................................. 53
4.2.1. Mathematical model ................................................................................................. 53
4.2.2. Global positive solutions and persistence and extinction ......................................... 54
4.2.3. Stationary distribution and ergodicity ...................................................................... 56
4.3. Numerical simulation ...................................................................................................... 56
4.4. Conclusions ..................................................................................................................... 61
Chapter 5 STOCHASTIC PERIODIC SOLUTION OF A NUTRIENT-PLANKTON MODEL
WITH SEASONAL FLUCTUATION ...................................................................................... 63
Abstract .................................................................................................................................. 63
5.1. Introduction ..................................................................................................................... 64
5.2. The mathematical model .................................................................................................. 66
5.3. The main results .............................................................................................................. 69
5.3.1. Survival analysis of model (5.2)............................................................................... 69
5.3.2. Existence of the nontrivial periodic solution of model (5.2) .................................... 70
5.4. Numerical simulation ...................................................................................................... 71
5.5. Conclusion....................................................................................................................... 78
Chapter 6 DYNAMICS OF A STOCHASTIC NUTRIENT-PLANKTON MODEL WITH
REGIME-SWITCHING.............................................................................................................. 81
Abstract .................................................................................................................................. 81
6.1. Introduction ..................................................................................................................... 82
6.2. The mathematical model ................................................................................................. 85
6.3. The main results .............................................................................................................. 90
6.3.1 Stochastically ultimately boundness.......................................................................... 90
6.3.2 Stochastic permanence .............................................................................................. 91
6.3.3 The persistence and extinction of plankton ............................................................... 92
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6.3.4 Ergodicity of model (6.2) .......................................................................................... 93
6.4. Numerical simulation ...................................................................................................... 94
6.4.1. Sensitivity analysis ................................................................................................... 94
6.4.2. Effects of regime shifting plankton mortality........................................................... 99
6.5. Discussion ..................................................................................................................... 104
6.6. Conclusions ................................................................................................................... 108
Chapter 7 DYNAMICS OF A STOCHASTIC NUTRIENT-PLANKTON MODEL WITH
IMPULSIVE CONTROL STRATEGY .................................................................................. 109
Abstract ................................................................................................................................ 109
7.1. Introduction ................................................................................................................... 110
7.2. The mathematical model ............................................................................................... 113
7.3. The main results ............................................................................................................ 116
7.3.1 Existence and uniqueness of the global positive solution ....................................... 116
7.3.2 Existence of positive -periodic solution ................................................................ 117
7.4. Numerical simulation .................................................................................................... 118
7.4.1 Effects of impulsive control .................................................................................... 120
7.4.2 Effects of stochastic fluctuation .............................................................................. 123
7.5. Conclusion and discussion ............................................................................................ 125
Chapter 8 CONCLUSION AND RECOMMENDATIONS.................................................... 128
8.1. Synthesis and conclusion .............................................................................................. 128
8.2. Limitations and future research..................................................................................... 132
References................................................................................................................................ 135
APPENDIX A. The proof of Theorem 3.1 .............................................................................. 170
APPENDIX B. The proof of Lemma 3.1................................................................................. 172
APPENDIX C. The proof of Lemma 3.2................................................................................. 173
APPENDIX D. The proof of Theorem 3.3 .............................................................................. 174
APPENDIX E. Auxiliar Lemmas ............................................................................................ 181
APPENDIX F. The proof of Lemma 4.1 ................................................................................. 182
APPENDIX G. The proof of Lemma 4.2 ................................................................................ 184
APPENDIX H. The proof of Theorem 4.1 .............................................................................. 187
APPENDIX I. The proof of Theorem 4.2................................................................................ 190
APPENDIX J. The proof of Lemma 5.1.................................................................................. 194
APPENDIX K. The proof of Theorem 5.1 .............................................................................. 195

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APPENDIX L. The proof of Theorem 5.2............................................................................... 198
APPENDIX M. The proof of Theorem 6.1.............................................................................. 202
APPENDIX N. The proof of Lemma 6.1 ................................................................................ 204
APPENDIX O. The proof of Theorem 6.2 .............................................................................. 207
APPENDIX P. The proof of Lemma 6.2 ................................................................................. 208
APPENDIX Q. The proof of Lemma 6.3 ................................................................................ 209
APPENDIX R. The proof of Theorem 6.3 .............................................................................. 210
APPENDIX S. The proof of Theorem 6.4 ............................................................................... 213
APPENDIX T. The proof of Theorem 6.5............................................................................... 218
APPENDIX U. Auxiliary results ............................................................................................. 219
APPENDIX V. The proof of Theorem 7.1 .............................................................................. 222
APPENDIX W. The proof of Theorem 7.2 ............................................................................. 228

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LIST OF TABLES
Table 3.1. Biological explanations of variables and parameters in model (3.1), and numerical
values used for simulation results ......................................................................................... 41
Table 4.1. Biological explanations of variables and parameters in model (4.1), and numerical
values used for simulation results ........................................................................................ 57
Table 5.1. Biological explanations of variables and parameters in model (5.1), and numerical
values used for simulation results ......................................................................................... 71
Table 6.1. Biological explanations of variables and parameters in model (6.1), and numerical
values used for simulation results ......................................................................................... 95
Table 7.1. Biological explanations of parameters in model (7.1), and their numerical values used
for simulation results .......................................................................................................... 119

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LIST OF FIGURES
Figure 1.1. Schematization of research content. ............................................................................ 6
Figure 3.1. The scheme representation of the model (3.1)........................................................... 38
Figure 3.2. Biomass distributions of (a) phytoplankton and (b) phytoplankton populations in model
(3.1) over time and space for = 0. .......................................................................................... 42
Figure 3.3. (a) The number of stability switches of the positive equilibrium of model (3.1) in the
− plane. Stability switches occur once in region III, twice in region IV, three times in region
V, four times in region VI, five times in region VII, six times in region VIII and seven times in
region IX. (b) The partial enlarge diagram of (a). ........................................................................... 42
Figure 3.4. Bifurcation diagrams of model (3.1) with respect to and for (a) = 0.45 and (b)
= 0.35. In the figures, the solid, dashed, dash-dot, and dotted curves represent the critical
values of for = 0, 1, 2, 3 respectively. .................................................................................. 43
Figure 3.5. Bifurcation diagrams of model (3.1) with respect to for (a) = 0.45 and = 0.1,
and (b) = 0.35 and = 0.05. In the figure, the solid blue line and the dotted red line represent
the stable positive equilibrium and the maximum biomass of phytoplankton, respectively...... 43
Figure 3.6. Biomass distribution of phytoplankton population in model (3.1) over time and space
for = 0.45, = 0.1 and different values of : (a) = 2, (b) = 10 and (c) = 18............ 44
Figure 3.7. Biomass distribution of phytoplankton population in model (3.1) over time and space
for = 0.35, = 0.05 and different values of : (a) = 3, (b) = 10, (c) = 18, (d) = 36,
(e) = 42 and (f) = 50........................................................................................................... 45
Figure 3.8. (a) Bifurcation diagram of model (3.1) with respect to for = 0.6 and = 0.1. In
the figure, the green, blue and yellow solid diamonds respectively represent the periodic−1, 2 &
3 solutions at = 10, 30 and 56; the black dashed line denotes chaotic solution at = 74. Figure
(b)-(e) show the biomass distributions of phytoplankton in model (3.1) over time and space at
= 10, = 30, = 56 and = 74, respectively. ................................................................... 46
Figure 4.1. The scheme representation of the model (4.1)........................................................... 54
Figure 4.2. The analysis of the threshold for the extinction and persistent of model (4.1) in ( −
− ) space. The space I corresponds to Theorem 4.1 (i); the space II corresponds to Theorem
4.1 (ii); the space III corresponds to Theorem 4.1 (iii); the space IV corresponds to Theorem 4.1
(iv); the space V corresponds to Theorem 4.1 (v)...................................................................... 58
Figure 4.3. Solutions of model (4.1) for = 0.2 with (a) = 0.7, = 0.5; (b) = 0.7, =
0.3; (c) = 0.3, = 0.5; (d) = 0.5, = 0.3; (e) = 0.15, = 0.35. ........................ 59
Figure 4.4. For model (4.1) with =
= 0.2, (a) the distribution of and (b) the distribution
of with respect to ............................................................................................................... 60
Figure 4.5. For model (4.1) with =
= 0.2, (a) the boxplot of and with respect to ;
(b) the mean and variance of and with respect to ......................................................... 60
Figure 4.6. For model (4.1) with =
= 0.2, (a) the sample path of with respect to =
0.3, 0.5, 0.7, 0.9; (b) the sample path of with respect to = 0.3, 0.5, 0.7, 0.9..................... 60
xii

Figure 5.1. The scheme representation of the model (5.1)........................................................... 68
Figure 5.2. For model (5.1) without noise, (a) the solutions of in the ( , ) plane and (b) the
solutions of in the ( , ) plane. For model (5.2) without noise, (c) the solutions of in the
( , ) plane and (d) the solutions of in the ( , ) plane. The white dash line and the red dash
line represents the critical value of ℜ = 0 and ℜ =
/ , respectively......................... 72
Figure 5.3. For model (5.1) without noise, (a) The sample paths of phytoplankton and (b)
zooplankton corresponding to
= 0.5 + 0.1sin( /25) , 3 + 0.1sin( /25) , 5.5 + 0.1sin( /
25) and 8 + 0.1sin( /25), respectively. The black line represents the solutions for model (5.2)
without noise. ............................................................................................................................. 73
Figure 5.4. For model (5.1), (a) the solutions of in the ( , ) plane and (b) the solutions of in
the ( , ) plane. For model (5.2), (c) the solutions of in the ( , ) plane and (d) the solutions
of in the ( , ) plane. The white dash line and the red dash line represent the critical value of
ℜ = 0 and ℜ = β β /m , respectively.................................................................................. 74
Figure 5.5. For model (5.2), (a) the sample paths of phytoplankton and (b) zooplankton
corresponding to
= 0.5 + 0.1sin( /25) , 3 + 0.1sin( /25) , 5.5 + 0.1sin( /25) and 8 +
0.1sin( /25), respectively. The black line represents the solutions for the deterministic model.
(c) Probability histograms for phytoplankton and (d) zooplankton with = 8 + 0.1sin( /25).
.................................................................................................................................................... 75
Figure 5.6. For model (5.1) and (5.2), (a) the mean and variance of phytoplankton biomass with
respect to nutrient input ; (b) The mean and variance of zooplankton biomass with respect to
nutrient input . ........................................................................................................................ 76
Figure 5.7. For model (5.2), (a) the distribution of phytoplankton biomass in different periods; (b)
the distribution of zooplankton biomass in different periods..................................................... 77
Figure 5.8. For model (5.2), (a) the solutions of in the ( , ) plane and (b) the solutions of in
the ( , ) plane with = 2 + 0.1sin( /25); (c) The solutions of in the ( , ) plane and (d)
the solutions of in the ( , ) plane with = 5 + 0.1sin( /25); (e) The solutions of in the
( , ) plane and (f) The solutions of in the ( , ) plane with = 8 + 0.1sin( /25)......... 78
Figure 6.1. Schematic diagram representing the dynamics of the considered nutrient–
phytoplankton–zooplankton model. In the figure, the black dashed lines represent the remineralizations of the dead biomasses of phytoplankton and zooplankton into the nutrient
concentration. ....................................................................................................................... 88
Figure 6.2. (a) Semi-relative sensitivity and (b) logarithmic sensitivity solutions of the model (6.2)
in the absence of environmental noise with respect to , , , ℎ, , and . Parameters are
at the same values as in Table 6.1 except =1.2, =0.5, =0.4 and =0.04....................... 96
Figure 6.3. For model (6.2) without noise and regime switching, the mean values of the solutions
with respect to nutrient input................................................................................................ 98
Figure 6.4. The solutions of model (6.1) in the ( , ) plane for (a) and (b) ; the solutions of
model (6.2) without environmental noise in the ( , ) plane for (c) and (d) ; the solutions
of model (6.2) in the ( , ) plane for (e) and (f) . The white and the red dashed line
xiii

respectively represent the critical value of ℜ = 0 and ℜ =
/ . .............................. 100
Figure 6.5. The paths of phytoplankton and zooplankton in the model (6.2) for (a) = = 0.5,
(b) = 0.2, = 1 and (c) = 0.5, = 12. .................................................................. 101
Figure 6.6. For model (6.2) with = 0.5, (a) the Markov chain with respect to
= 8; (b) the
sample path of phytoplankton; (c) the distribution of phytoplankton population with respect
to state 1, state 2 and hybrid system, respectively; (d) the distribution of phytoplankton
population at = 1000 with respect to nutrient input . ................................................. 102
Figure 6.7. (a) The mean and the variance for phytoplankton biomass with respect to nutrient input
; (b) The mean and the variance for zooplankton biomass with respect to nutrient input .
............................................................................................................................................ 103
Figure 6.8. For model (6.2), the sample paths of (a) phytoplankton and (b) zooplankton for =
1, 4, 7 and 10, respectively. The black line represents the solutions for determined model.
............................................................................................................................................ 104
Figure 7.1. Schematic diagram depicting the interplay among nutrients, phytoplankton and
zooplankton in an aquatic ecosystem. ................................................................................ 115
Figure 7.2. The evolution of a single path of solutions for model (7.2) and the corresponding
impulsive model without random noise. The figures depict the paths of (a) nutrient, (b)
phytoplankton, (c) zooplankton, and the (d) phase diagram. ............................................. 120
Figure 7.3. Under =
=
= 0.02 + 0.01sin( /20), the evolution of a single path of (a)
nutrient, (b) phytoplankton and (c) zooplankton in model (7.2) with respect to time for three
different sets of pulse intensity........................................................................................... 121
Figure 7.4. The evolutions of a single path of phytoplankton (first row) and zooplankton (second
row) in model (7.2) with respect to time for
= 0.1, 0.4, 0.7, and 1. In this figure, =
=
= 0.02 + 0.01sin( /20), and
= 0.1 (first column),
= 0.3 (second column)
and
= 0.5 (third column). ............................................................................................. 121
Figure 7.5. Under
=
=
= 0.02 + 0.01sin( /20) ,
= 0.5 and
= 0.1 , the
evolution of a single path of (a) nutrient, (b) phytoplankton and (c) zooplankton in model (7.2)
with respect to time for three different values of . ........................................................... 122
Figure 7.6. The evolution of a single path of (a) nutrient, (b) phytoplankton and (c) zooplankton
in model (7.2) with respect to time for
= 0.5 and
= 0.1. In the figure, the red, blue and
black lines respectively correspond to
=
=
= 0.02 + 0.01sin( /20), =
=
= 0.1 + 0.01sin( /20) and =
=
= 0.5 + 0.01sin( /20). ...................... 124

xiv

ACKNOWLEDGEMENT
First and foremost, I would like to express to my supervisor, Dr. Jianbing Li, and cosupervisor, Dr. Min Zhao, for their invaluable guidance, encouragement, and support throughout
this study. Without their advice and support, I would not have been able to complete this research.
I would also like to thank my supervisory committee members, Dr. Ron Thring and Dr. Youmin
Tang, for their insightful guidance and encouragement.
I would also like to extend my special appreciation to Dr. Chuanjun Dai from Wenzhou
University for generously sharing his expertise and insights in research and providing selfless assistance in
both my research and personal life. Dr. Dai consistently offered valuable suggestions and resources that
greatly contributed to my research. My heartfelt thanks also go to Dr. Bill McGill, Dr. Darwyn
Coxson, and Dr. Ellen Petticrew for their supervision during my first year of the PhD program.
Their guidance inspired me to approach my research from a broader perspective and instilled in
me a positive and optimistic outlook on life. Additionally, I am grateful to all the staff and faculty
members at UNBC for their help, which made my life and studies at UNBC more comfortable.
This research received support from multiple organizations, including the National Natural
Science Foundation of China, the National Key Research and Development Program of China, the
Zhejiang Provincial Natural Science Foundation of China, the UNBC Graduate Entrance Research
Award (GERA), Wenzhou University Joint Doctoral Stipend, Wenzhou Huang Mengqi Dandelion
Charity Foundation, and the PhD Dissertation Completion Award.
I would like to express my thanks to Dr. Youqin Wang for providing me with lab equipment,
supplies, and the necessary resources for my simulation work at UNBC. I am also grateful to Dr.
Yuan Tian, Dr. Jingjing Shi, and Dr. He Liu for their encouragement and selfless help. I would
like to acknowledge my classmates at UNBC for sharing their knowledge, skills, and thoughtful
discussions. Special thanks go to Dr. Xiufeng Yan, Dr. Mingjiang Wu, Dr. Zengling Ma, Dr.
xv

Xiangyong Zheng, Dr. Qi Wang, Dr. Min Wang, Dr. Wenli Qin, and Dr. Hengguo Yu for their
support and professional advice regarding water sampling and data analysis at Wenzhou University.
I extend my gratitude to all my friends in China and Canada, with a special thanks to Ms.
Jingjing Su, Ms. Lijun Wang, Mr. Lei Shen, Mr. Jess, and all my teammates who supported me
through many challenging times during my PhD studies. Lastly, I would like to express my
heartfelt thanks to my family for their unconditional support and love. I am especially grateful to
my parents, my elder brother, my cousins, and my relatives, who consistently support and
understand my choices, brightening my life with their presence.

xvi

Chapter 1
GENERAL INTRODUCTION
1.1. Background
Aquatic planktonic ecosystems are natural systems formed by the interaction between
planktonic organisms and the abiotic environment. They encompass a specific spatiotemporal
range, and are an integral part of aquatic ecosystems. Plankton, a group of organisms that passively
float in the water layer under water movement, form the foundation of productivity in the water.
Plankton can be classified into six distinct groups based on their size: picoplankton (< 2 m),
nanoplankton (2-20 m), microplankton (20-200 m), mesoplankton (0.2-2 mm), macroplankton
(2-20 mm), and megaplankton (> 20 mm) (Baretta-Bekker et al., 1998; Medvinsky et al., 2002;
Raymont, 1980). Additionally, the functional classification of plankton is significantly influenced
by their trophic level, size, and distribution (Medvinsky et al., 2002). Based on the nutritional
approach, plankton can be divided into two categories: phytoplankton and zooplankton species.
Phytoplankton are commonly known as floating plant species that inhabit various water
environments, including fresh-water lakes, larger rivers, and the pelagic zone of the sea (Moruff et
al., 2016). Phytoplankton are autotrophic plankton that produce organic matter through
photosynthesis by absorbing light energy and carbon dioxide. Phytoplankton, such as diatoms,
cyanobacteria, and methanogens, are unicellular and microscopic in size. They serve as the primary
producers in aquatic ecosystems and are the primary food source for zooplankton. In contrast,
zooplankton are consumers and secondary producers in aquatic ecosystems.
Apart from their role as the base of the food chain in aquatic ecosystems, phytoplankton also
contribute to our climate by absorbing harmful carbon dioxide from the atmosphere and producing
oxygen through photosynthesis. Additionally, phytoplankton recycle phosphorus, nitrogen, and
1

sulphur (Lobus et al., 2023). However, there is growing evidence of an increasing trend in the
frequency, intensity, and duration of algal blooms over the past decades. This includes the
emergence of more toxic species and the adverse effects on fisheries resources (Anderson, 2009;
Fang et al., 2022).
Algal blooms are characterized by rapid accumulation followed by an equally rapid decline
(Guo et al., 2023). Experimental and modeling studies suggest that excessive nutrient input (Zhang
et al., 2020), suitable temperature (Ralston and Moore, 2020), and favorable light intensity (Tian
et al., 2018) are likely to affect the characteristics of algal blooms, such as frequency, intensity,
and duration. Due to the abundance and diversity of phytoplankton, the scale and timing of blooms
are highly stochastic, with some occurring in spring and others in summer, and some localized in
bays or estuaries for a few weeks, while others are massive and last for years (Anderson, 1997;
Fang et al., 2019; Kim et al., 2022). These events can generally be categorized into two types:
spring blooms and red tides. Spring blooms are mainly induced by seasonal changes in temperature
or nutrient availability, influenced by seasonal variations in thermocline depth and strength (Ho
and Michalak, 2020; Leruste et al., 2019). In contrast, red tides occur independently of specific
regions or high water temperatures and are localized outbreaks commonly observed in coastal
waters, estuaries, and fronts (Li et al., 2021; Truscott and Brindley, 1994).
Algal blooms can have a negative impact on aquatic ecosystems (Huang et al., 2019). For
example, cyanobacteria can cause waterbodies to produce an unpleasant smell and form green
scum layers, significantly reducing water transparency and interfering with recreational activities
and the quality of drinking water reservoirs (Huisman et al., 2018; Sultana et al., 2022). In addition,
during the bloom phase, hypoxia and anoxia can be induced by a significant increase in sediment
oxygen consumption due to the microbial degradation of senescent blooms, resulting in the death
of fish and benthic invertebrates (Kumari et al., 2022). Some cyanobacterial blooms can produce
2

potent toxins that can cause various gastrointestinal and neurological illnesses in humans through
the food chain (Mello et al., 2018; Pradhan et al., 2022).
In recent decades, various countermeasures have been developed to control algal blooms,
including chemical treatments, physical contact between prey and predator, and biological control.
However, most of these methods have achieved limited success. For example, copper algicides
have proven effective in controlling harmful algal blooms in drinking water supplies, but they are
toxic to other aquatic microorganisms and can induce the release of toxins due to cyanobacterial
cell lysis (Shen et al., 2019). Similarly, while sonication is applicable for water treatment systems,
it is not suitable for large water bodies (Gallardo-Rodr ́guez et al., 2019). The underlying processes
of phytoplankton bloom formation are still not fully understood, and further knowledge is needed
to improve the understanding of algal blooms and develop effective control methods.
1.2. Statement of the problem
Due to the rapid development of industry and agriculture, a significant amount of industrial
wastewater and domestic sewage containing large amounts of nutrients are discharged into water,
resulting in severe eutrophication problems in recent years (Hansen et al., 2017). China has more
than 2300 natural lakes covering a surface area larger than 1 km2, mainly used for irrigation,
drinking water, aquaculture, and tourism (Liu and Qiu, 2007; Yang et al., 2016). Over the past

decades, increased nutrient inputs have significantly boosted the production of phytoplankton
and other microorganisms, leading to the deterioration of the normal functioning of water
bodies (Dubey and Dutta, 2020).
Under natural conditions, some phytoplankton species, such as Microcystis, the most common
freshwater cyanobacterial genus, typically form colonies (Xiao et al., 2018). Colony formation of
algae contributes to preventing algal species from being grazed by zooplankton, reducing damage
3

from ultraviolet radiation, resisting severe water turbulence, and providing protection from heavy
metals (Lürling, 2021; Xiao et al., 2018; Yamamoto et al., 2011). Additionally, the colony
formation of some phytoplankton species is important for their dominance and even bloom
formation (Yamamoto et al., 2011). Blooms are characterized by a sharp increase in the number of
large colonies, which is directly attributable to the population increase. Thus, restraining the
growth of phytoplankton is an effective way to prevent and control algal blooms. However, the
extent to which influencing factors affect algal growth mechanisms remains unclear.
In recent years, numerous field or experimental studies have been conducted to explore the
key drivers of the bloom phenomenon. Extensive studies support the notion that reducing nutrient
inputs to aquatic ecosystems is an efficient way to decrease the frequency and intensity of blooms
(Huisman et al., 2018). Additionally, the nitrogen to phosphorus ratio has been considered a key
driver in controlling phytoplankton growth and seasonal succession (Liu et al., 2021b). Bharathi
et al. (2018) experimentally suggest that low salinity and a high N:P ratio favor the growth of bluegreen algae. However, there is evidence suggesting that this may be an incomplete explanation for
the presence or absence of blooms. Bloom occurrence is ultimately a random phenomenon
resulting from a combination of climatic and hydrographic events (Guo et al., 2023; McGowan,
2017). For example, Ralston and Moore (2020) indicated that global warming expressed as an
increase in summer temperatures can increase the risk of phytoplankton bloom. Beaulieu et al.
(2013) suggested that some blooms, such as cyanobacterial blooms, are favored by warm
temperatures. Light intensity and carbon supply may also contribute to the formation of algal
blooms (Huang et al., 2019; Liao et al., 2018; Tian et al., 2018). These studies indicate that the
bloom phenomenon is much more complex than we expected.
Growing evidence reveals that rapid algal growth is a consequence of the interplay of various
hydrodynamics, chemical processes, and biological processes (Chen and Mynett, 2006; Guo et al.,
4

2023), making it difficult to understand the mechanism of phytoplankton growth based solely on
experimental and field observations. Moreover, the nonlinearity and complexity of aquatic
ecosystems make it challenging to develop management systems and improvement measures.
Therefore, mathematical modeling of plankton populations as an important alternative
methodology has drawn increasing attention (Dai et al., 2019).
One of the interesting problems in the dynamical analysis of phytoplankton growth models is
the oscillation behavior of plankton populations. While many phytoplankton growth models
suggest that plankton populations can coexist at equilibrium globally under certain conditions (Dai
et al., 2019; Deng et al., 2015), the results from Sherratt and Smith (2008) reveal that a constant
population density may not exist in reality due to the presence of environmental noise and physical
factors such as water temperature, light, and hydrodynamics. Moreover, experimental results
support that the physical factors, such as reduced vertical mixing, can generate oscillations and
chaos in phytoplankton biomass (Huisman et al., 2006).
In recent years, the dynamical behaviors, including regular, irregular, and chaotic
spatiotemporal oscillations, have gained significant attention in numerous experimental and
modeling studies of aquatic ecosystems. A long-term experimental study with a complex food web
was conducted by Beninc ̀ et al. (2008), and the results revealed that species interactions in food
webs can generate chaos. Furthermore, a nutrient-plankton model developed by Wang et al. (2016)
indicated that periodic oscillation and chaotic behaviors of plankton biomass can be induced by
delays resulting from phytoplankton absorbing nutrients. It is worth noting that compared to
experimental studies, mathematical models have the advantage of being able to capture these
ubiquitous stoichiometric constraints for modeling species growth and interaction.
Although various phytoplankton growth dynamical models have been explored to investigate
the mechanism of algal growth, the dynamic mechanism of phytoplankton growth remains a
5

challenge due to the complex and highly nonlinear nature of the planktonic ecosystem. It is
therefore important to gain a better understanding of the underlying mechanisms for this
phenomenon.

1.3. Objectives of the study
The main objective of this research is to investigate plankton dynamics using mathematical
models. Theoretical and numerical analyses are conducted to examine the dynamic properties of
the models. A summary schematization of the research content is shown in Figure 1.1.

Figure 1.1. Schematization of research content.

The specific objectives of this dissertation include:
⚫ Investigating how the delay caused by phytoplankton’s nutrient absorption affects the nutrientplankton interaction dynamics. Sufficient conditions for the stability switches of the positive
equilibrium were derived and the direction of Hopf bifurcation and the stability of the
bifurcation periodic solutions were tracked. In addition, chaotic behavior induced by delay was
6

investigated through simulation.
⚫ Examining the joint effect of prey refuge and stochastic fluctuation on plankton growth

dynamics. The existence of the global positive solution of the model was studied, and the
conditions for the model to admit an ergodic stationary distribution were derived. Moreover,
numerical results were presented to further explore the effect of noise density and prey refuge
on the intensity of planktonic oscillation.
⚫ Investigating how seasonal fluctuation affects the bloom events of phytoplankton under

stochastic environments. The survival analysis of plankton populations was conducted and the
sufficient conditions for the existence of a positive periodic solution were derived. Additionally,
numerical analysis was conducted to explore the effect of seasonal fluctuation on phytoplankton
growth.
⚫ Developing a stochastic nutrient-plankton model under regime switching to analyze the effect
of regime-switching plankton mortality on plankton dynamics. Theoretical analysis was
conducted to derive the sufficient conditions for the stationary distribution of the model.
Additionally, numerical simulations were conducted to investigate the sensitivity of plankton
dynamics to parameters. The plankton dynamics in response to shifting plankton mortality and
stochastic fluctuation were investigated.
⚫ Investigating a stochastic nutrient-plankton model with impulsive control both theoretically and
numerically. The sufficient conditions for the existence of a global positive solution and
periodic solutions were derived. Furthermore, numerical analysis was carried out to investigate
how plankton dynamics respond to stochastic fluctuation and impulsive control.
1.4. Organization of the dissertation
The dissertation is organized as follows: literature review is presented in Chapter 2; in
7

Chapters 3-7, the mathematical analyses for five different phytoplankton growth models are
developed, including the delayed nutrient-plankton model with diffusion (Chapter 3), the
stochastic Leslie-Gower phytoplankton-zooplankton model with prey refuge (Chapter 4), the
stochastic nutrient-plankton model with seasonal fluctuation (Chapter 5), the stochastic nutrientplankton model under regime switching (Chapter 6), and the stochastic nutrient-plankton model
with impulsive control (Chapter 7); in Chapter 8, some conclusions and recommendations for future
research are provided.

8

Chapter 2
LITERATURE REVIEW
2.1. The causes of algal blooms
Algal bloom is a significant sign of extreme eutrophication in aquatic ecosystems, drawing
increasing attention from researchers. Over the last several decades, the frequency, intensity, and
geographic distribution of such events have shown an increasing trend (Anderson et al., 2021), and
most coastal countries have experienced an escalating and worrisome trend in the incidence of
phytoplankton blooms. Extensive research has revealed that the formation of phytoplankton
blooms is a consequence of the interplay of various physical, geological, biological, chemical, and
hydrodynamic processes (Brandenburg et al., 2017; Chen and Mynett, 2006), contributing to the
difficulty in understanding how phytoplankton growth responds to environmental factors.

2.1.1. Nutrient supply of nitrogen (N) and phosphorus (P)
Phytoplankton growth process is primarily limited by the availability of nitrogen (N)
(demanded for protein synthesis) and phosphorus (P) (demanded for DNA, RNA, and energy
transfer) among multiple nutrients (Conley et al., 2009). Numerous studies have suggested that
eutrophication is typically caused by excess nitrogen and phosphorus inputs (Rathore et al., 2016),
resulting in the rate of primary production exceeding the rate at which it is utilized by secondary
consumers. Under high nutrient availability, specific algal species can become dominant through
nutrient uptake, leading to algal blooms. In addition to the overall level of nutrient input, the
nitrogen to phosphorus ratio is also an important driver of phytoplankton blooms, due to the
differing nutrient requirements of different algal species (Orihel et al., 2015). The results reported
by Paerl et al. (2018) indicated that while phytoplankton growth indeed requires nitrogen and
9

phosphorus, reducing nitrogen input (or phosphorus input) alone does not necessarily lead to
noticeable improvements in water quality related to eutrophication. Moreover, Zhang et al. (2020)
demonstrated that controlling both nitrogen (N) and phosphorus (P) in water is necessary for
improving the aquatic environment.

2.1.2. Light intensity
Light intensity is a crucial environmental factor that affects the growth rate and carrying
capacity of primary producers as algae are primarily photoautotrophs (Tian et al., 2018; Zohdi and
Abbaspour, 2019). Furthermore, Tian et al. (2018) revealed an important fact that different algal
species exhibit varying sensitivity to light intensity, as evidenced by their trace element uptake
(e.g., Mn, Zn, and Fe) and photosynthesis responses. While high light intensity is generally
assumed to increase water temperature and promote the photosynthetic rate of algae, ultimately
leading to a large increase in algal biomass. Yang et al. (2012) demonstrated that some
cyanobacteria prefer low irradiance and are vulnerable to photoinhibition. Moreover, increased
light intensity can enhance the growth rates of certain species such as Prorocentrum micans
Ehrenberg, Cryptomonas sp., and diatoms, but eventually reach a plateau (saturation) (Tian et al.,
2018). Growing evidence suggests that light intensity is closely linked to phytoplankton bloom
events, and the growth responses of algae to varying light intensities are highly variable (Cui et al.,
2015; Winder et al., 2012).

2.1.3. Water temperature
Water temperature, under nutrient-sufficient and adequate light conditions, is a key factor in
the occurrence of phytoplankton blooms (Araki et al., 2018). Reinl et al. (2023) demonstrated that
although increasing lake surface temperature can promote cyanobacterial growth, “cold-water
cyanobacterial bloom” was also observed when the water temperature is < 15℃. In addition,
10

Winder et al. (2012) reported that changes in water temperature are a key driver of the launch of
plankton blooms in spring, and higher water temperatures may result in earlier bloom events for
many functional plankton groups. It is widely observed that seasonal changes often lead to the
formation of different dominant phytoplankton species (Guo et al., 2014; Xiao et al., 2016), with
periodic changes in water temperature being one of the obvious reasons. To sum up, water
temperature is one of the main factors in triggering algal blooms.

2.1.4. Biotic factor
Limiting factors of phytoplankton blooms also include trophic cascades via top-down control
of phytoplankton by zooplankton (Gianuca et al., 2016; Mao et al., 2020). Some zooplankton such
as daphnia and copepod zooplankton, are effective predators of algae, which not only affect the
phytoplankton biomass but also the diversity of multiple trophic levels (Birtel and Mattews, 2016;
Duggan et al., 2015). Under resource limitations, herbivores may enhance primary production by
decreasing interspecific competitors (Ger et al., 2018). Thus, maintaining the dominance of such
zooplankton species may be an effective way to control algal growth. Furthermore, intraspecific
competition in algal populations can also affect the internal structure of the plankton community.
For example, toxin release from Microcystis can impact the growth of zooplankton and other algae,
resulting in their dominance in the waters (Jester et al., 2009). In fact, biotic factors affecting algal
blooms extend well beyond those mentioned above, and the mechanisms of coupling between
various biological factors affecting planktonic systems are still the focus of current research.

2.1.5. Climate and hydrology
Extensive research has suggested that the bloom phenomenon is subject to climate and
hydrographic events (Havens et al., 2019). For example, the inflow of trace elements and nutrients
into the water due to rainfall can result in excessive growth of algae. In addition, during the rainfall
11

season, high water temperature and air temperature can create favorable conditions for algae
growth. Hydrodynamic motion, such as horizontal transport, mixing, and stirring, also plays a
significant role in plankton dynamics (Hernández-Carrasco et al., 2018; Jaccod et al., 2021;
Sandulescu et al., 2007). Deng et al. (2016) demonstrated that horizontal migration can trigger
phytoplankton blooms and have an important influence on the competition and coexistence of
different plankton species. In fact, climate conditions, wave disturbances, and wind effects all
affect algal growth, and the seasonal fluctuation of environmental changes can result in the scale
and timing of blooms being highly stochastic. Some blooms occur in spring, while others in
summer or winter. Some blooms are localized, occurring in bays, while others are massive,
covering thousands of square kilometers. Some blooms occur at the same time and place each year,
lasting a few weeks, while others happen randomly, lasting years (Anderson, 1997; Silva et al.,
2021). Therefore, the variability of climate and hydrology poses a challenge for studying the
mechanisms of algal growth.

2.1.6. Human activities
Over the last few decades, excessive human activities, such as the discharge of untreated
domestic sewage, industrial wastewater, and agricultural wastewater, have seriously deteriorated
water quality, and the increase in nutrient loadings in lakes or rivers contributes to the process of
eutrophication. Zhao et al. (2019) demonstrated that due to the increased intensity of human
activities frequent outbreaks of algal blooms have become typical events affecting the Earth's
atmosphere. Furthermore, Bhagowati et al. (2020) suggested that natural eutrophication is a
gradual aging process of aquatic ecosystems, occurring over thousands of years. However, human
activities or cultural influences, such as high agricultural activities, sewage, and industrial
discharges, have significantly accelerated eutrophication, leading to profound changes in the
12

structure and function of water bodies (Ansari et al., 2011). Lane-Medeiros et al. (2023) also
revealed that overexploitation of piscivorous fisheries may favor phytoplankton blooms through
capturing top predators.
The factors affecting algal blooms are not limited to those mentioned. Other factors include
water pH and transparency (Amorim and Moura, 2021; Lane-Medeiros et al., 2023). Phytoplankton
blooms have been considered as the consequence of synergistic effects of internal and external
regulatory factors. Although extensive investigations have been conducted to study how these
factors affect the bloom phenomenon, the underlying mechanisms of phytoplankton growth remain
a challenge. Because aquatic ecosystems are much more complex than expected, there is still much
to investigate regarding the causes of algal blooms.
2.2. The characteristics of algal blooms
Phytoplankton blooms are characterized by high densities of algae, often reaching millions of
cells per liter due to the proliferation and occasional dominance of these phytoplankton species
under favorable environmental conditions (Sarkar, 2018). During the bloom phase, the number of
algae grows exponentially, rather than following the general population growth characteristics
(Menden-Deuer and Montalbano, 2015), and the blooms are often visible as a discoloration of the
water. For example, Microcystis blooms typically turn the water green. Algal blooms can be
divided into four phases: the initial phase, development phase, maintenance phase, and decline
phase.
In aquatic ecosystems, two important forms of algae are unicellular individuals and colonies.
Some phytoplankton species, such as Microcystis, normally form colonies under natural conditions,
which is thought to contribute to bloom formation; for example, colony formation promotes the
Microcystis dominated blooms (Xiao et al., 2018). Thus, colony formation may indeed play an
13

important role in the dominance and formation of blooms in some algae, which has been
extensively investigated by many researchers (Cao and Yang, 2010; Mars Brisbin and Mitarai,
2019; Yamamoto et al., 2011).
Interestingly, blooms are generally followed by a sudden decline in population numbers (Guo
et al., 2023). There is a growing body of literature that recognizes the effect of phytoplankton
mortality on the termination of algal blooms (Choi et al., 2017), and phytoplankton mortality
normally shows a dramatic increase in the decline phase, leading to blooms that are generally
followed shortly by a sudden collapse, whereby the phytoplankton population returns to its original
low level (Mukhopadhyay and Bhattacharyya, 2006). In addition, Garcé s et al. (2005) designed a
dilution experiment of natural single Alexandrium spp. to observe cells during the development,
maintenance, and decline phases of blooms, and they found the highest mortality rates during the
decline phase. One obvious reason for this is that fast-growing algae can cause a dramatic decrease
in dissolved oxygen levels, as evidenced by rapid oxygen consumption through metabolism of
algae and a sharp decrease in oxygen production in bottom water due to decreased light penetration
(Raj et al., 2020). Furthermore, excessive oxygen depletion is often responsible for massive
organism deaths (Chislock et al., 2013).
2.3. Algal bloom impacts
Algal blooms can be classified into two categories: toxic and non-toxic, both of which can
pose serious threats to public health and the function of aquatic ecosystems (Treuer et al., 2021).
Although only a few dozen toxic phytoplankton species exist, they have the capacity to produce
potent toxins that can cause severe diseases in humans (Curran and Richlen, 2019). After the toxic
phytoplankton is consumed by shellfish, the toxic substances can accumulate in shellfish bodies to
levels that can have human health impacts. For example, paralytic shellfish poisons (PSP) can be
14

caused by Gymnodinium catenatum, and it has been shown that PSP can easily accumulate in
shellfish and could be fatal to humans (Hallegraeff, 1993). Additionally, based on the symptoms
observed in human intoxications, algal toxins can be classified into several types, such as diarrhetic
shellfish poisoning (DSP), amnesic shellfish poisoning (ASP), neurotoxic shellfish poisoning
(NSP), Ciguatera fish poisoning, and azasporacid poisoning (AZP) (Grattan et al., 2016; Wang and
Wu, 2009). In recent years, there has been an increasing trend of human poisoning due to the
consumption of fish or shellfish during algal bloom events, which are becoming more frequent
around the world.
In contrast, non-toxic algal species do not produce toxins but can still cause harm in various
ways, such as marine fauna kills due to excessive oxygen depletion or disturbance of the marine
food web (Shen et al., 2012). Excessive growth of non-toxic algal species can also result in
unpleasant smells and discoloration of the water, forming a thick layer on the water surface and
preventing light penetration to the bottom. Additionally, during the decline phase of blooms, the
massive biomass decay can lead to a dramatic increase in oxygen depletion, leading to widespread
mortalities of plants and animals in the affected area (Anderson, 2009; He, 2015). Excessive
growth of non-toxic algae can also destroy the water body’s landscape and cause serious damage
to aquatic ecosystems. Villacorte et al. (2015) suggests that these “high biomass” blooms can occur
in pristine water or in response to excessive pollution inputs.
In addition to the public health and ecological risks, phytoplankton blooms can also have a
variety of economic impacts. The rapid growth of algae can have a significant impact on fisheries,
leading to reductions in seafood sales and an increase in wild and farmed aquatic product
mortalities (Sanseverino et al., 2016). Anderson et al. (2015) demonstrated that unfavorable
changes in the plankton ecosystems may have undesirable effects on fisheries. Moreover, algal
blooms can significantly impact tourism and tourism-related businesses in the bloom-affected
15

areas, and the increasing risk of human illness due to the consumption of contaminated shellfish
or fish leads to higher medical costs (Sanseverino et al., 2016). All these phenomena are
responsible for direct or indirect economic impacts of phytoplankton bloom events.

2.4. Bloom control
Although the prevention and control of phytoplankton blooms have drawn significant
attention in recent years, the diversity of blooms makes bloom control continuing to be one of the
most challenging and controversial tasks for bloom management (Nelson et al., 2018). Strategies
for controlling blooms can generally be classified into five categories: mechanical, biological,
chemical, genetic and environmental control (Anderson et al., 2017).
Recently, a variety of mechanical control methods have been investigated and applied in field
applications. For example, clay can be used to remove algal cells (Alshahri et al., 2021; Li et al.,
2023). Since the clay particles can aggregate with phytoplankton cells and sink to the bottom
through sedimentation, this method has already been applied in several countries such as Japan,
Australia, and South Korea to control HABs in the field (Ibrahim et al., 2022). However, the clay
application can threaten benthic fauna and burial of resident populations (Anderson et al., 2017).
Furthermore, the higher cost, increased ecological impact, and greater logistic challenges limit the
application of this method outside the laboratory (Anderson, 2009; Yu et al., 2017).
Biological control (biocontrol) agents represent another approach for controlling
phytoplankton blooms. Although a variety of biological methods have been reported in recent years,
laboratory success is predominant, and the success rate of field management appears to be quite
low (Pal et al., 2020). Biocontrol has not been explored in algal bloom control due to logistical
problems, although it has been widely explored in field applications in agriculture, such as insect
pest control. Although there are many successful cases where such an approach has been both
16

effective and environmentally benign on land, considerable efforts are needed in aquatic
ecosystems.
In contrast, chemical control of algal blooms is operated by releasing toxic chemicals,
including titanium dioxide (Chang et al., 2018), sophorolipids (Balaji-Prasath et al., 2022), copper
(Schoffman et al., 2016). Although chemical treatments have been widely reported and have shown
massive success in small systems, some chemical precipitants could become problematic. For
example, the release of endotoxins can be induced by massive algal cell lysis caused by copper
sulfate (Nwankwegu et al., 2019). Additionally, Schoffman et al. (2016) found that the sensitivity
of algal blooms to copper ions differs significantly among different phytoplankton species. In
practice, chemical control has not been actively pursued for algal bloom control since it may cause
the widespread mortality of other aquatic organisms, and the full range of sensitive environmental
concerns is still challenging (Nwankwegu et al., 2019).
Genetic control is a method of introducing purposefully designed species to alter the habits
of harmful species, such as environmental tolerances and reproduction. Similar to biological
control, genetic control in terrestrial agriculture has shown remarkable success, but issues occur in
aquatic ecosystems, as evidenced by the possible negative impacts of introducing a non-indigenous
organism to a water area. The issue of genetic control may be more of a societal concern, and the
hypothetical impacts would make it exceedingly difficult to obtain approval for such approaches
in the near future (Anderson, 2009).
Finally, for environmental manipulation, the aim of such approaches is to affect the target
species or introduce bio-controlled species through chemical or physical modifications of the
environment (Anderson, 2009). Nutrient level may be an important indicator of environmental
manipulation and might involve large-scale manipulation. However, due to long-term storage of
nutrients in sediments and alteration in the biogeochemistry of systems after years of nutrient
17

loading, such an approach may not achieve the desired effect on shorter time scales (Tang et al.,
2016).
In summary, while some treatments like copper ions can effectively control phytoplankton
blooms, they may also lead to the release of endotoxins. Other treatments like clay may be more
environmentally friendly but have high costs. It has now been broadly acknowledged that a
sustainable technique for algal bloom control needs to be both environmentally effective and
economically affordable (Li et al., 2013; Zhang et al., 2018). However, most current strategies for
bloom control do not represent a sustainable development approach, and one obvious reason is that
the diversity and complexity of the bloom phenomenon resulting in the dynamics of phytoplankton
growth are currently not well understood. It is therefore important to gain a better understanding
of these factors to develop new techniques for controlling blooms that provide additional
environmental and economic benefits.
2.5. Nonlinear dynamics of planktonic ecosystems
Phytoplankton and herbivorous zooplankton are known to be the basis for all food chains and
webs of aquatic ecosystems (Titocci et al., 2022; Yannawar, 2022). Considerable field and
experimental studies have focused on plankton growth. However, the complexity and nonlinearity
of aquatic ecosystems contribute to the difficulty in understanding phytoplankton growth dynamics
based solely on field and experimental studies (Dai et al., 2019). Mathematical models have made
important contributions to the study of phytoplankton growth by providing quantitative insights
into the dynamic mechanism of phytoplankton growth (Guo et al., 2020; Malerba et al., 2012). In
recent years, there has been a growing body of literature that recognizes the importance of
mathematical models in the study of plankton dynamics (Sekerci and Petrovskii, 2015; Thakur and
Ojha, 2020).
18

Contemporary mathematical modeling of phytoplankton growth dates back to Fleming (1939),
who provided the first mathematical model of a planktonic system investigating phytoplankton
blooming controlled by zooplankton grazing. Later, Riley et al. (1949) established a vertical, onedimensional ecodynamic model to simulate the seasonal variation of aquatic plankton in the North
Sea of Europe. Based on the work of Fleming (1939) and Riley et al. (1949), numerous
phytoplankton growth models have been developed, such as bottom-up control (nutrientphytoplankton) and top-down control (zooplankton-phytoplankton) plankton models (Mao et al.,
2020; Sandhu et al., 2020; Ward et al., 2014).
Additionally, the functional response describing the response of the consumption of prey by
individual predators (Dawes and Souza, 2013) has been shown to play an important role in
modeling studies of plankton. Holling (1959a) presented the first functional response named the
Holling-type, followed by various forms of functional response such as the Beddington–DeAngelis
functional response (Beddington, 1975; DeAngelis et al., 1975), Crowley-Martin functional
response (Crowley and Martin, 1989), and so on. In recent years, various plankton growth models
have been developed by introducing various functional responses in the classical predator-prey
models (Meng and Li, 2020; Thakur et al., 2020).
The aim of investigating phytoplankton growth models is to reveal the key factors that affect
algal growth through dynamic studies from a quantitative analysis perspective. Two main model
structures are used to describe population growth, mortality, predation or prey relationships,
intraspecific competition, and other factors. They include ordinary differential equations and
partial differential equations. Growing evidence suggests that the planktonic ecosystem’s response
to impact factors is regulated by complex nonlinear processes, and nonlinearity has been
recognized as an inherent property of planktonic ecosystems (Cael et al., 2021; McGillicuddy Jr,
2010).
19

In the past decades, a variety of phytoplankton growth models have been developed to provide
a deeper understanding of the structure of planktonic ecosystems (Camara et al., 2019; Dai et al.,
2019). Extensive research has demonstrated that various dynamic behaviors exist in plankton
models, such as steady-state behavior (Chen et al., 2019; Li and Lin, 2010), bifurcation behaviors
(Chen et al., 2020a; Dai and Zhao, 2020), pattern formation (Han and Dai, 2019; Righetti et al.,
2019), among others. Although a considerable number of phytoplankton growth models have been
developed, only a few models are able to predict longer-term dynamical behavior due to the
existence of chaotic behavior (Song et al., 2014; Tian, 2012; Wang et al., 2016). Furthermore,
numerous mathematical models have predicted that chaotic behavior in plankton ecosystems can
be generated by species interactions, such as competition and predation (Benincà et al., 2015;
Huisman et al., 2006).
In natural aquatic ecosystems, phytoplankton growth is extremely complex, with many
physical-chemical processes affecting the distribution of phytoplankton populations, such as
delayed nutrient recycling, diffusion process, environmental fluctuations, impulsive control, and
others. For instance, changes in plankton population biomass can be attributed to the daily variation
in temperature and light. Many researchers have dedicated their efforts to investigating plankton
dynamics, and many interesting developments have been reported.
2.5.1. Delay-induced nonlinear dynamics
Extensive studies have demonstrated that under certain conditions, plankton populations can
coexist around equilibrium globally (Deng et al., 2015; Lv et al., 2014). However, due to the
existence of random fluctuations, phytoplankton population density usually exhibits oscillatory
behavior, indicating that the phytoplankton population density is not constant but changes over
time (Fussmann et al., 2000; Huisman et al., 2006). Additionally, most studies of plankton models
20

assume that phytoplankton populations respond instantaneously to interactions with other species,
such as the conversion of nutrients by phytoplankton (Chakraborty et al., 2015; Chen et al., 2020c;
Yu et al., 2019a). It may be doubtful whether there is a time lag in the growth response of
phytoplankton over a large area. However, the pioneering work conducted by Caperon (1969)
suggested that the growth response of Isochrysis galbana is subject to a time lag in a variable
nitrate environment, supporting the idea that time delay may indeed exist in the phytoplankton
growth process.
Time delay can be classified into two categories: distributed and discrete time-delays, both of
which have been extensively investigated by many researchers. For example, Ruan (1995) revised
the model of Beretta et al. (1990) with a discrete time delay due to gestation and a distributed time
delay that describes nutrient recycling. The author demonstrated that the effect of the distributed
time delay is somewhat weaker than that of the discrete time delay since nutrient-plankton
oscillation can be induced by the discrete time delay rather than the distributed time delay.
Moreover, it is shown that both the discrete time delay and distributed time delay are “harmless”
in some models, as the conditions for uniform persistence are the same as those for the
instantaneous case (Li and Liu, 2010).
Time delay not only plays a crucial role in the growth of phytoplankton but also significantly
influences the oscillatory behavior of phytoplankton biomass (Rehim and Imran, 2012; Singh et
al., 2023). It is now well recognized from a variety of studies, that delay can destabilize the positive
equilibrium via Hopf bifurcation and induce various dynamic behaviors, such as periodic
oscillation (Meng et al., 2020), stability switch (Thakur et al., 2021), and chaos (Wang et al., 2016).
Dai et al. (2019) investigated a nutrient-phytoplankton model with multiple delays, and the results
showed that the unique positive equilibrium is globally asymptotically stable when there is no
delay. However, if the delay is beyond a critical value, the unique positive equilibrium may lose
21

its stability via Hopf bifurcation, and then periodic solutions emerge.
Moreover, in most studies of delayed phytoplankton growth models, many authors
demonstrated that the equilibrium is always unstable when the delay increases beyond a critical
value (Shi et al., 2020; Zhao et al., 2015). However, the results reported by Song et al. (2014)
implied that delay can not only generate but also suppress the instability coexistence of species
populations, that is, the stability switch of species coexistence. Furthermore, the stability switch of
species coexistence has been explored by many authors (Du and Yang, 2022; Zhuang et al., 2021)
to control the stable coexistence of species populations.
In real natural ecosystems, chaotic behavior may exist in population dynamics due to the
inherent nonlinearities of the system (Song et al., 2014). However, empirical evidence of chaos in
real ecosystems is relatively scarce, and one possible explanation for this might be that the food
webs contain many weak links between species, which may stabilize food-web dynamics (Neutel
et al., 2002; Wootton and Stouffer, 2016). Additionally, external fluctuations, such as seasonal
perturbations, contribute to the lack of suitable data to test for chaos in food webs (Benincà et al.,
2008; Song et al., 2014). Benincà et al. (2008) designed the first long-term experiment with a
complex food web consisting of bacteria, several phytoplankton species, herbivorous and
predatory zooplankton species, and detritivores, and their results implied that species interactions
in food webs can generate chaos, indicating that chaotic behavior may indeed exist in real aquatic
ecosystems.
In contrast to experimental attention, the studies of chaos in population dynamical models
date back to the work of May (1974), in which the author found that complex chaotic dynamics
can be generated in a simple population model. Growing evidence reveals that species interactions
such as competition for limiting resources and predation, can generate chaotic behavior (Chen et
al., 2017; Levy et al., 2016). In recent years, many different routes to chaotic dynamics have been
22

explored, and evidence suggests that delay is one of the most important factors for the emergence
of chaos in population dynamical models. Many researchers have shown increasing interest in
delay-induced chaotic behavior (Song et al., 2014; Wang et al., 2016). For instance, a nutrientplankton model with a single delay can generate chaotic behavior with increasing delay (Sharma
et al., 2014), while gestation delays induce chaos in a plankton-fish model as studied by Thakur et
al. (2020). Dai et al. (2019) examined chaos in a nutrient-phytoplankton model with multiple delays.
Furthermore, there is evidence that typical chaotic behavior is characterized by the sensitivity of
the plankton oscillations to initial conditions due to the coexistence of a chaotic attractor and a
limit cycle (Sadhu and Thakur, 2017), suggesting that it might be fundamentally impossible to
predict phytoplankton biomass.
The study of phytoplankton growth models also includes an interesting topic of delay-induced
spatiotemporal dynamics known as “patchiness”. It has been recognized that the emergence of
patchiness is closely connected to the spatial heterogeneity of plankton distribution (Medvinsky et
al., 2002). The spatiotemporal pattern formation was accidentally discovered by Turing (1952),
who demonstrated that spatial structure can be induced by the nonlinear interaction of two or more
agents with different diffusion coefficients, which is called Turing instability (Giricheva, 2019).
Segel and Jackson (1972) first introduced Turing’s ideas into population dynamics to investigate
the dissipative instability in the prey-predator interaction of phytoplankton and herbivorous
copepods with higher herbivore motility, and later Levin and Segel (1976) suggested that this
scenario of spatial pattern formation might be a possible explanation for planktonic patchiness.
Furthermore, Medvinsky et al. (2001) demonstrated that conceptual reaction-diffusion
mathematical models are appropriate tools for understanding the underlying mechanisms of
plankton pattern formation and complex spatiotemporal plankton dynamics. They also revealed
that turbulence, advective currents, and wind, etc., can affect the formation of spatial structures in
23

plankton communities. Extensive research has been carried out in recent years to better understand
the mechanisms resulting in plankton spatial patterns (Sekerci and Petrovskii, 2018; Tian and Ruan,
2019).
Despite the Turing mechanism, non-Turing mechanisms underlying spatial pattern formation
have also drawn growing attention (Hu et al., 2015). Banerjee and Petrovskii (2011) demonstrated
that irregular spatiotemporal patterns can emerge around the Turing-Hopf bifurcation in a ratiodependent prey-predator model. Delay-induced spatiotemporal patterns have also attracted a lot of
attention due to the prevalence of the delay effect in aquatic ecosystems (Dai et al., 2016; Tian,
2012; Tian and Zhang, 2013). Grill et al. (1995) experimentally studied the effect of delay feedback
on pattern formation using the light-sensitive Belousov-Zhabotinsky reaction. Tian (2012)
revealed that delay can induce spatial patterns in a delayed plankton allelopathic system, and the
results showed that the delay significantly affects pattern selection. Additionally, Dai et al. (2016)
suggested that the delay can promote the formation of patchiness (an irregular pattern) via Hopf
bifurcation.
2.5.2. Stochastic population dynamics
In real ecosystems, population systems are often subject to environmental noise (Dean and
Shnerb, 2020; Lee, 2020). May (2019) pointed out that carrying capacities, birth rates, and other
parameters exhibit random fluctuations. Thus, models that ignore the stochastic nature may not
reflect the true dynamical behavior of phytoplankton growth. Moreover, there is a growing body
of literature that recognizes the key role of environmental noises in ecological systems, such as
white noise (Belabbas et al., 2021) and colored noise (Guo et al., 2023). White noise is
characterized by many small, independent random fluctuations such as rainfall, wind, or day to
night transitions. Colored noise, say telegraph noise, is considered a switching between two or
24

more environmental regimes. For example, in aquatic ecosystems, the growth environment in
winter will be much different from that in summer, resulting in seasonal changes in the diversity
of phytoplankton. All of these perturbations have an impact on changes in population density, and
one effective method to study this problem is to develop stochastic dynamics models that can
capture the pattern of random effects on the population and provide a more accurate quantitative
insight into population growth.
The study of stochastic population dynamics dates back to Hasminskii (1980), who introduced
two white noise sources to stabilize an unstable two-dimensional linear system. Later, Mao et al.
(2002) revealed an important fact that stochastic noise can suppress potential explosions in
population dynamics, while Deng et al. (2008) pointed out that noise can suppress or express
exponential growth. Furthermore, it is now recognized that stochastic effects, such as those
resulting from rainfall, windy conditions, and temperature variations, are widely present in the
phytoplankton growth process (Song et al., 2020a). In recent years, stochastic prey-predator
plankton models have been widely explored (Camara et al., 2019; Chen et al., 2020b; Song et al.,
2020b). Stochastic plankton growth models incorporating constant and periodic toxin-producing
phytoplankton have been analyzed by Wang and Liu (2020) and Jang and Allen (2015),
respectively. Zhao et al. (2016) developed a non-autonomous toxic-producing phytoplankton
allelopathy model with white noise fluctuations and derived sufficient conditions for the existence
of the boundary periodic solution and the nontrivial positive stochastically periodic solution.
Stochastic chemostat models with white noise fluctuations have also been widely explored (Sun et
al., 2017; Xu et al., 2021; Xu and Yuan, 2016).
In addition, seasonal fluctuation is a crucial factor among stochastic effects. It is widely
accepted that the vital rates of plankton populations often undergo seasonal variations due to the
periodicity of the natural environment (Song et al., 2019). Growing evidence suggests that seasonal
25

changes in abiotic factors such as temperature, light intensity, and nutrient enrichment (Baek et al.,
2019; Lind et al., 2016; Vanderley et al., 2021) result in apparent seasonality of rapid
phytoplankton growth. Recently, many studies have been reported on how seasonal fluctuations
affect the dynamics of phytoplankton growth within random environments (Guo et al., 2022; Zhao
et al., 2020). The results from Huisman (2006) suggest that seasonal environments can trigger more
complex dynamics than constant environments.
While many stochastic phytoplankton growth models have focused on white noise, it has been
discovered that phytoplankton growth can switch between two or more environmental regimes,
which differ by factors such as nutrition or rainfall. For example, algal blooms are often followed
by rapid collapses (Ma et al., 2018), and phytoplankton mortality has been found to be much higher
in the decline phase compared to the development phase (Garcés et al., 2005), which cannot be
adequately described by white noise. Moreover, the decomposition of dead algae quickly leads to
a rapid depletion of dissolved oxygen, subsequently leading to hypoxic or anoxic “dead zone”
lacking sufficient oxygen to support most organisms (Morro et al., 2022). This phenomenon may
result in an increasing trend of zooplankton mortality in the bloom phase (Anderson et al., 2021).
It is widely accepted that the switching phenomenon described above can be mathematically
described by the so-called colored noise, namely telegraph noise, which can be described by a
finite-state Markov chain switching between two or more environmental regimes. Typically, the
switching among different environments is memoryless, and the waiting time for the next switch
follows an exponential distribution (Pan et al., 2015; Wang and Jiang, 2018; Zou and Wang, 2014).
Zhao et al. (2016) proposed a stochastic phytoplankton allelopathy model under regime switching
and demonstrated that both white and colored noises have great impacts on the evolution of
phytoplankton populations. Additionally, Yu et al. (2019a) studied a nutrient-plankton model with
the effect of regime switching and the results showed that the Markov chain is beneficial for the
26

survival of plankton.
In summary, researchers have long sought to understand how phytoplankton growth responds
to random fluctuations in aquatic ecosystems. Stochastic models of plankton populations have been
meaningfully used to investigate the mechanisms of bloom formation (Cai et al., 2020; Liu et al.,
2021a; Yu et al., 2019b). Modeling studies not only deepen our quantitative understanding of
plankton growth but also provide theoretical foundations for the prevention and control of algal
blooms. Despite significant progress in the field of stochastic phytoplankton growth models, the
complexity and nonlinearity of aquatic ecosystems under random fluctuations present ongoing
challenges for further research.
2.5.3. The dynamics of impulsive control system under environmental fluctuation
In natural ecosystems, the state of ecosystems can change abruptly due to environmental
fluctuations and human activity interventions, such as catching adult fish and releasing fry.
Typically, these changes occur over a relatively short period, and are often described
mathematically as impulsive control. In addition, many impulsive controls involve human
interventions that introduce or remove some members from a population at a given time or when
the state of the species satisfies certain criteria (Akhmet et al., 2006). Extensive research has
suggested that impulsive differential equations are useful tools for studying these phenomena
because they have the advantage of reflecting the features of transient changes in the system’s
status.
In recent years, impulsive dynamical systems have received considerable attention due to the
increasing frequency of human interventions in natural ecosystems, which often leads to transient
changes. The study of impulsive dynamical systems can be traced back to Milman and Myshkis
(1960). The impulsive control has since been widely used in many scientific fields in recent
27

decades, such as orbital transfer of satellites, epidemic control, ecological systems, and population
models. Recently, impulsive dynamical systems have been widely developed in the study of
population dynamics. Notably, impulses control may contribute to the changes in the dynamics of
a model (Li et al., 2019). Specifically, Li et al. (2019) demonstrated an important finding that
appropriate impulse control can promote species survival.
It is important to note that various strategies have been implemented to prevent and control
phytoplankton blooms, including planting and harvesting, which can result in transient changes in
the state of aquatic ecosystems. Therefore, the study of prevention and control of phytoplankton
blooms has incorporated impulsive differential equations as they provide a natural description of
observed phenomena in real-world ecosystems. In addition, it is of great interest to investigate
complex dynamics for impulsive perturbations in plankton dynamics. For instance, Lürling et al.
(2018) reported that nutrient impulses are crucial for the survival of phytoplankton species.
However, these models primarily focused on how impulsive control affects phytoplankton growth,
while not accounting for the prevalent random perturbations in aquatic ecosystems. In reality, both
environmental fluctuation and impulsive control significantly influence plankton dynamics, and
therefore, the stochastic model coupled with impulsive control has received growing attention.
Feng et al. (2021) demonstrated that nutrient pulses play an important role in the stochastic
dynamics of phytoplankton growth. Furthermore, Liu et al. (2023) revealed an important fact that
both environmental fluctuation and impulsive control can directly influence population extinction
and persistence in the mean (the population can survive in the system).
Although some research studies have been conducted on the coupling effects between
environmental fluctuation and impulsive control on plankton dynamics, there is still very little
scientific understanding of how plankton dynamics respond to impulsive perturbations under
environmental fluctuation. Understanding these dynamical behaviors is crucial for developing
28

effective strategies to prevent and control algal blooms. Therefore, gaining insight into the
coupling effects between environmental fluctuation and impulsive control on plankton dynamics
is of great importance.
2.6. Summary of literature review
Phytoplankton, as the primary producer in aquatic ecosystems, plays crucial role in assessing
water quality and determining the production capacity of water bodies. Phytoplankton is
inextricably linked to various aspects of human life, such as fisheries, medicine, agriculture, and
more. However, in recent years, there has been an increasing discharge of domestic sewage, as
well as industrial and agricultural wastewater, leading to accelerated eutrophication in many water
bodies. Phytoplankton blooms are considered the primary symptom of eutrophication, disrupting
the aquatic food webs and upsetting the balance of ecological structure and function. While various
treatments for bloom control have been explored, achieving a balance between cost and
environmental health has proven challenging due to a limited understanding of the underlying
mechanisms of algal blooms.
To investigate the dynamics of algal growth, phytoplankton growth models have attracted
widespread attention as an effective method due to the high nonlinearity and complexity of aquatic
ecosystems. Phytoplankton growth models incorporate various functional responses into classical
prey-predator models have been widely adopted and provide a broader perspective on
phytoplankton growth dynamics. Furthermore, nonlinear dynamics induced by delay, stochastic
fluctuations, and impulsive control have emerged as particularly important factors, as delay and
random disturbances are inherent in the phytoplankton growth process. Despite considerable
efforts to investigate the effects of delay, stochastic fluctuations, and impulsive control on the
underlying mechanism of phytoplankton growth, the extent of these effects on plankton growth
29

dynamics remains unclear due to the diversity of aquatic ecosystems and the complexity of habitat
environments.
Previous studies on chaotic behavior induced by delay have primarily focused on plankton
dynamics without considering diffusion. Thus, in Chapter 3, the spatiotemporal dynamics are
investigated and the results show that chaotic behavior appear in a spatiotemporal nutrientplankton model. In addition, most stochastic phytoplankton growth models have been investigated
without considering the effects of prey refuge and seasonal fluctuations. In the present research,
the effects of prey refuge and seasonal fluctuations under noise turbulation are studied (Chapter 4
& 5). Previous research has mostly focused on constant plankton mortality rather than regimeswitching plankton mortality. However, due to the impact of algal blooms on aquatic ecosystems,
regime-switching plankton mortality is more realistic, which is investigated in Chapter 6. Most
studies on nutrient-plankton models with impulsive control have been conducted in a constant
environment, ignoring the stochastic effects on aquatic ecosystems. Thus, a stochastic nutrientplankton model with impulsive control is developed and analyzed in Chapter 7. The results
obtained from Chapter 7 may provide valuable insights into the possible management of excessive
algal growth in aquatic systems.
It is important to note that some phytoplankton growth models may only offer abstractions of
real-world phenomena and may not be applicable for predicting bloom events. However, they can
provide an understanding of specific key processes underlying the growth of phytoplankton.
Particularly, research on bloom-triggering mechanisms contributes to the development of effective
prevention and control measures to improve water quality and restore aquatic ecosystems.

30

Chapter 3
STABILITY SWITCHES AND CHAOS INDUCED BY DELAY IN A NUTRIENTPLANKTON MODEL WITH DIFFUSION1
Abstract
In this chapter, a reaction-diffusion model considering nutrient, phytoplankton, and
zooplankton as dynamic variables was investigated. The impact of time delay that involved in the
growth of phytoplankton after the uptake of nutrients was studied. The theoretical findings
indicated that delay in the growth of phytoplankton could trigger the emergence of persistent
oscillations in the model via a Hopf bifurcation. In addition, the theoretical analysis tracked the
direction of Hopf bifurcation and the stability of bifurcating periodic solutions. The simulation
results showed that stability switches occur for the positive equilibrium with an increasing value
of the time lag. The findings showed that the model experienced homogeneous periodic-2 & 3
solutions as well as chaos. Overall, the results showed that the presence of time delay in the growth
of phytoplankton could bring forth dynamical complexity in the nutrient-plankton model of the
aquatic habitat.
Keywords: Nutrient-plankton model, time delay, Hopf bifurcation, stability switches, multiple
periodic solutions, chaos

_____________________________
1

Guo, Q., Wang, L.J., Liu, H., Wang, Y., Li, J., Tiwari, P.K., Zhao, M., Dai, C.J., 2022. Stability switches and
chaos induced by delay in a nutrient-plankton model with diffusion. Submitted to Journal of Biological Dynamics
on October 9, 2022 (Revision required on June 27, 2023).
31

3.1. Introduction
Algal blooms occur in aquatic systems due to the massive growth of phytoplankton, which
can lead to significant water-quality problems and have adverse effects on the human health
(Hallegraeff, 1993; Medvinsky et al., 2002; Mukhopadhyay and Bhattacharyya, 2006). The
occurrence of algal blooms is the result of the interplay of various hydrodynamics, chemical
processes, and biological processes (Chen and Mynett, 2006). It is well established that algal
blooms are influenced by nutrients level and the predation of zooplankton (Dacey and Wakeham,
1986). Experiment conducted by Vanni (1987) has shown that even small changes in zooplankton
size can have a significant impact on the phytoplankton community, although nutrient levels play
a more substantial role in phytoplankton growth compared to zooplankton. Despite extensive
research on phytoplankton growth dynamics, the process by which phytoplankton bloom occurs is
not clearly understood yet. A deeper understanding of the nutrient-plankton interaction is necessary.
However, studying the mechanisms of phytoplankton growth solely through experimental or field
observations is challenging (Dai et al., 2019).
Researchers have employed mathematical models to gain quantitative insights into the
dynamics of planktonic blooms in aquatic reservoirs (Edwards and Brindley, 1999; Guo et al.,
2020; Guo et al., 2023; Mandal et al., 2021a). Tiwari et al. (2022) observed that the nutrientplankton model destabilizes with an increase in nutrient input and recycling of dead phytoplankton
biomass into nutrients. The toxins liberation by phytoplankton species and the intraspecific
competition among themselves have the potentials to terminate persistent oscillations and stabilize
the ecosystem. Biswas et al. (2020b) found that the presence of free-viruses and environmental
toxins in aquatic systems could drive the zooplankton population to a very low equilibrium value,
but the ecological balance of the aquatic food web could be maintained by modulating the decay
(depletion) rate of free-viruses (environmental toxins). Mandal et al. (2021b) demonstrated that
32

environmental toxins could be reduced to a low level, thereby maintaining the equilibrium of the
planktonic ecosystem. Through sophisticated sensitivity analysis technique, Guo et al. (2023)
reported that the biomass of phytoplankton in aquatic reservoirs was highly sensitive to the uptake
rate by zooplankton and least sensitive to the re-mineralization of dead plankton biomass into
nutrients. Their numerical results also revealed that the persistence and extinction of plankton
populations highly depended on nutrient levels in the aquatic system.
Experimental evidence suggests that a time lag occurs in the growth response of Isochrysis
galbana under high nutrient availability (Caperon, 1969), indicating a delay exists in the growth
of phytoplankton. therefore, it is reasonable to consider the effect of time delay in phytoplankton
growth models. Both experimental and field observations have shown that changes in plankton
population density often exhibit oscillatory behavior due to the existence of internal factors
(Fussmann et al., 2000; Huisman et al., 2006), indicating that the population density of
phytoplankton in aquatic reservoirs is not constant but changes over time in the realistic scenario
(Sherrattand and Smith, 2008). Several previous studies have introduced time delay into the study
of phytoplankton growth dynamics (Agnihotri and Kaur, 2019; Chen et al., 2020a; Das and Ray,
2008; Sharma et al., 2014; Yuan, 2012). The growing body of evidence suggests that plankton
models with time delay exhibit rich and complex dynamical behaviors. Previous studies have
demonstrated that delay can destabilize a model via Hopf bifurcation, induce periodic solutions
(Bentounsi et al., 2018; Chen et al., 2013; Song and Wei, 2005), create stability switches (Thakur
et al., 2021; Wang et al., 2016), and even lead to chaotic dynamics (Adak et al., 2020; Shu et al.,
2015).
Moreover, it is well established that nutrients and plankton are able to disperse in aquatic
reservoirs due to currents and turbulent diffusion. Previous studies have suggested that plankton
growth and distribution can be characterized by spatial variation. Based on the work of Medvinsky
33

et al. (2002) and Holmes et al. (1994), it is recognized that appropriate partial differential equations
can be used to model a variety of ecological phenomena, including complex spatio-temporal
dynamics of plankton populations. Segel and Jackson (1972) were the first to introduce spatial
structure in population dynamical models., and since then, numerous plankton models have
considered the effect of diffusion (Agmour et al., 2021; Dai et al., 2015; Han and Dai, 2019;
Upadhyay et al., 2012; Zhao et al., 2019). Chakraborty et al. (2015) investigated the spatial
dynamics of a nutrient-phytoplankton model considering the toxicity of phytoplankton. Their
findings showed that in the presence of toxicity, the distributions of nutrient and phytoplankton in
the aquatic ecosystem become spatially heterogeneous, resulting in different patterns such as
stripes, spots, and their mixtures, depending on the level of toxicity. Chakraborty et al. (2015) also
observed spatio-temporal oscillations in the distributions of nutrient and phytoplankton under
certain toxicity level.
Ecological systems possess all the elements to produce chaotic dynamics (May, 1987). The
chaotic situations may arise from an equilibrium state for various reasons. Although chaos is
commonly predicted by mathematical models, evidence for its existence in the natural world is
scarce and inconclusive. The characteristics of chaos and its presence in nature have been
extensively discussed in the field of ecology (Godfray and Grenfell, 1993; Hastings et al., 1993;
Jorgensen, 1995; Perry et al., 1993). Recent developments in dynamical modeling theory consider
chaotic fluctuations of a model as highly desirable because they allow for easy control. To assess
the ecological implications of chaotic dynamics in different natural systems, it is important to
explore changes in the dynamics when structural assumptions of the system are varied. One
approach to studying of the dynamics of ecological communities is through their food webs and
the coupling of interacting species (Hastings and Powell, 1991). Chattopadhyay and Sarkar (2003)
applied the idea of Hastings and Powell (1991) to a plankton model with toxic effects. Jorgensen
34

(1995) demonstrated that chaos can occur in planktonic systems due to variation in the size of
zooplankton species. More information on the ecological implications of body size can be found
in Peters (1983). Mandal et al. (2006) applied thermodynamic principles to a plankton-fish model
and found that gradually decreasing the size of zooplankton changes the model’s dynamics from
an equilibrium state to chaotic conditions. Biswas et al. (2020a) found that when viral infection
triggers chaotic dynamics in an ecosystem, a high avoidance intensity of zooplankton can stabilize
the system. Biswas et al. (2022) considered a time delay in the viral replication process in a lysis
event, and found that infection and replication of free viruses were seasonally forced. They found
that time delay could account for recurrent stability switching event in the model. They also
observed chaotic oscillations in the seasonally forced delayed model, which indicates the
emergence of harmful algal blooms. Their investigations suggested that increasing the strength of
toxic compounds exuded by phytoplankton species may suppress chaotic disorder and drive the
system into a zooplankton-free zone (Biswas et al., 2022). However, the intensity of selective
grazing by zooplankton can change the state of chaos to order and promote a disease-free aquatic
ecosystem.
It is worth noting that although the spatio-temporal dynamics of nutrient-plankton models
have been widely investigated, the influence of time delay in the growth of phytoplankton after
nutrient uptake on the spatial distributions of nutrient, phytoplankton, and zooplankton in aquatic
ecosystems remains unclear. Therefore, in this study, the role of such time delay in a diffusive
nutrient-plankton model is explored by considering the fact that some species of phytoplankton
may release toxic chemicals to reduce predation pressure by zooplankton. The remaining portion
of this chapter is organized as follows: Chapter 3.2 proposes a model for the combined actions of
delay and diffusion in an aquatic ecosystem is proposed. Chapter 3.3 analyzes the dynamics of the
model mathematically, including the equilibrium points in the absence of time delay, and the
35

stability behavior of the model in the presence of delay and diffusion. Existence of Hopf bifurcation
and direction and stability of the bifurcating periodic solutions are also analyzed. In Chapter 3.4,
numerical simulations are performed to illustrate the analytical findings and gain further insights
into the dynamics of the delayed-diffusive nutrient-plankton model. In the end, conclusions of this
study are presented in Chapter 3.5.
3.2. The mathematical model
In this chapter, a mathematical model is constructed to examine the impact of delay in the
growth of phytoplankton after nutrient uptake in a diffusive nutrient-plankton model of an aquatic
ecosystem. The model incorporates nutrient concentration, phytoplankton biomass, and the
zooplankton biomass as dynamical variables. The growth dynamics of the phytoplankton and
zooplankton population mainly depend on the availability of nutrient and the phytoplankton,
respectively. The concentration of nutrient in the aquatic system depends on its input and washout
rates, as well as uptake by the phytoplankton population. For the phytoplankton population, its
biomass is determined by nutrient uptake, grazing by zooplankton, and mortality, either natural or
due to nutrient shortage. A more appropriate functional form to describe the predation of
phytoplankton by zooplankton is the Holling type II (Holling, 1959a). Finally, the zooplankton
biomass in the aquatic reservoir depends on the grazing of phytoplankton, toxin liberation by
phytoplankton, and loss due to mortality.
Let

( , ), ( , ), and ( , ) respectively represent the concentration of nutrient, biomass

of phytoplankton, and biomass of zooplankton at any time > 0 and position . The model is built
based on the following assumptions.
(1) There is a constant input of nutrient to the aquatic system. The nutrient depletes naturally
at a rate

and is uptaken by the phytoplankton following the bilinear law of interaction.
36

(2) Let

denote the maximum uptake rate of phytoplankton by zooplankton. The predation

rate of zooplankton on phytoplankton is described by the Holling type II functional form (Holling,
1959a),

, where ℎ is the half saturation constant.

(3) The parameters

and

represents the natural mortality rates of phytoplankton and

zooplankton species, respectively. Besides natural mortality, phytoplankton also experience death
due to competition for available nutrients in the aquatic system. Furthermore, intraspecies
competition among phytoplankton species, resulting from limited resources, is considered.
(4) The rate of conversion of phytoplankton biomass into zooplankton biomass is denoted by
. Toxin-producing phytoplankton species release toxic chemicals at a rate
reduction in the biomass of zooplankton at a rate

, leading to a

.

(5) A delay is incorporated in the growth response of phytoplankton after nutrient uptake.
(6) In aquatic system, nutrient and plankton disperse due to the currents and turbulent
diffusion.
Based on these assumptions, a schematic of the model expressing the interactions of nutrient
and plankton is presented in Figure 3.1. Then the following reaction-diffusion nutrient-plankton
model with delay is obtained:
⎧
⎪
⎨
⎪
⎩

=

−

=

( − ) −

=

−

In the above model, the parameter

−

−

+

∆ ,
−

+

−

+

(3.1)

∆ ,

∆ ,

represents the delay, which denotes the time required by

phytoplankton to absorb nutrients and reproduce. The constants

,

and

denote the self-

diffusion coefficients of nutrient, phytoplankton and zooplankton, respectively. Let
[0,

], and for ∈ [− , 0],

( , )=

( , ) ≥ 0, ( , ) =
37

∈

=

( , ) ≥ 0 and ( , ) =

( , ) ≥ 0. All the parameters in model (3.1) are assumed to be positive, and their biological
meanings are listed in Table 3.1.

Figure 3.1. The scheme representation of the model (3.1).

3.3. The main results
In this chapter, the existence of ecologically meaningful equilibria of model (3.1) is discussed
by ignoring the delay and diffusion factors. Then, the linear stability and bifurcation analysis are
performed by taking time delay as bifurcation parameter.
3.3.1. The existence of equilibria
Model (3.1) does not have any trivial solution, but it has a non-trivial solution
(

,

,

) , where

=

,

= 0 and

is the positive solution of the following

quadratic equation:
+(

)

−

−

= 0.

Obviously, the above equation has only one positive root that is given by
=

(

)

=

(

)

Additionally, if the following conditions are satisfied:
38

.

>

( (

)

)( (
(

)
)

then model (3.1) has a unique positive equilibrium

∗

ℎ
=
,
( − − )

∗

∗

∗

=

)

∗)

( +

,

(

=

∗

,

,

>

∗

∗

,
∗

(

+

) whose components are given by

−

∗

From an ecological point of view, the positive equilibrium

∗ )(ℎ

−

+

∗)

.

is very important as all the

considered dynamical variables are presented here. So, this research is mainly focus on the
dynamics of model (3.1) around this equilibrium point in the forthcoming chapters.

3.3.2. Stability and Hopf bifurcation
∗

In this subchapter, the stability of positive equilibrium
in model (3.1) are analyzed. Let (

(

∗

,

∗

,

∗

) and Hopf bifurcation
, and one can get the

):

<

and (

) holds, then the positive equilibrium

<

following result:
Theorem 3.1. If

> max

− ,

∗

of model

(3.1) without delay is locally asymptotically stable for any ∈ ℕ .
The proof is given in Appendix A. Furthermore, by direct computing, the following lemma
obtained
Lemma 3.1. Let

>

− ,

(i)

If

(ii)

If there exists a

=

,

−3

and (

≤ 0 for any ∈ ℕ , then Eq. (B2) does not have any positive roots.

,

∈ ℕ such that

positive roots if and only if

) holds. Then, following two cases are obtained.

,

> 0 and (

=

,

−3

,

> 0, then Eq. (B2) possesses two

, ) < 0.

The proof is given in Appendix B. Now, the following lemma, which states the transversality
condition required for the presence of Hopf bifurcation in model (3.1), is presented.
Lemma 3.2. If Lemma 3.1 (ii) holds, then
39

( )
( )

|

,

|

,

> 0,
< 0,

= 0, 1, 2, ….
The proof is given in Appendix C. By Lemma 3.2, Hopf bifurcation occurs in model (3.1) at
the critical value
is larger than

=

, which means that the positive equilibrium

∗

becomes unstable when

. Thus, the following theorem obtained.

Theorem 3.2. For model (3.1), one can obtain that
(i) If Lemma 3.1 (i) holds, then the positive equilibrium
asymptotically stable for all values of

∗

of model (3.1) is locally

≥ 0.

(ii) If Lemma 3.1 (ii) holds, then there exists a nonnegative integer
equilibrium

∗

, ,

∪ ⋯∪

,

,

∪

, , +∞

=

,

,

,

,

every

such that the positive

of model (3.1) is locally asymptotically stable when

for

,

,

, and is unstable when

∈

, ,

∪

,

∈ 0,
, ,

,

∪⋯∪

. Further, model (3.1) undergoes Hopf bifurcation around

= 1, 2 and

∪

,

∗

at

= 0,1,2, ⋯.

Now, the following results, which are based on the center manifold theorem and the normal
form theory, are stated.
Theorem 3.3. The Hopf bifurcation is characterized by the signs of

,

, and

(i) If

> 0(< 0), Hopf bifurcation is supercritical (subcritical) at

=

∗

(ii) If

< 0(> 0), the bifurcated periodic solutions are stable (unstable);

(iii) If

> 0(< 0), the period of the bifurcating periodic solutions increases (decreases).

A detailed proof of Theorem 3.3 is provided in Appendix D.

40

as follows.

;

Table 3.1. Biological explanations of variables and parameters in model (3.1), and numerical
values used for simulation results
Parameter Description

Unit

Nutrient concentration

/

Phytoplankton biomass

/

Zooplankton biomass

/

Nutrient input rate to the aquatic system

/ /

Source

[0.1]

2(a)

0.4

0.4(a)

Natural washout rate of nutrient

day-1

Uptake rate of nutrient by phytoplankton

/

/

0.09

0.1(b)

/

/

0.95

0.9677(c)

day-1

0.09

0.1(b)

day-1

0.95

0.9661(c)

/

3

3.5(b)

Natural mortality rate of phytoplankton

day-1

0.19

0.2-0.65(d)

Natural mortality rate of phytoplankton

day-1

0.2

0.2(a)

Conversion rate of nutrient concentration
into the biomass of phytoplankton
Uptake rate of phytoplankton by
zooplankton
Conversion rate of phytoplankton
biomass into the biomass of zooplankton
ℎ

Value

Half-saturation constant

Strength of interspecies competition
0.2
Assumed
/ /
among the phytoplankton population
Rate of toxin liberation by toxin
day-1
[0.1] 0.22/0.3/0.92(e)
producing phytoplankton
(a) Ruan (1993); (b) Wang et al. (2016); (c) Rehim et al. (2016); (d) Garcés et al. (2005);
(e) Javidi and Ahmad (2015).
3.4. Numerical simulation
In this chapter, the effect of time delay on the dynamics of model (3.1) is further discussed.
For the numerical investigations of model (3.1), the following parameter values are chosen, which
are in accordance with the values mentioned in Table 3.1:
= 0.4, = 0.09,

= 0.95, = 0.09, ℎ = 3,

= 0.01,

= 0.2,

= 0.2,

= 0.02.

= 0.95,

41

=

= 0.01,

The above set of parameter values is used throughout this chapter if not mentioned in the text.
The parameters , , and are chosen as control parameters in the nutrient-plankton model (3.1).
Furthermore, the initial values for nutrient, phytoplankton, and zooplankton are selected as 0.9,
0.9, and 22, respectively.

Figure 3.2. Biomass distributions of (a) phytoplankton and (b) phytoplankton populations in model (3.1)
over time and space for

= 0.

Figure 3.3. (a) The number of stability switches of the positive equilibrium of model (3.1) in the

−

plane. Stability switches occur once in region III, twice in region IV, three times in region V, four times in
region VI, five times in region VII, six times in region VIII and seven times in region IX. (b) The partial
enlarge diagram of (a).

42

Figure 3.4. Bifurcation diagrams of model (3.1) with respect to

and for (a)

= 0.45 and (b)

= 0.35.

In the figures, the solid, dashed, dash-dot, and dotted curves represent the critical values of

for =

0, 1, 2, 3 respectively.

Figure 3.5. Bifurcation diagrams of model (3.1) with respect to for (a)
= 0.35 and

= 0.45 and

= 0.1, and (b)

= 0.05. In the figure, the solid blue line and the dotted red line represent the stable positive

equilibrium and the maximum biomass of phytoplankton, respectively.

Firstly, the dynamics of model (3.1) in the absence of time delay are investigated. Figure 3.2
shows the biomasses of phytoplankton and zooplankton populations over time and space by setting
= 0 in model (3.1). It is evident from the figure that the corresponding equilibrium point is
asymptotically stable, indicating that the biomasses of plankton populations do not change over
time. However, theoretical analysis indicates that Hopf bifurcation may occur when there is a time
lag in the growth of phytoplankton after nutrient uptake. The nutrient-plankton model may also
exhibit stability switches with a gradual increase in such time delay. Furthermore, in the simulation
process, the results show that the natural mortality of phytoplankton species does not significantly
43

affect the model’s dynamics much. Therefore, to highlight the dynamical behaviors of stability
switches, Figure 3.3(a) shows the number of stability switches that the model experiences around
the positive equilibrium by simultaneously varying the parameters

and

under

= 0.19. In

Figure 3.3(a), region I corresponds to the unstable domain of the positive equilibrium. In region X,
the positive equilibrium is stable if it exists. In region II, the positive equilibrium loses its stability
when crosses a critical value

. It is important to note that there are no stability switches for

positive equilibrium in region II, despite the fact that Eq. (B2) has two positive roots. However,
stability switches emerge in region III. In Figure 3.3(b), an enlarged view of Figure 3.3(a) is
presented to explore the characteristics of the stability switches for the positive equilibrium of
model (3.1).

Figure 3.6. Biomass distribution of phytoplankton population in model (3.1) over time and space for
0.45,

= 0.1 and different values of : (a) = 2, (b) = 10 and (c)

=

= 18.

To further investigate the stability switches in model (3.1), Figure 3.4(a) illustrates the
bifurcation diagrams of the model in the

− plane for

= 0.45 (see Figure 3.4(a)) and

0.35. The figures depict that stability switches exist in the model before

=

enters the green zone.

In the green zone, i.e., region I, stability switches do not emerge, and the positive equilibrium
remains stable regardless of the value of time delay. However, the positive equilibrium disappears

44

Figure 3.7. Biomass distribution of phytoplankton population in model (3.1) over time and space for
0.35,

= 0.05 and different values of : (a) = 3, (b)

= 10, (c) = 18, (d)

=

= 36, (e) = 42 and (f)

= 50.

from the model for all values of if the value of

falls within region II. Corresponding to Figure

3.4, the bifurcation diagram of model (3.1) with respect to the delay parameter is given in Figure
3.5. The figures only show the biomass of the phytoplankton population by varying the time delay
along the −axis. For

= 0.45, Figure 3.5(a) shows that a stability switch occurs once, and

periodic-2 solutions emerge as the value of time delay increases. Figure 3.6 shows the biomass
distribution of phytoplankton over time and space at
3.6 that the positive equilibrium is stable at

= 2, 10, and 18. It is apparent from Figure

= 2 and 18 but unstable at

= 10. For

= 0.35,

Figure 3.5(b) shows that the model (3.1) goes through three transitions from stable to unstable via
Hopf bifurcation as the value of time delay gradually increase. That is, stability switches occur
twice in the model. The biomass distribution of phytoplankton over time and space is presented in
45

Figure 3.7. One can easily see in the figure that the positive equilibrium is stable at
42, but unstable at

= 3, 18, and

= 10, 36, and 50. Figures. 3.4 & 3.5 indicate that the nonlinear analysis is in

agreement with the results predicted by the linear analysis. Note that all the parameters in these
results indicate that delay in the growth of phytoplankton is the only reason for the rise in the
instability of the positive equilibrium.

Figure 3.8. (a) Bifurcation diagram of model (3.1) with respect to for

= 0.6 and

= 0.1. In the figure,

the green, blue and yellow solid diamonds respectively represent the periodic-1, 2 & 3 solutions at
30 and 56; the black dashed line denotes chaotic solution at

= 10,

= 74. Figure (b)-(e) show the biomass

distributions of phytoplankton in model (3.1) over time and space at

= 10,

= 30, = 56 and

= 74,

respectively.

Figure 3.8(a) shows a bifurcation diagram of model (3.1) for a wide range of the delay
parameter . The figure only shows the biomass of the phytoplankton as the model exhibits the
same dynamical behaviors of the zooplankton population. The figure clearly shows that the model
46

exhibits periodic-1, 2, and 3 solutions with a gradual increment in the value of time delay.
Moreover, for a very large value of time delay, the model shows chaotic dynamics. In the figure,
the periodic-1, 2, and 3 solutions are marked at

= 10, 30, and 56 with green, blue, and yellow

solid diamonds, respectively. Also, there exists a vertical black dashed line at

= 74, where the

model enters the chaotic regime. Furthermore, Figures 3.8(b)-3.8(e) illustrate the biomass
distributions of the phytoplankton population in model (3.1) over time and space at these critical
values of time delay.
3.5. Conclusions
Planktonic blooms have gained significant attention from ecologists and mathematicians in
recent decades (Mukhopadhyay and Bhattacharyya, 2006). Mathematical models have been widely
used by researchers as an important tool to explain these bloom phenomena. Huppert et al. (2002)
investigated a simple mathematical model and found that high nutrient concentrations can trigger
planktonic blooms in aquatic reservoirs. Furthermore, Steele and Henderson (1992) explored the
role of predation in plankton models. The influence of time delay in aquatic environments is also
inevitable. Researchers have noted that the diffusion of nutrients and plankton plays a crucial role
in the spatial distribution of phytoplankton in aquatic ecosystems (Tian, 2012). However, the
dynamics induced by time delay, such as oscillation, chaotic behavior, and stability switches in
phytoplankton growth models with diffusion, need to be further explored.
In this chapter, a nutrient-plankton model was investigated to reveal the effects of time delay
on the growth of phytoplankton and the interplay between nutrient and plankton populations. The
proposed model also considers the effect of diffusion to incorporate the spatial movements of
nutrient and plankton populations in the aquatic ecosystem. The theoretical results for the model
without delay and diffusion showed that the positive equilibrium was locally asymptotically stable
47

under certain conditions on the model parameters, implying the persistence of nutrient and
plankton populations in the ecosystem. However, the positive equilibrium may lose stability when
the time lag involved in the growth of phytoplankton exceeded the critical values. In this case,
periodic solutions emerge, and the model exhibited oscillatory dynamics of nutrient and plankton
populations, indicating that the biomass of plankton populations cannot maintain a certain level.
Thus, time delay plays a significant role in inducing oscillations in the nutrient-plankton model.
Moreover, the direction of Hopf bifurcation around the positive equilibrium and the stability of the
bifurcating periodic solutions were tracked using the center manifold theorem and the normal form
theory. Several studies have demonstrated that ecosystems exhibit unstable coexistence of species
when the time delay exceeds a critical value (Li and Liu, 2010; Meng and Li, 2020; Rehim et al.,
2016). However, simulation results showed that time delay could generate and suppress
oscillations in the model. This phenomenon is called stability switches of species coexistence in
ecosystems (Song et al., 2014). These results contribute to the control of plankton biomass in
aquatic reservoirs when the model exhibits stable coexistence of nutrient and plankton populations.
Furthermore, the effects of time delay on the model’s dynamics were also explored through
bifurcation diagrams. Figures 3.4 and 3.5 showed the number of stability switches that occur in the
model around the positive equilibrium. Unlike other nutrient-plankton models that only show
destabilizing role of time delay, the model demonstrated both destabilizing and stabilizing effects
of delay on the model’s dynamics. Specifically, the model showed the occurrence of multiple
stability switches around the positive equilibrium as the delay involved in the growth of
phytoplankton. Clearly, the model exhibited periodic-2 & 3 solutions and ultimately became
chaotic for larger values of time delay. Therefore, delay can be considered as the main factor in
the chaotic behavior of the model. It is worth noting that the emergence of chaos may signify the
unpredictability of plankton population biomass distributions over time and space.
48

The size of an organism affects virtually all aspects of its physiology and ecology (Lebedeva,
1972). The body size of zooplankton gradually decreases during equilibrium conditions compared
to chaos (Peters, 1983). Jorgensen et al. (2002) have demonstrated that the size combinations
between phytoplankton and zooplankton are crucial for the model’s self-organization. The model
cannot adapt to the gradual decrease in zooplankton size, resulting in a transition from an
equilibrium state to a chaotic condition. Low zooplankton populations benefit from rapid growth.
If rapid growth continues, phytoplankton will be rapidly exhausted, causing the zooplankton
population to plummet. As a result, the system undergoes violent oscillations and ultimately
reaches chaos. However, this behavior is not prevalent in many ecosystems because they are selforganizing and self-adapting (Odum, 1988). They tune themselves to a critical state (Kauffman,
1991) and exhibit a high extent of self-organization based on a hierarchy of feedback mechanisms.
The present study also showed that the plankton model may become chaotic if there is a large time
delay in the growth of phytoplankton after nutrient uptake. These results indicate that the
investigated nutrient-plankton model cannot accurately predict bloom events but it contributes to
enhancing our understanding of the influences of time delay and diffusion on the interplay between
nutrient and plankton populations in aquatic reservoirs. Overall, these findings may not provide
insights into the sustainability of biodiversity by explaining the emergence of chaos in the plankton
model, but they can be employed to develop management strategies to preserve and restore the
integrity of aquatic habitats.

49

Chapter 4
DYNAMICS OF A STOCHASTIC LESLIE-GOWER PHYTOPLANKTON-ZOOPLANKTON
MODEL WITH PREY REFUGE2
Abstract
In this chapter, the interaction dynamics of a Leslie-Gower phytoplankton-zooplankton model
under stochastic environment were investigated. The theoretical analysis explored the sufficient
conditions for the existence of a global positive solution of the model. The results concluded that
noise intensity and prey refuge play a significant role in the coexistence of plankton, and the model
exhibited an ergodic stationary distribution. The numerical results further indicated that refuge of
phytoplankton could significantly influence the intensity of planktonic oscillations, and prey refuge
that is too large or too small is detrimental to plankton coexistence. These findings provide insights
into the dynamics of phytoplankton-zooplankton models.
Keywords: Prey refuge, stable in mean, stationary distribution, ergodicity

_____________________________
2

Guo, Q., Dai, C.J., Wang, L.J., Liu, H., Wang, Y., Li, J., Zhao, M., 2021. Dynamics of a stochastic LeslieGower phytoplankton-zooplankton model with prey refuge. (prepared).

50

4.1. Introduction
The rapid change in plankton populations has drawn considerable attention in recent years as
planktonic blooms can significantly affect aquatic ecosystems and pose a human health risk
through the food web (Javidi and Ahmad, 2015). Experimental evidence has identified several
factors that affect plankton growth dynamics, including nutrient availability (Sandrini et al., 2020),
light intensity (Burson et al., 2019), and intraspecific competition (Passarge et al., 2006). Among
these factors, the interaction between phytoplankton and zooplankton has become an important
research area in aquatic ecosystems, as the effects of zooplankton grazers often extend well beyond
the phytoplankton community and influence the diversity of multiple trophic levels (Birtel and
Matthews, 2016; Duffy, 2002).
Recent experimental and field studies have successfully explored these interaction dynamics
(Landry and Hassett, 1982). For example, a grazing experiment supports that Daphnia, a filter
feeding cladoceran, can directly affect phytoplankton diversity (Berga et al., 2015; Verreydt et al.,
2012). In addition, the role of zooplankton grazing selection on phytoplankton dynamics has been
experimentally studied by Ger et al. (2018). However, the extent to which such interaction
dynamics are not well established. One potential reason is that nonlinear interactions between
plankton species widely exist in aquatic ecosystems, thereby leading to the growth process of
plankton considerably more complex. Consequently, studies on the dynamics of plankton growth
remain a challenge.
Mathematical modeling of plankton populations has been successfully developed and has
become an important field of investigation for providing quantitative insights into the dynamics of
plankton growth (Dai et al., 2016; Dai et al., 2019; Ojha and Thakur., 2020). Over the recent years,
extensive plankton growth models have been developed (Camara et al., 2019; Chen et al., 2019;
51

Guo et al., 2020; Kaur et al., 2021; Sarkar et al., 2005; Tian and Ruan., 2019; Zhao and Wei., 2015).
Moreover, a growing body of evidence suggests that refuge for plankton is a common phenomenon
in some aquatic ecosystems (Li et al., 2017). For example, submerged plants provide a refuge for
daphnids against predation (Beklioglu and Moss, 1998; Lauridsen and Lodge, 1996). Additionally,
phytoplankton also use benthic sediments as a refuge through the production of cysts. (Schindler
and Scheuerell, 2002). Wiles et al. (2006) claimed that stratification of the water column could
provide a temporary refuge for phytoplankton to recover. Therefore, considering prey refuge is
crucial in studying the dynamics of planktonic interactions.
Furthermore, there is growing evidence that aquatic ecosystems are inevitably affected by
environment fluctuations, as they are generally complex and open systems (Zhao et al., 2017b). In
fact, blooms occur in response to a combination of climatic and hydrographic events, resulting in
the stochastic scaling and timing of blooms (Anderson, 1997; Anderson et al., 2002; Bruno et al.,
1989; McGowan et al., 2017), which has led to an increasing interest in understanding how
plankton growth responds to environmental fluctuations (Anderies and Beisner, 2000; Liu et al.,
2021a; Majumder et al., 2021; Møller et al., 2011; Wang and Liu, 2020; Wei and Fu, 2020). Wang
and Liu (2020) improved the stochastic hybrid phytoplankton-zooplankton model proposed by Yu
et al. (2019b), in which a unique ergodic stationary distribution existed. Furthermore, the nontrivial
positive stochastically periodic solution was studied in a non-autonomous toxic-producing
phytoplankton allelopathy model with environmental fluctuation (Zhao et al., 2017a). Yu et al.
(2019a) developed a nutrient-plankton model with the effect of regime switching, and the results
showed that the Markov chain was beneficial for the survival of plankton. Although stochastic
population dynamic models have been widely explored in recent years, the investigation of the
related dynamics is still ongoing.
The aim of this chapter is to provide insights into the dynamics of phytoplankton-zooplankton
52

interactions. This research is organized as follows: in Chapter 4.2, the mathematical model is
proposed. Chapter 4.3 discusses the existence of a global positive solution of model (4.1), as well
as the persistence and extinction of plankton species. Chapter 4.4 derives the sufficient conditions
for the existence of a stationary distribution of model (4.1). Chapter 4.5 presents some numerical
results that provide an intuitive view of the effect of white noise and refuge on the interaction
between phytoplankton and zooplankton. Finally, the research ends with some conclusions in
Chapter 4.6.
4.2. The methods and results
4.2.1. Mathematical model
In this chapter, the noise effect and the refuge of phytoplankton are taken into account in a
Leslie-Gower Holling-type II plankton model. The model is formulated based on the following
assumptions.
(1) The growth of phytoplankton ( ) depends on zooplankton ( ) predation.
(2) The growth of plankton follows a logistic law, denoted by

and

, respectively.

denotes the strength of competition among individuals of phytoplankton.
(3) The capture rate of zooplankton on phytoplankton is denoted by , and

is the maximum

value of the per capita reduction rate of zooplankton.
(4) Similar to Mandal et al. (2021a)’s research, a constant proportion
taking refuge is considered, leaving (1 −

)

of phytoplankton

of unprotected phytoplankton available for

zooplankton grazing, which follows by Holling type II functional response,

(

)

( )

(

)

( )

.

Regarding these assumptions, a model schematic that expresses the interactions of nutrient
and plankton is depicted in Figure 4.1. The following stochastic nutrient-plankton model is
obtained:
53

where

( )=

( )

−

( )=

( )

−

( )−

(

)

( )

(

)

( )

( )
(

)

( ),

+

( )

( ) ,

+

( ) represents independent standard Brownian motions, and

(4.1)

is the intensities of the

white noise for = 1, 2, which is assumed to be a positive, bounded.

( ) and

( ) denotes

population density of phytoplankton and zooplankton population with time , respectively. All the
parameters are positive and have biological meanings as listed in Table 4.1.

Figure 4.1. The scheme representation of the model (4.1).

For simplicity, the following notations are introduced:
= 0.5

, = 1, 2;

={

∈

|

> 0, = 1, 2}.

Moreover, to derive the dynamical properties of model (4.1), some auxiliary lemmas are
presented in Appendix E.

4.2.2. Global positive solutions and persistence and extinction

In this subchapter, the results imply that model (4.1) exists a unique positive solution
for any positive initial value.
Lemma 4.1. For any given initial value ( (0),
54

(0)) ∈

, model (4.1) has a unique global

solution ( ( ),

for any ≥ 0 almost surely.

( )) ∈

The proof is given in Appendix F. Lemma 4.1 indicates that the solution of model (4.1) is
remain in

. Then, one can obtain the following Lemma:

Lemma 4.2. If

>

,

, then

>

( ) = 0, a.s.

ln

→

The proof is given in Appendix G. Then the extinction and persistence of model (4.1) are
explored. Thus, the following results are obtained.
Theorem 4.1. (i) If

<

and

, then both phytoplankton population

zooplankton population

( ) are extinct, i.e.,

<

( ) = 0,

→

→

( ) and

( ) = 0, almost surely

(a.s.);
(ii) If

and

<

population

, then phytoplankton population

>

( ) is stable in mean, i.e.,

and

>

population

>

,

>

and

,

>

( )

∫

<1−

)

. a.s.;
( ) is extinct and phytoplankton

(

)

=

. a.s.;

then phytoplankton population

( ) is extinct and

( ) is stable in mean, i.e.,
→

>

(

( ) is stable in mean, i.e.,

zooplankton population

(v) If

=

, then zooplankton population

<

→

(iv) If

( )

∫

→

(iii) If

( ) is extinct and zooplankton

and 1 >

( )

∫
>1−

(

=
)

the Leslie-Gower term are stable in mean:

55

(

)

. a.s.;

then both phytoplankton population

( ) and

( )

∫

→

−

( )

∫

→

=

(

)

( )

(

)(

)

,

.

=

The proof is given in Appendix H. From the previous theorem, one can conclude that the
extinction of population can be induced by high-intensity noise, indicating that excessive noise
intensity can threaten the survival of populations. Additionally, Theorem 4.1 implies that refuge
of phytoplankton plays an important role in extinction and coexistence of plankton.
4.2.3. Stationary distribution and ergodicity
In this subchapter, the existence of stationary distribution is investigated, and the following
results are obtained.
Theorem 4.2. If

(

>
(

< min

)

)

(

,
∗

+ (

Then the stochastic process ( ( ),
( ∗,

∗

=

)

and

> 0, = 1, 2 such that

(

) )

∗

+

∗

,

( ∗) .

( )) is ergodic and has a stationary distribution (⋅) in

.

) is defined as in (I1) and
∗

16(

−

(1 −

) )

+

∗

+

1
2

+ 1−

∗

∗

∗

+

∗

.

The proof is given is Appendix I. The results in Theorem 4.2 show that model (4.1) exists an
ergodic stationary distribution, implying the stability in stochastic sense.
4.3. Numerical simulation
In this chapter, the effects of prey refuge on the plankton dynamics are further investigated.
Unless otherwise mentioned, the parameter values used for numerical results are the same as in
Table 4.1. The parameters

,

, and

are chosen as control parameters.

56

Table 4.1. Biological explanations of variables and parameters in model (4.1), and numerical
values used for simulation results
Parameter

Description

Unit

Value

Source

Phytoplankton biomass

/

Zooplankton biomass

/

Intrinsic growth rate of
phytoplankton

day-1

0.2

0.2(a)

Intrinsic growth rate of zooplankton

day-1

0.1

[0.05,0.25](b)

/

0.05

0.08/0.1(c)

0.3

0.2/1(d)

-

[0,1]

Assumed

day-1

0.1

0.1(e)

-

0.3

0.3(d)

Phytoplankton death rate due to
intraspecific competition
Saturation constant for the uptake
of phytoplankton by zooplankton
Refuge protection to phytoplankton
population
Phytoplankton uptake rate by
zooplankton
Phytoplankton toxin release rate

/
/

(a) Kartal et al. (2016); (b) Hopcroft et al. (2005); (c) Sajan and Dubey (2021);
(d) Lv et al. (2010); (e) Wang et al. (2016).
Theorem 4.1 demonstrates the critical role of prey refuge and noise density in the persistence
of plankton. It is clear from Figure 4.2 that there exist five spaces in (

−

−

) plane. Space I

correspond to Theorem 4.1 (i), indicating that both phytoplankton and zooplankton are tended to
go extinct (see Figure 4.3(a)). In space II,

<

and

>

hold, showing that the

phytoplankton is extinct while the zooplankton is stable in mean (see Figure 4.3(b)), and the space
III represents the opposite scenario to space II (see Figure 4.3(c)). In space IV, the phytoplankton
is extinct while the zooplankton is stable in mean (see Figure 4.3(d)). In space V, both
phytoplankton population and the Leslie-Gower term are stable in mean (see Figure 4.3(e)).

57

Figure 4.2. The analysis of the threshold for the extinction and persistent of model (4.1) in (

−

−

)

space. The space I corresponds to Theorem 4.1 (i); the space II corresponds to Theorem 4.1 (ii); the space
III corresponds to Theorem 4.1 (iii); the space IV corresponds to Theorem 4.1 (iv); the space V corresponds
to Theorem 4.1 (v).

Moreover, by choosing

=

= 0.2, the distribution of

and

are presented in Figure

4.4(a) and Figure 4.4(b), respectively. Figure 4.4(a) clearly shows an increase in variance of
phytoplankton population density, while Figure 4.4(b) displays an increasing trend followed by a
decreasing trend in the variance of zooplankton population density. In addition, the boxplot of
and

under

value of

=

= 0.2 are shown in Figure 4.5(a), revealing that the maximum and minimum

increase with increasing prey refuge, and

shows an increasing trend followed by a

decreasing trend in the maximum and minimum value. Figure 4.5(b) displays the mean and
variance of plankton population density, and the results reveal that increasing prey refuge leads to
an increase in the mean and variance of

, which indicates increasing prey refuge can enhance

phytoplankton population oscillation. Meanwhile, the results in Figure 4.5(b) further indicate that
the mean and variance of

show an increasing trend followed by a decreasing trend, suggesting

that prey refuge can enhance the zooplankton population oscillation within the threshold, but
oscillation tends to recede after exceeding the threshold. Furthermore, Figure 4.6 illustrates the
58

sample path of plankton with respect to

=0.3, 0.5, 0.7, and 0.9. Clearly, the results in Figure 4.6

are consistent with those in Figure 4.4 and 4.5.

Figure 4.3. Solutions of model (4.1) for
= 0.3,

= 0.5; (d)

= 0.5,

= 0.2 with (a)

= 0.3; (e)

= 0.15,

59

= 0.7,
= 0.35.

= 0.5; (b)

= 0.7,

= 0.3; (c)

Figure 4.4. For model (4.1) with
respect to

= 0.2, (a) the distribution of

=

and (b) the distribution of

with

.

Figure 4.5. For model (4.1) with
mean and variance of

and

= 0.2, (a) the boxplot of

=

with respect to

Figure 4.6. For model (4.1) with
0.3, 0.5, 0.7, 0.9; (b) the sample path of

=

and

with respect to

; (b) the

.

= 0.2 , (a) the sample path of
with respect to

60

= 0.3, 0.5, 0.7, 0.9.

with respect to

=

4.4. Conclusions
Previous studies have suggested that environmental fluctuations have a considerable impact
on the dynamics of plankton populations. Furthermore, growing evidence suggests that different
species respond differently to these fluctuations, resulting in phytoplankton-zooplankton
interactions becomes more complex and not well understood. Therefore, it is crucial to investigate
the collective response of plankton to environmental changes (Lee et al., 2020). Moreover, due to
the prevalence of prey refuge as a common phenomenon in aquatic ecosystems, it is essential to
account for prey refuge in modeling studies.
In this chapter, a stochastic Leslie-Gower plankton model incorporating prey refuge was
developed. Firstly, the theoretical results concluded that model (4.1) existed a global positive
solution for any positive initial values. Then, the sufficient conditions for the persistence and
extinction of plankton were derived. The results implied that noise intensity and prey refuge played
a crucial role in determining the coexistence of plankton. Specifically, when the noise intensity
was within a critical value and the prey refuge was beyond a threshold, phytoplankton and
zooplankton could coexist indefinitely. Furthermore, the analysis revealed that model (4.1) has an
ergodic stationary distribution, which improves the understanding of how environmental noise
affects plankton interaction dynamics by not only indicating random weak stability but also
providing a better description of persistence (Gao et al., 2019; Liu et al., 2013).
The numerical results further showed how environmental noise and prey refuge affected the
interaction dynamics of model (4.1). As shown in Figure 4.2, the (

−

−

) plane could be

divided into five spaces, each corresponding to one of the cases in Theorem 4.1. Particularly, space
V indicated that plankton populations can coexist. Figure 4.4 illustrated the distribution of the
plankton population, revealing that increasing prey refuge leads to an increase in the variance of
phytoplankton, which indicates that prey refuge can enhance the oscillation range of the
phytoplankton population. However, the variance of zooplankton tended to increase and then
61

decrease as prey refuge increased, suggesting that excessive prey refuge may be detrimental to
zooplankton populations. Figure 4.5(a) showed the boxplot of plankton with respect to prey refuge,
and the mean and variance of the plankton were shown in Figure 4.5(b). The numerical results
from Figure 4.5 were consistent with Figure 4.4. Clearly, when the refuge of phytoplankton was
too small, the phytoplankton population went extinct. When the prey refuge increased within the
threshold, both phytoplankton and zooplankton populations showed an increasing trend. However,
when the prey refuge exceeded the threshold, the phytoplankton population showed a slower rate
of increase in oscillation intensity, whereas the zooplankton population showed a decreasing trend,
indicating that excessive refuge of phytoplankton may lead to a decrease in zooplankton
populations. Obviously, phytoplankton refuge that is too large or too small is not conducive to
plankton coexistence. The present study provides insights into how phytoplankton refuge affects
plankton interaction dynamics under environmental fluctuations.

62

Chapter 5
STOCHASTIC PERIODIC SOLUTION OF A NUTRIENT-PLANKTON MODEL WITH
SEASONAL FLUCTUATION3
Abstract
In this chapter, a stochastic nutrient–plankton model with seasonal fluctuation was developed
to investigate how seasonality and environmental noise affect the dynamics of aquatic ecosystems.
Firstly, the survival analysis of plankton was proposed. Then, by using Lyapunov function and
Khasminskii’s theory for periodic Markov processes, the sufficient conditions for the existence of
positive periodic solution were derived. Numerical simulations were carried out to provide a better
understanding of the model, and the results indicated that seasonal fluctuation was beneficial to
the coexistence of plankton species.
Keywords: Extinction and persistence, survival analysis, periodic solution, white noise

_____________________________
3Guo, Q., Dai, C.J., Wang, L.J., Liu, H., Wang, Y., Li, J., Zhao, M., 2022. Stochastic periodic solution of a

nutrient-plankton model with seasonal fluctuation. Journal of Biological Systems. 30, 695-720. DOI:
10.1142/S0218339022500255.
63

5.1. Introduction
Phytoplankton is the basis of aquatic food webs and can directly affect large-scale global
processes by absorbing carbon dioxide from the atmosphere (Huppert et al., 2002). However,
excessive growth of phytoplankton may disrupt food webs and create large hypoxic zones leading to
fish deaths. Furthermore, excessive growth of phytoplankton can disturb the balance of ecological
structure and function (Otten and Paerl, 2011). Experimental and mathematical studies have revealed
an important fact that temperature (Paerl and Huisman, 2008), light (Anderson et al., 1994), and
nutrient supply (Conley et al., 2009) are all responsible for the algae growth. However, due to the
complexity and high nonlinearity of aquatic ecosystems, the dynamics of phytoplankton growth are
not well understood (Dai et al., 2016).
Mathematical modeling is regarded as an efficient method to explain the complex dynamical
behaviors of phytoplankton growth because mathematical models can provide quantitative insights
into the dynamics of phytoplankton growth (Dai et al., 2019; Guo et al., 2020). Additionally, the two
classical ways to study phytoplankton growth dynamics are the bottom-up control and top-down
control approaches, which have been widely explored in recent years (Dai et al., 2015; Sekerci and
Ozarslan, 2020; Tian and Ruan, 2019; Zhao and Wei, 2015). For example, a simple nutrientphytoplankton model was proposed and studied theoretically and numerically in Huppert et al. (2002),
and the results revealed that a bloom can only be triggered by nutrients. Furthermore, the effect of
spatial heterogeneity on phytoplankton allelopathy was studied in a nutrient-plankton model by
Mukhopadhyay and Bhattacharyya (2006).
All the forementioned literatures assumed an unvarying deterministic environment, where all the
biological parameters were constants. However, it has been observed that environmental fluctuations
underlie key processes of the structure and function of ecosystems (Vellend et al., 2014), such as
carrying capacities, birth rates, and other parameters, to some extent, exhibit random fluctuations
64

(Black and McKane, 2012; May, 1973). Since the pioneering work of Haminskii (1980), many
explanations relying on stochastic population dynamics have been proposed (Cai et al., 2020; Dean
and Shnerb, 2020; Dobramysl et al., 2018; Liu and Deng, 2020). Importantly, Mao et al. (2002)
revealed an important fact that stochastic noise can suppress potential explosions in population
dynamics, while Deng et al. (2008) pointed out that noise can suppress or express exponential growth.
In fact, blooms occur in response to a combination of physical, chemical, and biological
processes. In addition, the climatic and hydrographic events contribute to the unpredictability of algal
blooms (Anderson et al., 2002; Bruno et al., 1989; McGowan et al., 2017). Some blooms occur in
spring (Hunter-Cevera et al., 2016), others in summer (Kahrua et al., 2020), and even in winter (Ma
et al., 2016). Moreover, the high degree of variability in the scale and timing of blooms results in the
occurrence of blooms characterized as ultimately random phenomena, which makes understanding
the growth of phytoplankton challenging. In recent years, numerous stochastic phytoplankton growth
models have been proposed to investigate the bloom phenomenon (Chen et al., 2020c; Song et al.,
2020a; Wang and Liu, 2020; Yu et al., 2019b; Zhao et al., 2016). Imhof and Walcher (2005)
constructed a stochastic single-substrate chemostat model and demonstrated that white noise may lead
to extinction in certain scenarios while the deterministic model predicts persistence. Yu et al. (2019a)
investigated a nutrient-plankton model involving regime switching effects, and the results suggested
that the Markov chain was beneficial for the survival of plankton.
Moreover, one of the interesting problems in phytoplankton growth studies is the seasonal
fluctuation in the plankton community. Extensive research has been conducted on the mechanistic
principles underlying the real-world randomness caused by external perturbations like seasonal
differences in water temperature, nutrient supply, etc. (Yuan and Zhang, 2012a), resulting in changes
in population density exhibiting a more or less periodicity. Smayda (1998) demonstrated that
irrespective of highly variable planktonic habitats, there are impressive, quasiregular, predictable
65

annual occurrences of major blooms, in seasonal cycles. For example, the seasonal changes in water
temperature result in seasonal blooms of Microcystis spp. in Lake Taihu, China, lasting from May to
October (Otten and Paerl, 2011). Additionally, Wu et al. (2013) analyzed harmful algal blooms (HABs)
data for over 11 years in the southwest Bohai Sea and found that the occurrences of HABs exhibit
significant seasonality. Due to its evident importance, the effect of seasonal fluctuation on plankton
dynamics under stochastic environments has been extensively studied in recent years (Jang and Allen,
2015; Yuan and Zhang, 2012b; Zhao et al., 2017a).
This chapter aims to improve the understanding of phytoplankton growth dynamics under
periodic environmental and stochastic effects. The rest of the chapter is arranged as follows: The
mathematical model is presented in Chapter 5.2, the survival analysis and conditions for the existence
of nontrivial periodic solutions of model (5.2) are proposed in Chapter 5.3. In Chapter 5.4, some
numerical simulations are performed to illustrate the analysis results. In the end, the conclusion is
provided in Chapter 5.5.
5.2. The mathematical model
In this chapter, the interactions among nutrients, phytoplankton, and zooplankton are
considered. Motivated by the mathematical models of Ruan (1993) and Yu et al. (2019a), a
phytoplankton growth model consisting of nutrient ( ), phytoplankton ( ) and zooplankton ( ) is
developed. The model is formulated based on the following assumptions.
(1) Zooplankton graze on phytoplankton and the growth of phytoplankton depends on nutrient.
(2) The nutrient uptake of phytoplankton follows a logistic law, and the zooplankton predation
function is described by the Beddington-DeAngelis form (Beddington, 1975; DeAngelis et al.,
1975),

, where

represents the consumption rate, and

and

denote the mutual

interference between the zooplankton and phytoplankton saturation constant, respectively.
66

(3)

and

denote the constant nutrient input and washout rate of nutrient, respectively. In

addition, the constant

describes the maximal nutrient uptake.

(4) Referring to the model of Ruan (1993), nutrient recycling is considered as an instantaneous
term, where the nutrient recycling rate from the dead phytoplankton and zooplankton is denoted as
ℎ and , respectively.
(5) Parameters

and

represent the mortality of phytoplankton and zooplankton,

respectively.
(6) Furthermore, white noise is incorporated into the model to describe the random
fluctuations in the real world, as environmental noise is ubiquitous in aquatic ecosystems (Liu and
Wang, 2010).
Based on the above assumptions, a model schematic diagram is shown in Figure 5.1.
Following the approach used in Liu and Bai (2016), Li and Mao (2009), and Zhu and Yin (2009),
i.e., the white noise is assumed to mainly affect the growth rates of the plankton. This approach
has been claimed to be reasonable and well justified biologically (Braumann, 2002; Liu and Bai,
2016). Thus, the following nutrient-plankton model, which accounts for white noise:
⎧
⎪
⎨
⎪
⎩
where

( )=[
( )=
( )=

− ( ) −
( ) ( )−

( ) ( ) + ℎ) ( ) +
( )−

( ) ( )
−
( )
( )

( )

( ) ( )
( )
( )

+

( )

+

( )]
( )

( )

+
( ),

( ),
(5.1)

( ),

( ) are independent standard Brownian motions, and

are the intensities of the white

noise for = 1, 2, 3. Throughout this chapter, the processes

( ), = 1, 2, 3 are defined on a

complete probability space (Ω, , {

}

, ℙ). In addition, the biological meanings of the variables

and parameters involved in model (5.1) are listed in Table 5.1.
Additionally, it is widely recognized that the growth of plankton is evidently affected by
seasonal fluctuations (e.g., periodic fluctuation with the seasons (Zhao et al., 2017a)). Thus,
67

seasonal fluctuations are incorporated into model (5.1), yielding the following model:

⎧
⎪
⎨
⎪
⎩

( )=[ ( )
( )=

( ) − ( ) − ( ) ( ) ( ) + ℎ( )) ( ) + ( ) ( )]
( ),
+ ( ) ( )

( ) ( ) ( )− ( ) ( )−
( ) ( ) ( )
−
( ) ( ) ( ) ( )

( )=

( ) ( )

( ) ( ) ( )
( ) ( ) ( ) ( )

+

( ) ( )

( ),

+

( ) ( )

( ), (5.2)

where all the parameters are positive, bounded, continuous -periodic functions.
By now, a stochastic nutrient–plankton model is developed focusing on the effects of
seasonal fluctuation and the stochastic environment on phytoplankton growth dynamics.

Figure 5.1. The scheme representation of the model (5.1).

Denote
=

sup
∈[ ,

( )

=

)

inf

∈[ ,

)

( ),

where ( ) is a bounded function on [0, +∞). In addition, in view of Gray et al. (2011)’s research,
the existence of the global positive solution of model (5.2) is obtained. Furthermore, according to
Lemma 2.3 in Zhao (2016), the following relations obtained:
lim sup[ ( ) + ( ) + ( )] < ∞ , lim

→

lim

→

→

∫

( )

( )

( ) = 0 , lim
→

∫

( )

( )

( ) = 0,

(5.3)

∫

( )

( )

( ) = 0,

(5.4)

hold almost surely. For convenience, the following function is defined:
68

( )=

( )

exp ∫

∫

( )

1 − exp − ∫

It is easy to check that ( ) is the unique

.

( )

−periodic solution of equation

( )= ( ) ( )−

( ). In addition, the following notations are introduced:
( ) = ( ) + [ ( ) − ℎ( ) − ( )] ( ),
( ) = ( ) + [ ( ) − ( ) − ( )] ( ),
ℜ =

1

ℜ =−
ℜ =

1
2

( ) ( ) ( )−

( )+

1
2

( )−

( )
( )

( )+

( )

1

( )+

ℜ −

∫

( )

,

,
,

and two lemmas are presented in Appendix E.
5.3. The main results
In this chapter, the dynamical behaviors of the model are investigated. First, the survival
analysis of model (5.2) is investigated, and then sufficient conditions for the existence of nontrivial
periodic solution are derived.
5.3.1. Survival analysis of model (5.2)
In this chapter, the survival analysis of model (5.2) is given. First, one can get the following
lemma.
Lemma 5.1. For model (5.2), if

→

( ) = 0 a.s., then

→

( ) = 0 a.s.

The proof is given in Appendix J. The result in Lemma 5.1 suggests that if phytoplankton
species are extinct with sufficient large time, then the zooplankton population follows.
Then, the following assumptions are presented:
Assumption 1. For arbitrary 0 < < 1 , there exists a set
69

⊂

with ℙ(Ω ) ≥ 1 − and a

constant

= ( ) such that lim

( )

∫

→

( )+

− lim ∫
→

( )

≤ , for

∈

and > .
Thus, the following theorem is obtained.
Theorem 5.1. Denote ( ( ), ( ), ( )) to be the solution of model (5.2) with initial values
( (0), (0), (0)) ∈ ℝ . If

{ ( ) − ℎ( ) − ( )} ≥ 0,

{ ( ) − ( ) − ( )} ≥ 0 holds,

then one can obtain
(i) If ℜ < 0, then

( )=

→

→

( ) = 0 almost surely;

(ii) If ℜ > 0 and ℜ < 0, then
ℜ

1

≤

( )

→

≤

, ( − ℎ) , ( − ) } > max{(

) ,(

→

( ) = 0 almost surely;

(iii) Under Assumption 1, if min{
ℜ >

ℜ

( )

→

and

1

≤

> 0 hold, there exist a constant
1

( )

) ,(

) } and

> 0 such that
1

,

→

( )

≥

>0

→

almost surely.
The proof is given in Appendix K. Theorem 5.1 provides the information about the survival
analysis of plankton in periodic environment, that is, when noise intensity is small, the plankton
populations can coexist under certain conditions.
5.3.2. Existence of the nontrivial periodic solution of model (5.2)
In this subchapter, the existence of positive
in Theorem 5.2.

70

−periodic solution is studied. The result is given

Theorem 5.2. If min{

, ( − ℎ) , ( − ) } > max{(

Then model (5.2) has at least one positive

) ,(

) ,(

) } and ℜ > 0 hold.

−periodic solution.

The proof is given is Appendix L. The results in Theorem 5.2 suggest that if the model has
low noise intensity, the model admits a positive

−periodic solution when ℜ > 0, which reveals

that periodic blooms of phytoplankton can be triggered.
Table 5.1. Biological explanations of variables and parameters in model (5.1), and numerical
values used for simulation results
Parameter Description

ℎ

Unit

Nutrient concentration

/

Phytoplankton biomass

/

Zooplankton biomass

/

Value

Source

0.8

0.6(a)

Maximal nutrient uptake rate
of phytoplankton
Nutrient recycling rate from
dead phytoplankton
Nutrient recycling rate from
dead zooplankton

day-1

0.08

0.08(b)

day-1

0.01

0.01(f)

Death rate for phytoplankton

day-1

0.6

[0.2,0.65](c)

Death rate for zooplankton

day-1

0.5

[0.08,0.6](d)

Consumption rate of phytoplankton
by zooplankton

day-1

0.4

0.5(e)

Phytoplankton saturation constant

/

0.5

Assumed

Mutual interference between
the zooplankton

/

[0,5]

Assumed

Washout rate for nutrient

day-1

0.5

0.4(g)

/

[0,10]

[0,12.5](b)

/

Input concentration of nutrient

/

(a) Jang and Baglama (2005); (b) Ruan (2001); (c) Garcés et al. (2005); (d) Turner et al. (2014);
(e) Jang and Allen (2015); (f) Yu et al. (2019a); (g) Rehim et al. (2016).

5.4. Numerical simulation
In this chapter, some numerical results are presented to further investigate how seasonal
71

fluctuation affects model (5.2). For the stochastic model (5.2), it is assumed that all the parameters
are periodic and have a common period of 50 days. The parameter set in the periodic environment
is chosen as follows:
= 0.5 + 0.01 sin( ⁄25),

= 0.8 + 0.1 sin( ⁄25), ℎ = 0.08 + 0.1 sin( ⁄25),

= 0.01 + 0.1 sin( ⁄25) ,

= 0.4 + 0.1 sin( ⁄25) , = 0.6 + 0.01 sin( ⁄25),

= 0.5 + 0.01 sin( ⁄25) ,

= 0.5 + 0.1 sin( ⁄25) ,

= 0.15 + 0.01 sin( ⁄25) ,
The parameters

and

= 0.15 + 0.01 sin( ⁄25),

= 0.15 + 0.01 sin( ⁄25).

are selected as control parameters.

Figure 5.2. For model (5.1) without noise, (a) the solutions of

in the ( , ) plane and (b) the solutions of

in the ( , ) plane. For model (5.2) without noise, (c) the solutions of
solutions of

in the ( , ) plane and (d) the

in the ( , ) plane. The white dash line and the red dash line represents the critical value of

ℜ = 0 and ℜ =

/

, respectively.
72

First, under

= 0.2 + 0.01sin ( /25), Figure 5.2 displays the solution of phytoplankton and

zooplankton populations for the deterministic model with respect to nutrient input

. The results

from Figure 5.2 show that the phytoplankton and zooplankton species can coexist after the nutrient
input exceeds the threshold. Different from the deterministic model, the periodic environment can
cause periodic bloom events of phytoplankton, and the zooplankton biomass changes periodically
over time. Additionally, the sample paths of plankton populations for the deterministic model are
given in Figure 5.3, which reveal that the increase in nutrient input level leads to a marked increase
in the amplitude of planktonic oscillation.

Figure 5.3. For model (5.1) without noise, (a) The sample paths of phytoplankton and (b) zooplankton
corresponding to

= 0.5 + 0.1 sin( /25) , 3 + 0.1 sin( /25) , 5.5 + 0.1 sin( /25) and 8 + 0.1 sin( /

25), respectively. The black line represents the solutions for model (5.2) without noise.

For model (5.2), the theoretical analysis reveals that there exists a nontrivial periodic solution
under certain conditions. Thus, the solutions of model (5.1) and (5.2) with respect to nutrient input
are presented, respectively (see Figure 5.4). Figure 5.4 shows that when the parameters are periodic
functions, periodic coexistence of phytoplankton and zooplankton exists after the nutrient input
exceeds the critical value. Furthermore, the results imply that the patterns in Figure 5.4 are in
agreement with those obtained in Figure 5.2. However, it is worth mentioning that the plankton
density for the stochastic model is always oscillating over time instead of maintaining a certain
level (see Figure 5.4). More importantly, the results obtained in Figure 5.4 imply that
73

environmental fluctuation and nutrient input can significantly increase the distribution and
oscillation intensity of plankton biomass.
Furthermore, the sample paths with respect to nutrient input are given in Figure 5.5 by
choosing

= 0.5 + 0.1 sin( /25), 3 + 0.1 sin( /25), 5.5 + 0.1 sin( /25), and 8 + 0.1 sin( /

25) . Additionally, for model (5.2), by choosing

= 8 + 0.1 sin( /25) , the probability

histograms for phytoplankton and zooplankton species are illustrated in Figures 5.5(b) and 5.5(d),
which can be regarded as approximate probability density functions of the stationary distribution.
From Figure 5.5, it is obvious that nutrient input can enhance the oscillation intensity of plankton
biomass, and the results show that the overall trend of stochastic periodic solutions of model (5.2)
exhibits obvious periodicity.

Figure 5.4. For model (5.1), (a) the solutions of

in the ( , ) plane and (b) the solutions of

in the ( , )

plane. For model (5.2), (c) the solutions of

in the ( , ) plane and (d) the solutions of

in the ( , )

plane. The white dash line and the red dash line represent the critical value of ℜ = 0 and ℜ =
respectively.
74

/

,

Figure 5.5. For model (5.2), (a) the sample paths of phytoplankton and (c) zooplankton corresponding to
= 0.5 + 0.1 sin( /25), 3 + 0.1 sin( /25) , 5.5 + 0.1 sin( /25) and 8 + 0.1 sin( /25) , respectively.
The black line represents the solutions for the deterministic model. (b) Probability histograms for
phytoplankton and (d) zooplankton with

= 8 + 0.1 sin( /25).

To further investigate the effect of seasonal fluctuations on the distribution of plankton, Figure
5.6 is given, which describes the changes in the mean and variance of plankton biomass. Clearly,
with the increase in nutrient input, phytoplankton biomass shows a steady increasing trend, but
there is no significant difference between model (5.2) and (5.1) in the mean and variance of
phytoplankton biomass (see Figure 5.6(a)). Actually, in contrast to seasonal fluctuation, the
concentration of available nutrients is one of the decisive factors controlling phytoplankton
productivity because nutrient limitation is directly related to phytoplankton growth (Hder and Gao,
2015). This is consistent with the results in Figure 5.6(a) that seasonal fluctuation shows less
impact than nutrient input on phytoplankton growth. By contrast, zooplankton growth does not
depend directly on nutrients (Gajbhiye, 2002), but is directly affected by phytoplankton and higher75

level predators. In addition, the effect of seasonal fluctuation on the zooplankton community is
both intense and prolonged compared to their life spans (Mackas et al., 2012), which results in
zooplankton populations being more easily affected by seasonal fluctuation than phytoplankton.
The results from Figure 5.6(b) suggest that seasonal fluctuation significantly increases the
distribution and oscillation intensity of zooplankton biomass. From an ecological point of view,
the periodic environment is beneficial to the survival of zooplankton and the coexistence of
plankton. Furthermore, under

= 8 + 0.1 sin( /25), the distribution of plankton of model (5.2)

in different periods is given in Figure 5.7, which shows that the distribution of phytoplankton of
model (5.2) is almost identical in different periods (see Figure 5.7(a)). In contrast, the results from
Figure 5.7(b) show that there is a significant change in the distribution of zooplankton with respect
to periods.

Figure 5.6. For model (5.1) and (5.2), (a) the mean and variance of phytoplankton biomass with respect to
nutrient input

; (b) The mean and variance of zooplankton biomass with respect to nutrient input

.

In recent years, the effect of mutual interference of zooplankton on the dynamics of plankton
growth models has been explored. For example, it has been reported that the mutual interference
of zooplankton can stabilize the population interactions (De Silva and Jang, 2017; De Silva and
Jang, 2018). Thus, the plankton biomass distribution with respect to the mutual interference of
zooplankton is given in Figure 5.8. Under

= 2 + 0.1 sin( /25), the solutions of plankton in
76

model (5.2) are given in Figures 5.8(a) and 5.8(b), respectively. Clearly, there is an obvious
periodic motion for phytoplankton biomass, but the zooplankton population tends to go extinct.
Then, by fixing nutrient input

= 5 + 0.1 sin( /25), Figures 5.8(c) and 5.8(d) show the solutions

of phytoplankton and zooplankton, respectively. Obviously, both phytoplankton and zooplankton
show periodic trends over time, and the increase in the mutual interference of zooplankton causes
a decreasing trend in the oscillation intensity of zooplankton biomass. In the end, under nutrient
input

= 5 + 0.1 sin( /25), the solutions of phytoplankton and zooplankton are given in Figures

5.8(e) and 5.8(f), respectively. Figures 5.8(e) and 5.8(f) show similar results to Figures 5.8(c) and
5.8(d).

Figure 5.7. For model (5.2), (a) the distribution of phytoplankton biomass in different periods; (b) the
distribution of zooplankton biomass in different periods.

The results from Figure 5.8 imply that excessive mutual interference of zooplankton may not
be beneficial to the survival of the zooplankton population. Furthermore, Figure 5.8 reveals a point
that the distribution of phytoplankton biomass is more sensitive to nutrient input than the mutual
interference of zooplankton because nutrient limitation is directly related to phytoplankton growth
(Hder and Gao, 2015). Notably, although the mutual interference of zooplankton can affect the
oscillation intensity of zooplankton biomass, increasing the mutual interference of zooplankton
does not really influence the dynamic properties of the model under this condition.
77

Figure 5.8. For model (5.2), (a) the solutions of
plane with

= 2 + 0.1 sin( /25); (c) The solutions of

the ( , ) plane with
solutions of

in the ( , ) plane and (b) the solutions of

in the ( , ) plane and (d) the solutions of

= 5 + 0.1 sin( /25); (e) The solutions of

in the ( , ) plane with

in the ( , )
in

in the ( , ) plane and (f) The

= 8 + 0.1 sin( /25).

5.5. Conclusion
Understanding the dynamics of plankton growth is one of the major goals for ecologists and
mathematicians, as phytoplankton communities are known to have a significant impact on aquatic
78

ecosystems (Huppert et al., 2002). It is now well established that both seasonality and stochasticity
are important factors influencing the dynamics of phytoplankton growth (Freund et al., 2006). In
recent years, there has been increasing interest in stochastic phytoplankton growth models
incorporating seasonal fluctuation (Ghosh et al., 2019; Wei and Fu, 2020).
In this chapter, a stochastic nutrient–plankton model with seasonal fluctuation was developed.
The theoretical analysis derived the sufficient condition for the survival of plankton populations.
The results implied that if the mortality rate of plankton was large and the noise intensity was small
enough, the extinction and persistence in mean of plankton depend on ℜ and ℜ . Then, the
sufficient condition for the existence of a stochastic positive periodic solution was carried out using
Khasminskii’s theory for periodic Markov processes.
The numerical simulation further showed the effect of white noise and seasonal fluctuations
on the distribution of plankton biomass. Figures 5.2 and 5.4 displayed the changes in plankton
biomass for the deterministic model and the stochastic model with respect to nutrient input,
respectively. Comparing the patterns of phytoplankton and zooplankton species in the stochastic
model (see Figure 5.4) with their corresponding deterministic model in Figure 5.2, it is obvious
that the oscillation phenomenon in the stochastic model became pronounced, demonstrating that
environmental fluctuations significantly increase the oscillation intensity of plankton populations.
Figure 5.4 showed that under seasonal fluctuations, the model (5.2) was persistent in mean and
existed a positive periodic solution within certain conditions, which implies that seasonality and
stochasticity may cause the periodic bloom of phytoplankton. Additionally, the sample paths of
plankton with respect to nutrient input implied that a high nutrient input level could enhance the
oscillation intensity of plankton biomass, and the overall trend of stochastic periodic solutions of
model (5.2) presented obvious periodicity after the nutrient input went beyond a certain level.
It would also important to note that with an increase in nutrient input, the mean and variance
79

of phytoplankton biomass did not constitute a visible difference between constant parameters and
periodic parameters. However, the variance of zooplankton biomass in model (5.2) with periodic
parameters was much higher than that in model (5.2) with constant parameters, while the mean
remained at the same level. One possible explanation is that the phytoplankton population is more
sensitive to nutrient availability than seasonal fluctuation, while the zooplankton population is
more easily affected by seasonal fluctuation. Figure 5.6 suggested that seasonal fluctuations may
contribute to the survival of zooplankton and further promoted the coexistence of plankton.
Furthermore, the distribution of plankton biomass of model (5.2) with respect to the period was
carried out, and the results demonstrated that there was no significant difference in the distribution
of phytoplankton biomass at various periods from 0 to 100 , but the distribution of zooplankton
biomass was sensitive to changes in periodicity (see Figure 5.7). This study provides insights into
how seasonal fluctuations influence phytoplankton growth dynamics under stochastic environment.
Additionally, it is generally accepted that the mutual interference of zooplankton may affect
the stability of population interactions (De Silva and Jang, 2017; De Silva and Jang, 2018). The
changes in plankton biomass with respect to the mutual interference of zooplankton were presented
in Figure 5.8. The results from Figure 5.8 revealed that with mutual interference of zooplankton,
the oscillation intensity of zooplankton biomass showed a decreasing trend, but there was no
significant change in the dynamic properties of the model. However, it is evident from Figure 5.8
that increased nutrient input could have a significant impact on phytoplankton biomass distribution.
From a biological point of view, the changes in phytoplankton biomass are more sensitive to
nutrient input in contrast to the mutual interference of zooplankton. This study may provide a better
understanding of the role of seasonal fluctuation and nutrient input for phytoplankton growth under
environmental fluctuation, and the results provide insights into the dynamics of phytoplankton
growth.
80

Chapter 6
DYNAMICS OF A STOCHASTIC NUTRIENT-PLANKTON MODEL WITH REGIMESWITCHING4
Abstract
In this chapter, a stochastic nutrient–plankton model with regime switching was proposed,
where the regime-switching plankton mortality was described by a continuous time Markov chain
with different states. Firstly, the effects of regime-switching plankton mortality on the distribution
of plankton biomass, as well as the persistence and extinction of plankton populations, were
examined. Moreover, theoretical analysis showed that the model existed a unique stationary
distribution, which is ergodic, indicating that the plankton populations will survive forever. By
applying a sophisticated sensitivity analysis technique, it was observed that the phytoplankton
biomass was highly sensitive to the grazing rate by zooplankton and least sensitive to the remineralization of dead biomass of plankton into nutrients concentration. The numerical results
showed that the persistence and extinction of plankton populations were sensitive to variations in
nutrient input. Additionally, the numerical analysis showed that noise can enhance the oscillations
of plankton biomass, and the regime-switching plankton mortality has the capacity to decrease the
amplitudes of the oscillations in the bloom phase. The results emphasized that regime-switching
plankton mortality contributes to the survival of plankton populations in the aquatic system.
Keywords: Nutrient input, regime-switching plankton mortality, stochastic permanence,
stationary distribution, sensitivity

_____________________________
4

Guo, Q., Wang, Y., Dai, C.J., Wang, L.J., Liu, H., Li, J., Tiwari, P.K., Zhao, M., 2023. Dynamics of stochastic
nutrient-plankton model with regime-switching. Ecological Modelling. 477, 110249. DOI:
10.1016/j.ecolmodel.2022.110249.

81

6.1. Introduction
Phytoplankton blooms are the rapid accumulation of algae to sufficient numbers, sometimes
reaching millions of cells per liter, resulting in severe negative consequences such as oxygen
depletion, fish mortality, and human illness (Anderson,1997). Results from both experimental and
field observations have demonstrated that planktonic ecosystems are affected by many factors,
such as light (Burson et al., 2019), temperature (Righetti et al., 2019), nutrients (Burson et al.,
2018), and zooplankton (Huisman et al., 2018). The complexity of planktonic ecosystems
contributes to the difficulty in preventing the occurrence of massive phytoplankton blooms (Dai et
al., 2019). For instance, the results reported by Jiang et al. (2015) suggested that the phytoplankton
community in different areas of Lake Erie shows different sensitivities to nitrogen and phosphorus
concentrations. In fact, blooms are likely to occur in response to a combination of climatic and
hydrographic events, and the nonlinearity of ecological dynamics results in unpredictable algal
blooms (Anderson et al., 2002; Bruno et al., 1989; McGowan et al., 2017). Actually, the growth
responses of plankton are inevitably affected by random fluctuations (Freund et al., 2006), which
result in the stochastic scale and timing of blooms. For example, blooms may occur in bays or
estuaries covering thousands of square kilometers, and last for a few weeks to years (Anderson,
1997).
The appearance of algal blooms is often considered one of the key signals of eutrophication
(Huppert et al., 2002), indicating the balance between the processes of algae production and
consumption is broken (Cloern, 2001). Growing evidence supports that the phytoplankton blooms
are generally followed by a sudden termination within a few days (Abada et al., 2021; Aberle et
al., 2007), demonstrating that mortality rates are higher during the decline phase compared to the
development phase of blooms (Garcés et al., 2005), and the underlying cause remains a mystery.
In recent years, the sudden collapse of phytoplankton blooms has drawn increasing attention
82

among research scientists. Abada et al. (2021) reported that the secretion of NO by algae and the
dramatic decrease in dissolved oxygen levels contributes to higher phytoplankton mortality and
further promote the collapse of the entire algal population. Specifically, Børsheim et al. (2005)
demonstrated that phytoplankton mortality varies between 8 to 18% of present biomass per day in
the development phase and reaches 45% in the bloom phase, followed by a collapse of the bloom,
indicating that phytoplankton mortality is an important factor regulating bloom termination.
Phytoplankton mortality has been measured in several experimental studies. The mortality of
natural single Alexandrium spp. cells during the development, maintenance, and decline phases of
blooms was studied by Garcés et al. (2005), using a dilution experiment. Their results indicated
that the mortality rates of Alexandrium taylori, one of the single Alexandrium spp., ranged from
0.2 to 0.65 d−1, with the highest mortality rates during the decline phase. This may be attributed to
the fast-growing algae causing a dramatic decrease in dissolved oxygen levels due to the
metabolism of algae (Cloern, 2001). Additionally, decreased light penetration in the bottom of the
water also results in a decrease in oxygen production (Møhlenberg, 1999). Furthermore, the rapid
decomposition of dead algae leads to the depletion of dissolved oxygen and creates a hypoxic or
anoxic ‘dead zone’ lacking sufficient oxygen to support aquatic organisms (Anderson, 1997;
Chislock et al., 2013). Zooplankton mortality varies substantially in the decline phase of bloom
events after the massive death of algae (Lin et al., 2014). Boyd et al. (1975) measured the average
density of Anabaena variabilis filaments and the concentration of dissolved oxygen before and
after the phytoplankton die-off. Their findings showed that the average density of Anabaena
variabilis filaments increased from 2440 cells ml−1 to 3600 cells ml−1 in the development phase,
with the peaks reaching 37300 cells ml−1, and then suddenly decreased to less than 1000 cells ml−1
in the following days, while the concentration of dissolved oxygen dramatically decreased from 9
mg/L to 0.15 mg/L.
83

The sharp increase in plankton mortality during the decline phase can be described by the socalled colored noise, specifically the telegraph noise that can be represented by a finite-state
Markov chain switching between two or more environmental regimes (Du et al., 2004; Liu and
Wang, 2010). Usually, the switching among different environments is memoryless, and the waiting
time for the next switch follows an exponential distribution (Li et al., 2009; Luo and Mao, 2007;
Zou and Wang, 2014). Over the last few decades, a number of stochastic models have been used
to explore the dynamics of phytoplankton growth. Additionally, stochastic noise has been proven
to suppress or express exponential growth in population dynamics (Deng et al., 2008; Mao et al.,
2002). In addition to white noise, aquatic ecosystems are greatly affected by colored noise. For
example, the growth environment in winter is much different from that in summer, resulting in
seasonal changes in the diversity of phytoplankton. In fact, the changes between different regimes
can be described by the Markov chain and their effects on population dynamics have been
extensively explored. Yu et al. (2019a) studied a nutrient–plankton food chain model with regime
switching, and found that the Markov chain is beneficial for the survival of plankton. The study by
Chen et al. (2020) demonstrated that the Markov chain could balance the density of the population
under different regimes. Although considerable efforts have been devoted to understanding the
factors regulating the development of phytoplankton blooms, the dynamical behaviors induced by
regime-switching plankton mortality remain largely unexplored.
In the study of plankton dynamics, constant phytoplankton mortality is commonly used.
However, constant phytoplankton mortality cannot explain the sudden decline of phytoplankton
blooms. Considering the variability in the mortality of phytoplankton as observed in previous
experimental studies, it is unrealistic to represent phytoplankton mortality with a constant. In this
chapter, regime-switching plankton mortality is described using a continuous time Markov chain
with different states, and a stochastic nutrient–plankton model is developed to investigate the
84

effects of regime-switching plankton mortality on the phytoplankton growth dynamics. The present
chapter applies colored noise to describe the changes in the mortality of plankton over different
phases of blooms. The objectives of the present work are to provide insight into phytoplankton
responses to stochastic environment under regime switching by using a nutrient-phytoplankton
zooplankton model.
The rest of the chapter is organized as follows. The mathematical model is presented in the
next chapter. In Chapter 6.3, the dynamical behaviors of the proposed model are investigated.
Some numerical simulations are performed in Chapter 6.4 to explore the model’s dynamics and
complexity in the aquatic system. Sensitivity analyses are also presented to help us identify the
crucial parameters causing/terminating the planktonic bloom. In Chapter 6.5, a discussion about
the implications of the results obtained in this study is presented. Lastly, some conclusions are
provided in Chapter 6.6.
6.2. The mathematical model
In recent years, nutrient-phytoplankton-zooplankton type food web models have been widely
employed in aquatic ecosystem studies, with the basic assumption that phytoplankton absorbs
nutrients and is grazed upon by herbivorous zooplankton (Jang et al., 2006; Mitra, 2009). In aquatic
ecosystems, a number of factors such as light and temperature are responsible for the growth of
phytoplankton. To avoid model complexity, this research does not consider the effects of light and
temperature on phytoplankton growth and assumes that the growth of phytoplankton fully depends
on the availability of nutrients. Meanwhile, all other environmental factors (e.g., light, temperature,
etc.) are assumed to be sufficiently abundant. The model is developed by modifying the aquatic
planktonic ecosystem model presented by Ruan (2001), which has been widely implemented. The
model is composed of nutrients, phytoplankton, and zooplankton. The nutrient concentration
85

consists of nutrient input, nutrient washout, and its uptake by phytoplankton. Furthermore, the
process of nutrient regeneration due to bacterial decomposition of the dead biomass of plankton is
involved, which has been widely reported (Guo et al., 2020; Ruan, 1993; Zhuang et al., 2021). The
phytoplankton biomass is determined by two factors: the growth caused by the conversion of
nutrients, and the loss caused by predation and natural death. In addition, the predation is described
by the Beddington-DeAngelis functional response (Beddington, 1975; DeAngelis et al., 1975). The
zooplankton biomass is determined by the growth caused by the conversion of phytoplankton and
the loss caused by natural death.
At any time > 0,

( ), ( ), and ( ) represent the nutrient concentration, phytoplankton

density, and zooplankton density, respectively. The model is formulated based on the following
assumptions.
(1) Phytoplankton growth depends on nutrient concentration and zooplankton predation.
(2) The nutrient uptake by phytoplankton follows a logistic law, and predation is described
by the Beddington-DeAngelis form (Beddington, 1975; DeAngelis et al., 1975),
represents the consumption rate, and

and

(

, where

)

denote the mutual interference between

zooplankton and the phytoplankton saturation constant, respectively.
(3) Following Ruan (1993), the constant nutrient input and washout rates to the lake
ecosystem are denoted by

and , respectively. Let

stand for the maximal nutrient uptake by

the phytoplankton.
(4) Both phytoplankton and zooplankton die naturally in the aquatic ecosystem at constant
rates

and , respectively.
(5) The dead biomass of phytoplankton/zooplankton is termed as detritus by ecologists

worldwide. Indeed, the dead biomass of phytoplankton and zooplankton is first converted into
detritus. Then, they are decomposed by micro-organisms and at the end of the process, nutrients
86

are regenerated. To avoid any model complexity, this research does not consider an explicit
dynamic of detritus in the nutrient-plankton model (Thakur et al., 2021).
(6) The nutrients are partially recycled from the dead biomass of plankton by bacterial
decomposition. For simplicity, the time required to regenerate nutrients is neglected, and nutrient
recycling is considered an instantaneous term. The nutrient recycling rates from the dead biomass
of phytoplankton and zooplankton are denoted by ℎ and , respectively.
(7) Aquatic ecosystems are inevitably affected by random perturbations coming from the
environment. In this research, environmental variables can act as noise sources because of their
random fluctuations, resulting in a stochastic dynamic to the plankton model (Denaro et al., 2013;
Valenti et al., 2016). Following previous studies (Luo and Mao, 2007; Zhao, 2016; Zou and Wang,
2014), white noise is considered to represent the random fluctuations in the natural world.
Based on the above assumptions, a model schematic diagram is shown in Figure 6.1. In the
figure, the dashed lines reflect the fact that the dead biomass of phytoplankton and zooplankton
does not directly convert into nutrients. Instead, they are first converted into an intermediate
substance, detritus, and then into nutrient concentration. The following stochastic nutrient–
plankton model is developed:
⎧
⎪
⎨
⎪
⎩
where

( )=
( )=
( )=

− ( ) −
( ) ( )−

( ) ( )+ℎ ( )+
( )−

( ) ( )
−
( )
( )

( )

( ) ( )
( )
( )

+

( )

( )
+

+
( )

( )
( ),

( ),
(6.1)

( ),

( ) are the independent standard Brownian motions and

are the intensities of the white

noise for = 1, 2, 3. In addition, model (6.1) shares the common biological meaning with other
phytoplankton growth models. The biological meanings of variables and parameters involved in
the model (6.1) are listed in Table 6.1.

87

Figure 6.1. Schematic diagram representing the dynamics of the considered nutrient–phytoplankton–
zooplankton model. In the figure, the black dashed lines represent the re-mineralization of the dead
biomasses of phytoplankton and zooplankton into the nutrient concentration.

The constant plankton mortality is widely accepted in most aquatic ecosystem models.
However, in natural aquatic ecosystems, the peak of phytoplankton biomass is generally followed
by a sudden decline within a few days, and then it moves to the next bloom cycle. More importantly,
several empirical evidences support the fact that plankton mortality varies in different phases of
algal blooms. For example, Choi et al. (2017) measured the loss of biomass of Alexandrium
fundyense in the bloom and decline phases. The results showed that the peak concentrations of
Alexandrium fundyense reached 1 × 10 cells L−1 in the bloom phase and then dramatically
decreased in the decline phase, eventually dropped to less than 1000 cells L−1. Furthermore, Boyd
et al. (1975) measured the average density of Anabaena variabilis filaments and the concentration
of dissolved oxygen in different phases of a bloom. The experimental evidence showed that the
average density of Anabaena variabilis filaments increased from 3600 cells ml−1 to 37300 cells
ml−1 in the bloom phase and then sharply decreased to less than 1000 cells ml−1 in the decline phase
of the bloom, while the concentration of dissolved oxygen dramatically decreased from 9 mg/L to
0.15 mg/L, resulting in the massive death of aerobic organisms.
Obviously, plankton mortality shows significant differences between the bloom and decline
88

phases of bloom events. Thus, shifting plankton mortality can be described as a random switching
between two or more environmental regimes (Du et al., 2004; Liu and Wang, 2010). Considering
the experimental evidence (Børsheim et al., 2005; Boyd et al., 1975), regime-switching plankton
mortality is described as a continuous time Markov chain with different states, and the mortality
of phytoplankton and zooplankton is denoted as ( (1), (2)) and ( (1), (2)), respectively.
Therefore, the presented model, which accounts for both colored and white noise, is shown as
follows:
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩

( )=[

( )

( ) − ( ) −

( )

( ) ( )+ℎ ( )

( )

+

( )

+
( )=

( )

( ) ( )−

( )−

( )

( )

( )

( ( )) ( ) ( )
−
( ) ( ) ( ( )) ( )

( )

( )

( )

( )

( )

+

( )

where ( ) represents a continuous time Markov chain with state space
∞. Throughout this chapter, the process
(Ω, , {

}

( )

( ( )) ( ) ( )
( ) ( ) ( ( )) ( )

( )

+
( )=

( )]

( )

(6.2)

= {1, 2, ⋯ , }, 1 ≤

<

( ) and ( ) are defined on a complete probability space

, ℙ), and ( ) is independent of

( ), = 1, 2, 3. From a biological point of view,

the initial value ( (0), (0), (0)) ∈ ℝ , and all parameters are assumed to be positive. The
model (6.2) shares the same biological meaning as the previous model (6.1).
ℝ is denoted as the positive cone in ℝ . For a vector
= min{ ( )},
∈

= ( (1),

(2), ⋯ ,

( )), denote

= max{ ( )}. Let ( ) be a right-continuous Markov chain with state space
∈

= {1, 2, ⋯ , }, 1 ≤

< ∞, and (

) × is the -matrix of ( ) satisfying the following for a

sufficiently small ∆ > 0:

ℙ{ ( + ∆ ) = | ( ) = } =
89

∆ + (∆ ),
1 + ∆ + (∆ ),

≠ ,
= ,

where

is the transition rate from state to state , , ∈ ,

≥ 0 if ≠ , and

.

= −∑

In addition, the Markov chain ( ) is assumed to be irreducible and independent of the Brownian
motion

( ) ( = 1, 2, 3). Hence, the Markov chain ( ) is ergodic and has a unique stationary

distribution

=(

,

,⋯,

), which is the solution of the following equation:
= 0, ∑ ∈

= 1 and

> 0, ∀ ∈ .

By now, a nutrient-plankton model has been developed to investigate the effects of regimeswitching plankton mortality on phytoplankton growth dynamics.
6.3. The main results
In view of Gray et al. (2011), for any given initial value ( (0),
the stochastic model (6.2) has a unique positive solution ( ( ),

(0),
( ),

(0), (0)) ∈ ℝ × ,
( ), ( )) for all ≥ 0.

Furthermore, according to Lemma 2.3 in Zhao (2016), the following relations hold almost surely:
lim sup[ ( ) + ( ) + ( )] < ∞ , lim

→

lim

→

→

∫

( )

( )

( ) = 0 , lim
→

∫

( )

( )

( ) = 0,

(6.3)

∫

( )

( )

( ) = 0.

(6.4)

6.3.1 Stochastically ultimately boundness
Regarding the stochastically ultimate boundedness of the solution of model (6.2), the
following theorem is presented.
Theorem 6.1. For any initial value ( (0), (0), (0)) ∈ ℝ , there exist χ > 0 and

∈ (0, 1)

such that
lim sup ℙ |(N(t), P(t), Z(t))| =

→

N (t) + P (t) + Z (t) > χ < ε.

Thus, the solution of model (6.2) is stochastic ultimately bounded.
The proof is given in Appendix M. Theorem 6.1 shows that the solutions of model (6.2) is
90

stochastically ultimately bounded.
6.3.2 Stochastic permanence
Denote ℎ = max

. For the stochastic permanence of the model (6.2), the

, ̌ − ℎ, −

following result is obtained.
Lemma 6.1. If

is a positive constant such that
{

0<

,

}<

,

− ℎ，

then the solution of model (6.2) has the following property:
lim sup | ( )|

≤ Π,

)

+
2

→

where

Π=

3 (4

+
4

=

max 1,

−ℎ−

2

+

+1
max{
2

= [ℎ + max{

,

,

,

+4

,

} −

.

> 0,

}] + 2 > 0.

The prove of Lemma 6.1 is given in Appendix N. Applying the Chebyshev's inequality, one
can derive the following result from Theorem 6.1 and Lemma 6.1.
Theorem 6.2. if
max{σ , σ , σ } < 2

−ℎ ,

then the solution of model (6.2) is stochastically permanent.
The proof is given in Appendix O. Theorem 6.2 indicates that model (6.2) is stochastic
permanent if the environment fluctuation is less than the threshold. The results from Theorem 6.2
imply that the stochastic permanence can be obtained by controlling noise intensity.

91

6.3.3 The persistence and extinction of plankton
In this subchapter, the extinction and persistence of plankton in the aquatic system is
investigated. For the sake of convenience, the following notations for all

∈

are introduced:

( ) = ( ) + [ ( ) − ℎ( ) − ( )] ( ),
( ) = ( ) + [ ( ) − ( ) − ( )] ( ),
ℜ =

( ) ( ) ( )−

( )+

1
2

( )
,
( )

∈

( )+

ℜ =−
∈

( )−

1
2

( )

,

where ( ) is the solution of Eq. (6.5). Furthermore, the following assumption is carried out.
Assumption 1.

> 0,

≠ , , ∈ .

To begin with, two lemmas are presented as follows.
Lemma 6.2. The linear system
( ) ( )−∑∈
admits a unique solution

≡ ( (1),

( ) − ( ) = 0,

(2), ⋯ ,

∈ ,

(6.5)

( )) .

The proof is given in Appendix P. Then, the following lemma is given by using two auxiliary
results in Appendix E.
Lemma 6.3. For model (6.2), if

→

( ) = 0 a.s., then

→

( ) = 0 a.s.

The proof is given in Appendix Q. From the above auxiliary results, one can derive the
following theorem regarding the persistence and extinction of plankton:
Theorem 6.3. If

∈ { (

) − ℎ( ) − ( )} ≥ 0 and

the solution ( ( ),

( ),

( ), ( )) of model (6.2) has the following properties:

(i) If ℜ < 0, then

92

∈ {

( ) − ( ) − ( )} ≥ 0, then

lim

→

( ) = 0 and lim
→

( ) = 0 a.s,

Both phytoplankton and zooplankton are extinct from the aquatic system.
(ii) If ℜ > 0 and ℜ < 0, then
ℜ

≤ lim inf ∫
→

( )

≤ lim sup ∫
→

( )

≤

ℜ

and lim
→

( ) = 0 a.s.,

Phytoplankton is persistent in mean and zooplankton is extinct from the aquatic system.
The proof is given in Appendix R. Theorem 6.3 indicates that the extinction and persistence
in mean of plankton depend on ℜ and ℜ . For ℜ < 0, both phytoplankton and zooplankton are
extinct for sufficiently large time. Meanwhile, when ℜ < 0 and ℜ > 0, the phytoplankton is
persistent in mean and zooplankton is extinct from the aquatic system.
6.3.4 Ergodicity of model (6.2)
In this subchapter, the sufficient conditions for the existence and uniqueness of the stationary
distribution of model (6.2) is derived. Following Yu et al. (2019a) and Zhu and Yin (2007), one
can get the following theorem.
Theorem 6.4. Under Assumption 1, if for given initial condition ( (0), (0), (0), (0)) ∈
ℝ ×

,

∈ { (

{( ) , ( ) , ( ) } and ℜ > 0 , then the

) − ℎ( ) − ( )} >

stochastic process ( ( ), ( ), ( ), ( )) defined by the solution of model (6.2) is ergodic and
has a unique stationary distribution in ℝ × .
The proof is given in Appendix S. Theorem 6.4 provides the sufficient conditions for the
existence of ergodic stationary distribution. It shows that if the intensities of the white noise are
within the threshold and ℜ > 0, the stochastic process ( ( ), ( ), ( ), ( )) is ergodic and
admits a unique stationary distribution. Next, the explicit lower bound for phytoplankton is studied.
The following assumption from Eq. (6.4) is carried out.
93

Assumption 2. For arbitrary 0 < < 1, there exist a set
ℱ = ℱ( ) such that for
lim
→

∈

and > ℱ

∫

( )

− lim ∫
→

⊂

with ℙ(Ω ) ≥ 1 − and a constant

( ) +

( ( ))

≤ .

Now, one can obtain the following theorem.
Theorem 6.5. Let Assumptions 1 and 2 hold. If
, ̂ − ℎ, −
then for model (6.2) whenever ℜ >

→

∫

( )

≥

/

{( ) , ( ) , ( ) },

>

there exists a constant ς such that

> 0 and

→

∫

( )

≥ℜ −

> 0 a.s.,

Thus, both phytoplankton and zooplankton are persistent in mean.
The proof is given in Appendix T. Biologically, Theorem 6.5 tells that if the environmental
noise is within the threshold and ℜ >

/ , then both phytoplankton and zooplankton can

survive forever.
6.4. Numerical simulation
In this chapter, some numerical results are presented to further investigate how regimeswitching plankton mortality affects the distribution of plankton biomass in the aquatic system.
Unless otherwise mentioned, the parameter values used for numerical results are the same as those
listed in Table 6.1. Let the transition rate from state 1 to state 2 be the same as the transition rate
from state 2 to state 1, that is,

= (−1/2, 1/2; 1/2, −1/2). The nutrient input

rate , phytoplankton mortality , and zooplankton mortality

, consumption

are the main control parameters.

6.4.1. Sensitivity analysis
In comparison with simply varying the parameters to observe the outcome of the model, the
techniques of sensitivity analysis are mathematically more sophisticated. Thus, a basic differential
94

Table 6.1. Biological explanations of variables and parameters in model (6.1), and numerical
values used for simulation results
Parameter Description

Unit

Nutrient concentration

/

Phytoplankton biomass

/

Zooplankton biomass

/

Maximal nutrient uptake rate by
/ /
phytoplankton
Nutrient recycle rate from the dead biomass
day-1
of phytoplankton
Nutrient recycle rate from the dead biomass
day-1
of zooplankton

ℎ

Value

Source

0.8

0.6(a)

0.08

0.08(b)

0.01

0.01(f)

Death rate for phytoplankton

day-1

[0.4,0.7]

[0.2,0.65](c)

Death rate for zooplankton

day-1

[0.4,0.7]

[0.08,0.6](d)

Consumption rate of phytoplankton by
zooplankton

day-1

0.2/0.5

0.5(e)

Phytoplankton saturation constant

/

0.5

Assumed

Mutual interference between the
zooplankton

/

0.2

Assumed

day-1

0.25

0.2/0.3(f)

/

[0,12]

[0,12.5](b)

Washout rate for nutrient
Input concentration of nutrient

(a) Jang and Baglama (2005); (b) Ruan (2001); (c) Garcés et al. (2005); (d) Turner et al. (2014);
(e) Jang and Allen (2015); (f) Yu et al. (2019a).
analysis approach is adopted to determine the semi-relative and logarithmic sensitivity solutions
of model (6.1) for the plankton populations in the absence of environmental noise (Bortz and
Nelson, 2004, Misra et al., 2016). The semi-relative sensitivity of the model solutions for a variable
to a parameter
with respect to

is given by

( , )

and is computed by formally differentiating model (6.1)

and interchanging the order of time and parameter derivative. Since model (6.1)

has 3 state variables, one can obtain a system of 3 equations for the sensitivity functions
( , )
95

( , )=

Figure 6.2. (a) Semi-relative sensitivity and (b) logarithmic sensitivity solutions of the model (6.2) in the
absence of environmental noise with respect to
in Table 6.1 except

, , , ℎ, , and . Parameters are at the same values as

=1.2, =0.5, =0.4 and =0.04.
( )

with initial conditions

(0)/

without environmental noise and

( )

=

= 0 , where
/

/

+

,

(6.6)

represents the Jacobian of model (6.1)

is the derivative of the right side of model (6.1)

(represented by ( , , )) with respect to . To obtain the sensitivity functions, first, model (6.6)
is solved for

( , ) by coupling it with the original model (6.1) without environmental noise

(total 3+3 equations). The values of the sensitivity functions provide the rates of change of the
variables with respect to the change in the chosen parameter ( ) as time flows. For example,
96

(80, 0.5) = −0.112 means that the derivative of
0.5 per day is −0.112

with respect to

at = 80 days and

=

day/L. Finally, the semi-relative sensitivity solutions are calculated by

multiplying the unmodified sensitivity functions with respect to the parameter , i.e.,
which provides the amount the state will change when the parameter

( , ),

is doubled (i.e., a

perturbation on the order of ).
The sensitivities of plankton densities are plotted for the seven most sensitive relevant
parameters (

,

,

, ℎ, ,

and ) in Figure 6.2. Among these parameters, only

can be

influenced by human activities. From the graph, it is clear that the perturbations of the parameters
exhibit their greatest influences early in the simulation, with a large initial expected variation in
the plankton densities. It is clear that the doubling of

and

can yield a sudden increase in

plankton densities around = 43 days. There is a huge increase in the density of phytoplankton at
43 days, followed by a sudden huge decrease just a few days later. This decrease in the density of
phytoplankton is due to the less availability of nutrients as they are taken up by the phytoplankton.
On the other hand, on doubling the uptake rate of phytoplankton by zooplankton ( ), the biomass
of phytoplankton decreases whereas that of zooplankton increases in the initial phase of time. But
as time flows, the biomass of zooplankton decreases and that of phytoplankton increases. Again,
there is an increment in zooplankton and a decrement in phytoplankton biomass as time flows.
Doubling the mortality rate of phytoplankton ( ), the biomasses of both the plankton significantly
changes in the aquatic system. The densities of phytoplankton increase while the zooplankton
densities decrease significantly in the initial stage on doubling the mortality rate of the latter. The
plankton densities in the aquatic system are least sensitive to the re-mineralization of dead
biomasses of phytoplankton and zooplankton. However, different parameters have different
strengths of influences,

has the greatest influence on phytoplankton as well as zooplankton

densities. It is worth noting that these outcomes greatly depend on the parameterization of the
model.
97

Figure 6.3. For model (6.2) without noise and regime switching, the mean values of the solutions with
respect to nutrient input.

Next, the logarithmic sensitivity solutions
log( )
( )=
( , )
log( )

( , )

with respect to all of the previously mentioned parameters are presented in Figure 6.2(b). These
quantities are dimensionless and indicate what percentage change in the variables can be expected
from a doubling of a parameter . Thus, to get a complete idea about the sensitivity of the solutions
to a particular parameter, it is best to calculate both semi-relative and logarithmic sensitivity
solutions. Figure 6.2(b) shows that doubling the parameters

, , ℎ, , and

results in a 34.62%,

34.62%, 1.42%, 1.42% and 98.01% increment in the density of phytoplankton, respectively at the
end of 80 days. On the other hand, the density of phytoplankton reduces by 112.7% and 19.22%
on doubling the parameters

and , respectively at 80 days. Similarly, the figure displays that at

the end of 80 days, the zooplankton density reduces by 71.46%, 128.1% and 48.06% on doubling
the parameters

, , and , respectively. Furthermore, it is apparent from the figure that the

zooplankton density boosts up by 235.7%, 190.4%, 7.275% and 7.275% at the end of 80 days on
doubling the values of model parameters

, , ℎ, and . In addition, the mean value of the solution

of the deterministic model is presented by choosing constant plankton mortality rate and fixing
98

=

=

= 0.5, with respect to nutrient input in Figure 6.3. Obviously, the mean of plankton

biomass increases with an increase in the nutrient input level. As the nutrient input exceeds a
threshold value, the plankton populations can coexist in the aquatic system.

6.4.2. Effects of regime shifting plankton mortality
Figure 6.4 shows three different types of the model (model (6.1), model (6.2) without noise,
and model (6.2)) for their sensitivity to changes in the nutrient input, since the concentration of
available nutrient is one of the decisive factors controlling phytoplankton productivity (Häder and
Gao, 2015). Instead of increasing and decreasing parameter values by a defined percentage, the
entire range of possible nutrient input values is determined by previous studies. The other
parameter values used for the model analysis are the same as those in Table 6.1. The sensitivity
analysis involves changing the nutrient input to observe its effects on the biomass of plankton
populations. The analysis shows that the persistence and extinction of plankton populations can be
divided into three stages (see Figure 6.4). When the nutrient input is low (below the white dashed
line), both phytoplankton and zooplankton populations eventually die out. With an increase in the
nutrient input, the phytoplankton population persists, while the zooplankton population tends to go
extinct (above the white dashed line and below the red dashed line). Once the nutrient input exceeds
the critical value (above the red dashed line), both phytoplankton and zooplankton populations can
coexist indefinitely. Moreover, it is worthy to note that nutrient input enhances the amplitude of
oscillation because higher plankton biomass is achieved with increased nutrient input. The results
from Figure 6.4 imply that the persistence and extinction of plankton populations are sensitive to
variations in nutrient inputs.

99

Figure 6.4. The solutions of model (6.1) in the ( , ) plane for (a)
without environmental noise in the ( , ) plane for (c)
( , ) plane for (e)

and (b) ; the solutions of model (6.2)

and (d) ; the solutions of model (6.2) in the

and (f) . The white and the red dashed line respectively represent the critical value

of ℜ = 0 and ℜ =

/ .

Additionally, Figure 6.4 presents another important result. When white noise is applied to the
deterministic model, Figures 6.4(a) and 6.4(b) show that the plankton density always oscillates
instead of maintaining a constant level. Furthermore, the shifting plankton mortality is described
by the right continuous Markov chain ( ) on state space
plankton is set at 0.4

= {1, 2}, and the low mortality of

while the high mortality of plankton is set at 0.7
100

. Figures 6.4(c) and

6.4(d) display the solutions of model (6.2) without noise, and the results indicate that the plankton
biomass oscillates within a smaller range but shows obvious switching between different states.
Clearly, the switching phenomenon is induced by shifting plankton mortality. In addition, Figures
6.4(e) and 6.4(f) show the effect of regime-switching plankton mortality on the distribution of
plankton biomass in a stochastic environment. It is evident from Figures 6.4(e) and 6.4(f) that the
oscillation range of plankton biomass becomes larger, and the switching between different states
also becomes more pronounced. Apparently, the regime-switching plankton mortality significantly
affects the planktonic ecosystem in a stochastic environment.

Figure 6.5. The paths of phytoplankton and zooplankton in the model (6.2) for (a)
0.2,

= 1 and (c)

= 0.5,

=

= 0.5, (b)

=

= 12.

In the study of stochastic population dynamics, persistence and extinction are important topics.
From Theorem 3.3, it is obvious that the extinction and persistence in mean of plankton depend on
ℜ and ℜ . For

=

= 0.5 , one can get ℜ = −0.195 < 0 , which means that both

phytoplankton and zooplankton become extinct according to Theorem 6.3(i) (see Figure 6.5(a)).
Next, by fixing

= 0.2,

= 1, one can find that ℜ = 0.2050 > 0, ℜ = −0.195 < 0, which

indicates that phytoplankton is persistent in mean while zooplankton becomes extinct according to
Theorem 6.3(ii) (see Figure 6.5(b)). The results from Figures 6.5(a) and 6.5(b) reveal that a higher
nutrient input and less consumption of phytoplankton by zooplankton contribute to the survival of
the phytoplankton population in the aquatic system. Similarly, by selecting
can get ℜ = 3.32202 >

/

= 0.5,

= 12, one

= 1.25 > 0. According to Theorems 6.4 and 6.5, the model (6.2)
101

is persistent in mean and exhibits a unique stationary distribution, i.e., the plankton species can
survive forever (see Figure 6.5(c)).

Figure 6.6. For model (6.2) with

= 0.5, (a) the Markov chain with respect to

= 8; (b) the sample path

of phytoplankton; (c) the distribution of phytoplankton population with respect to state 1, state 2 and hybrid
system, respectively; (d) the distribution of phytoplankton population at = 1000 with respect to nutrient
input

.

In order to investigate the distributions of phytoplankton biomass in different states, Figure
6.6(a) illustrates the path of the Markov chain by choosing

= 8,

= 0.5 in model (6.2).

Additionally, Figure 6.6(b) displays the sample paths for the phytoplankton population in state 1,
state 2, and the hybrid model, respectively. It is apparent that the phytoplankton population
becomes extinct in state 1 but persists in state 2 as well as in the hybrid model. Moreover, the
distributions of phytoplankton biomass in state 1, state 2, and the hybrid model demonstrate that
the Markov chain has the potential to enhance the oscillation cycles of the plankton (see Figure
102

6.6(c)). Figure 6.6(d) displays the distribution of phytoplankton biomass with respect to the
nutrient input

, which indicates that an increasing nutrient input leads to a significant increase in

the variance of phytoplankton biomass.

Figure 6.7. (a) The mean and the variance for phytoplankton biomass with respect to nutrient input
The mean and the variance for zooplankton biomass with respect to nutrient input

; (b)

.

Figure 6.7 shows the mean and variance of plankton biomass under three conditions for

=

1, 2, … , 10. The results imply that both the variance and the mean of plankton biomass show an
overall increasing trend as nutrient inputs increase. When the plankton community suffers from
nutrient deficiency, the mean and variance of phytoplankton biomass are lower for shifting
plankton mortality than for constant plankton mortality. However, the variance of phytoplankton
biomass in model (6.2) shows a comparatively large increasing trend. In contrast, the zooplankton
tends to become extinct when the nutrient input is low. Moreover, with an increase in the nutrient
input, the variances of plankton biomass with shifting plankton mortality show a faster increasing
trend. Obviously, regime-switching plankton mortality significantly increases the ranges of
distribution and oscillation intensity of phytoplankton biomass, suggesting that such plankton
mortality is beneficial for the survival of phytoplankton under high nutrient levels.

103

Figure 6.8. For model (6.2), the sample paths of (a) phytoplankton and (b) zooplankton for

= 1, 4, 7 and

10, respectively. The black line represents the solutions for determined model.

Numerous research studies support the fact that the growth rate of phytoplankton may not be
able to maintain a high level of plankton in the aquatic system due to zooplankton grazing and
intraspecific competition among the plankton for available resources such as light and nutrients
(Sommer, 1991; Sommer, 2012). The results from Figure 6.7 imply that when the nutrient input
exceeds a critical level, zooplankton appear in the aquatic system, and the mean and variance of
phytoplankton biomass show relatively slower increasing trends, while the mean of zooplankton
biomass has a larger increasing trend. A possible explanation behind this behavior of plankton
might be the predation by zooplankton and the intraspecific competition of phytoplankton for
resources. Finally, the sample paths of model (6.2) for plankton are shown in Figure 6.8, revealing
that an increasing input of nutrients leads to a marked increase in the amplitude of planktonic
oscillations.
6.5. Discussion
The sudden changes in plankton mortality have been observed in several experimental studies.
Thus, considering the impact of shifting plankton mortality on the population growth of planktonic
algae is crucial. In this study, a nutrient-plankton model considering shifting plankton mortality is
developed. The theoretical results imply that when the minimum plankton mortality is beyond a
104

threshold value, the extinction and persistence in the mean of plankton depend on ℜ and ℜ . It is
observed that model (6.2) possesses an ergodic stationary distribution, indicating that the plankton
populations can survive forever. Moving beyond the theoretical analysis, several numerical
simulations have been carried out to explore plankton dynamics. The sensitivity results indicate
that the biomasses of plankton in the aquatic system are highly sensitive to the input rates of
nutrients, the grazing rate of phytoplankton by zooplankton, and the mortality rates of planktons.
These results emphasize the need to focus on controlling the input rate of nutrients coming into the
lakes from outside during the bloom phase.
Furthermore, three different types of the models (model (6.1), model (6.2) without noise, and
model (6.2)) for their sensitivity to changes in the nutrient input is shown in Figure 6.4. It has been
well documented that nutrient availability is directly related to phytoplankton growth (Häder and
Gao, 2015; Elliott et al., 2006), and the distribution of phytoplankton biomass is sensitive to
nutrient input (Elliott et al., 2006). Figure 6.4 clearly shows that when the nutrient input is less
than a certain level, both phytoplankton and zooplankton populations tend to go extinct. If the
nutrient input is between the white and red dashed lines, the phytoplankton population is persistent,
and the zooplankton population disappears from the aquatic system. As the nutrient input exceeds
the red dashed line, both phytoplankton and zooplankton populations can coexist forever. Figure
6.4 suggests that nutrient input plays an important role in determining the survival of plankton; the
persistence and extinction of plankton populations are sensitive to variations in nutrient input.
These findings are consistent with the results reported by Elliott et al. (2006).
Additionally, Figure 6.4 shows a comparison of the plankton biomass in three different
models. The differences in the dynamics of the models are governed by nutrient input. The results
from Sherratt and Smith (2008) support the idea that population density always exhibits oscillatory
behavior due to the existence of noise and other factors. As shown in Figures 6.4(a) and 6.4(b), the
105

plankton population density always oscillates instead of maintaining a certain level owing to the
presence of environmental noise, which is consistent with the findings of Sherratt and Smith (2008).
By contrast, for the model (6.2) without noise (see Figures 6.4(c) and 6.4(d)), the oscillation
intensity of plankton biomass remains lower than that in Figures 6.4(a) and 6.4(b) due to the lack
of environmental noise. However, the distribution of plankton biomass switches between different
states, which is not observed in Figures 6.4(a) and 6.4(b). One possible explanation is that the
increasing plankton mortality leads to a sudden collapse of plankton populations. Figures 6.4(e)
and 6.4(f) show that the switching becomes more pronounced due to the coupling between noise
and shifting plankton mortality, and the oscillation intensity of plankton biomass is significantly
stronger than in Figures 6.4(c) and 6.4(d). These results further indicate that noise can enhance the
oscillation of plankton biomass, and the high plankton mortality in the decline phase results in a
dramatic collapse in the phytoplankton population, which agrees with the experimental results
mentioned in Børsheim et al. (2005), Boyd et al. (1975), and Choi et al. (2017).
In the study of stochastic population dynamics, the persistence and extinction of interacting
populations are the most important topics (de la Hoz and Vadillo, 2012; Liu and Wang, 2010).
From the biological points of view, the stochastic persistence means that the population will
survive forever. Figure 6.5 displays the persistence and extinction of the plankton populations in
model (6.2). Apparently, the plankton populations can survive forever under certain conditions
(see Figure 6.5(c)). Besides the stochastic persistence of the model, the numerical analysis has
shown how shifting plankton mortality affects the survival of the phytoplankton population in the
aquatic habitat (see Figure 6.6). Figure 6.6(b) depicts that the phytoplankton population goes
extinct in state 1 but is persistent in state 2 when the nutrient input is low; the phytoplankton
population in the hybrid model is persistent. It is inferred from Figure 6.6(b) that under the
influence of the Markov chain, although the phytoplankton population tends to go extinct in one
106

state, it still has a chance to persist. Under high nutrient input, regime-switching plankton mortality
can enhance the oscillation intensity of phytoplankton density, which means the model with
regime-switching plankton mortality is more beneficial for the survival of plankton populations
compared to the case when plankton mortality is constant (see Figure 6.6(c)). When compared with
regime-switching plankton mortality, nutrients have been proven to be an important factor
influencing the growth of phytoplankton in the aquatic system. The distribution of phytoplankton
biomass with respect to nutrient input suggests that increasing nutrient levels enhance the biomass
of phytoplankton in the model (see Figure 6.6(d)). The capability of nutrient loading to stimulate
the growth of phytoplankton is well demonstrated in Ramin et al. (2012).
Indeed, extensive research has established that total phytoplankton biomass may ultimately
respond to changes in nutrient availability (Carpenter et al., 2016; Salmaso, 2010), and that external
nutrient loading is a significant factor positively related to the total phytoplankton biomass (Ramin
et al., 2012). The simulation results reveal that nutrient input shows a positive control on the mean
as well as variance of phytoplankton biomass (see Figure 6.7). Additionally, nutrient input can
significantly increase zooplankton biomass as the increased phytoplankton biomass due to nutrient
input provides sufficient food for grazing zooplankton. However, the growth rate of phytoplankton
shows a decreasing trend after the nutrient input exceeds a threshold value. It is well documented
that phytoplankton abundance moderately increases rather than being maintained at a high level,
following nutrient enrichment in the aquatic system (Ramin et al., 2012). Moreover, Figure 6.7
shows that the plankton biomass in model (6.2) has the highest mean and variance values in
contrast to model (6.1) and the model (6.2) without noise. This suggests that shifting plankton
mortality is more beneficial for the survival of plankton populations in the aquatic system
compared to when plankton experiences constant mortality. This might be due to the sudden
collapse of the algal population, which creates a suitable environment for plankton to grow and
107

provides chances for their survival.
6.6. Conclusions
A stochastic nutrient-plankton model incorporating regime-switching plankton mortality was
implemented in this chapter. Several theoretical and numerical analyses were presented aiming to
understand how shifting plankton mortality affects phytoplankton growth dynamics. Analytically,
the threshold value for the survival of plankton species was derived, and the theoretical analysis
provided conditions under which the model was persistent or one of the plankton populations went
extinct. Furthermore, the results showed that model (6.2) admitted an ergodic stationary
distribution. In addition, the numerical results revealed that the coupling between noise and
regime-switching plankton mortality was not only capable of enhancing the oscillatory behavior
of plankton populations, but also decreasing the amplitude of the plankton oscillations in the bloom
phase. The simulation results indicated that nutrient loading in the aquatic system was an important
factor in controlling phytoplankton biomass, while regime-switching plankton mortality provided
a higher survival chance for plankton populations. It is worth noting that the importance of the
proposed model lies not in the precision with which it predicts specific events within a particular
lake/ocean, but in its contribution to the studies on how regime-switching plankton mortality
influences phytoplankton growth in lakes and oceans. Additionally, the results obtained in this
study may provide a reasonable explanation for the observed sharp collapse of a phytoplankton
bloom in experiments. Furthermore, it is worth noting that considering the explicit role of detritus,
the dead biomasses of phytoplankton and zooplankton, in model (6.2) will mimic a more realistic
situation in the nutrient-plankton model.

108

Chapter 7
DYNAMICS OF A STOCHASTIC NUTRIENT-PLANKTON MODEL WITH IMPULSIVE
CONTROL STRATEGY5
Abstract
In this paper, an impulsive nutrient-plankton model in a stochastic environment is investigated.
The model dynamics have been studied by using theoretical and numerical approaches. The results
show the existence of a global positive solution, indicating that the population size will remain
non-negative for a sufficiently large time. By employing a Lyapunov function, sufficient
conditions for the existence of a positive -periodic solution are derived. The numerical results
indicate that the coupling between environmental stochasticity and impulsive control plays an
important role in regulating the interplay between nutrients concentration and plankton densities
in aquatic ecosystems. Overall, the results obtained in this study may provide insight into how
plankton dynamics respond to impulsive perturbations under environmental fluctuations.
Keywords: Nutrient-plankton model, Stochastic periodic solutions, Impulsive control,
Environmental fluctuation

_____________________________
5Guo, Q., Liu, H., Wang, Y., Li, J., Zhao, M., Tiwari, P.K., Jin, Z., Dai, C.J., 2023. Dynamics of a stochastic

nutrient-plankton model with impulsive control strategy. European Physical Journal Plus. 138, 470. DOI:
10.1140/epjp/s13360-023-04111-0.

109

7.1. Introduction
Investigating the dynamics of phytoplankton in aquatic systems has become an essential topic
of research among scientists around the globe. Although phytoplankton is a vital component of the
aquatic ecosystem (Song et al., 2020a), the excessive growth of phytoplankton can cause decay
and death of many aquatic plants. The decomposition of dead algae may lead to a consumption of
dissolved oxygen by the bacterial pool, resulting in massive mortality events in fish populations
(Boyd et al., 1975; Leng, 2009; Mishra et al., 2022). Moreover, certain species of phytoplankton
produce toxins, thereby killing the marine life and also affecting the health of people through the
food web (Hallegraeff, 2010). Significant research attention has been devoted to investigating the
controlling factors responsible for the rise or decline of phytoplankton populations in aquatic
systems (Dai et al., 2016; Huisman et al., 1999; Tiwari et al., 2019). It has been well documented
that high nutrient load is one of the major triggers for algal blooms (e.g., cyanobacteria) (Downing
et al., 2001; McCarthy et al., 2009). Experimental studies have demonstrated that both nutrients
and zooplankton can significantly affect the phytoplankton community (Vanni, 1987). However,
the actual mechanism by which phytoplankton form blooms is far more complex than expected.
Thus, understanding the factors that contribute to algal blooms has attracted considerable scientific
attention in recent years (Anderson et al., 2012; Li et al., 2020).
Several studies have supported the fact that algal blooms are consequences of the interplay of
various hydrodynamical, chemical processes, and biological processes in aquatic systems (Chen
and Mynett, 2006). However, the mechanisms regulating plankton dynamics and diversity are still
not well understood by researchers (Dai et al., 2019; Guo et al., 2020). Moreover, the nonlinearity
and complexity of aquatic ecosystems make it challenging to gain insight into plankton dynamics
solely depended on experiments or field observations. Thus, mathematical models of ecological
population dynamics have been developed in the study of plankton dynamics as it not only captures
110

the ubiquitous stoichiometric constraints for the growth and interactions of species (Alijani et al.,
2015), but also provide quantitative insights into population growth dynamics (Dai et al., 2016).
Following the pioneering works of Volterra (1926) and Lotka (1925), numerous plankton models
have been developed by introducing various types of functional responses into classical predatorprey models, with the aim of understanding the triggering mechanisms of phytoplankton blooms
in aquatic ecosystems (Chakraborty et al., 2015; Liu et al., 2023; Ruan and He, 1998; Yu et al.,
2019a).
Although a number of studies have focused on the dynamic interactions of prey and predator
populations in constant environments (Das et al., 2022; Paul et al., 2021; Nie et al., 2011), real
ecological communities are inherently random, and biological phenomena in real ecosystems are
inevitably affected by environmental fluctuations (May, 2019; Guo et al., 2023), such as white
noise. White noise is characterized by many small, independent random fluctuations, such as
rainfall, wind, or day-to-night variations. Despite the existence of environmental fluctuations in
aquatic ecosystems, algal blooms are highly correlated with complex interactions of climatic
factors and hydrographic processes (McGowan et al., 2017), which can cause stochastic scaling
and timing of algal blooms (Chen et al., 2014). Moreover, stochastic population models have
attracted a great deal of attention as they can capture evolutionary trend of populations in a
randomly varying environment (Dobramysl et al., 2018; Meerson and Sasorov, 2008; Zhou et al.,
2022). In Mao et al., (2002) and Deng et al., (2008), the authors concluded that stochastic noise
has the potential to suppress or boost the exponential growth of populations in ecological systems.
In addition to environmental fluctuations, natural aquatic ecosystems are inevitably subject to
pulse perturbations (Stelzer et al., 2022), which result in an abrupt change in nutrient
concentrations and the abundances of phytoplankton populations. For instance, heavy rainfalls
trigger nutrient input through river discharge, resulting in algal blooms in coastal areas (Han et al.,
111

2023). In Tang et al. (2020), the authors reported that sediment resuspension caused by windinduced wave shear stress and the stochastic nature of turbulence may also contribute to the release
of internal nutrients, thereby maintaining a significant potential for obstinate eutrophication and
algal blooms. The findings of Cui et al. (2016) demonstrated that wind can mix floating
phytoplankton cells away from the water surface when the wind speed exceeds a critical value.
These phenomena can instantaneously change the status of the system and are commonly described
as pulsed instantaneous behavior, which cannot be considered continuous (Li et al., 2019).
Impulsive differential equations can accurately capture the instantaneous state of a system, and
have been widely used to describe such instantaneous perturbations in the plankton system. The
effects of impulsive perturbations have gained considerable interest among mathematicians as well
as ecologists in recent years (Liu and Rohlf, 1998; Zhao et al., 2017b). Authors such as Li et al.
(2019), Guo et al. (2015), and Yang and Zhao (2012) have concluded that impulsive perturbations
may alter the dynamical behavior of an ecosystem; however, some proper impulsive perturbations
can maintain the rhythm of the ecosystem.
The synergistic effects of stochasticity and impulsive perturbations on the interacting
dynamics of predator-prey systems in ecological communities have received considerable attention
among research scientists (Liu et al., 2018; Liu and Liu, 2019; Yu et al., 2019b). A stochastic nonautonomous Lotka–Volterra predator–prey model with impulsive effects was analyzed in Zhang
et al. (2017a). The results indicate that both stochastic noises and impulsive perturbations play
crucial roles in the persistence as well as the extinction of species. Zuo and Jiang (2016)
investigated the dynamics of a stochastic non-autonomous Holling-Tanner predator-prey model
involving impulsive effects and reported that the model can experience the extinction of predator
populations due to the presence of environmental fluctuations (impulses). Despite numerous
advancements in understanding plankton dynamics, the responses of plankton populations to
112

impulsive perturbation under environmental fluctuation have not been established yet.
In view of the importance of impulsive control and environmental fluctuation, this chapter
aims to develop a stochastic model for the nutrient-plankton system with impulsive control in order
to investigate and understand the effects of impulsive perturbation on plankton dynamics in a
random environment. The remainder of this article is organized as follows: a stochastic nutrientplankton model with impulsive control is constructed in the next chapter and all the main results
are presented in Chapter 7.3. In Chapter 7.4, numerical simulations are conducted to test the
sensitivity of the model parameters and comprehend the effects of environmental noise and
impulsive control on plankton dynamics. Finally, a comprehensive discussion and conclusion on
the dynamics of plankton populations is provided, and their responses to the environmental
fluctuations and impulsive effects are shown at the end of the paper.
7.2. The mathematical model
Mathematical models have been widely implemented in the study of plankton dynamics over
the decades as they allow for qualitative capture of the field observed instantaneous dynamics of
ecosystems. Here, a nutrient-plankton model with the effects of impulsive control and stochastic
fluctuation on the phytoplankton growth dynamics is developed. Previously, mathematical models
of food web types have been widely used to describe the dynamics of nutrient-phytoplanktonzooplankton interactions in aquatic habitats (Mukhopadhyay and Bhattacharyya, 2006; Ruan, 2001;
Thakur et al., 2021). For the model formulation, I consider an aquatic system and assume that the
growth of the phytoplankton population fully depends on the availability of nutrients, with other
environmental factors such as light, temperature, etc., being sufficiently abundant for their growth.
Indeed, the proposed model is a modification of the plankton model proposed in Ruan (2001),
where the authors considered nutrient, phytoplankton, and zooplankton as dynamic variables. It
113

should be noted that the nutrient concentration in the aquatic ecosystem includes nutrients washout
and its uptake by phytoplankton. In addition, the phytoplankton biomass is determined by its
growth through nutrient conversion, and loss due to predation by zooplankton species and natural
death (Holling, 1959b). The zooplankton biomass is determined by its growth through
phytoplankton consumption, and loss due to natural death and intraspecific competition due to the
limitation of resources in the aquatic system.
At any instant of time

> 0,

( ), ( ), and ( ) denote the concentration of nutrients,

phytoplankton population, and zooplankton population, respectively. The model is formulated
based on the following ecological assumptions.
(1) There is washout of nutrients in the aquatic system at the rate

. The uptake of nutrients

by phytoplankton follows the Holling type II functional response described by
represents the half-saturation constant and

, where

denotes the rate of consumption. I denote by

a

constant that represents the conversion of nutrients into phytoplankton biomass.
(2) The phytoplankton population faces natural mortality at a constant rate . The predation
function of phytoplankton by zooplankton follows the Holling type II form Holling (1959b), which
is by

. Here,
(3) I denote

represents the grazing rate of phytoplankton by zooplankton.
as the biomass conversion rate of phytoplankton into zooplankton. The

zooplankton population die naturally at a constant rate , and also face intraspecific competition
for the available food sources in the aquatic system. Let

be the strength of such competition

among the zooplankton species.
(4) Considering the ubiquitous presence of stochastic fluctuations in the natural world (May,
1973), I employ white noise to represent random fluctuations in the aquatic ecosystem.

114

Figure 7.1. Schematic diagram depicting the interplay among nutrients, phytoplankton, and zooplankton in
an aquatic ecosystem.

In view of the above assumptions, a schematic diagram is presented in Figure 7.1, and I come
up with the following nutrient-plankton model with stochastic fluctuations (Luo and Mao, 2007;
Zou and Wang, 2014):
( )= − ( ) ( )−

⎧
⎪
⎨
⎪
⎩

( )=
( )=

In model (7.1),

( ) ( ) ( )
( )

( ) ( )

+

( )

( ) ( ) ( )

( ) ( ) ( )

( )
( ) ( ) ( )

( )

− ( ) ( )−
( )

( )

( )

( )

− ( ) ( )− ( ) ( )

( ),

+

( ) ( )

( ),

+

( ) ( )

( ).

( ) represents the independent standard Brownian motions and

(7.1)

stands for the

intensities of the white noise for = 1, 2, 3. Model (7.1) shares the common biological meaning
with the usual phytoplankton growth models. All the parameters involved in model (7.1) are
positive, bounded and continuous

−periodic functions. The biological meanings of variables and

parameters describing model (7.1) are provided in Table 7.1.
Moreover, natural aquatic ecosystems are inevitably affected by pulse perturbations (Stelzer
et al., 2022). For instance, wind-induced wave motion can trigger internal nutrient release and
vertical migration of phytoplankton populations (Tang et al., 2020; Cui et al., 2016), leading to an
abrupt change in nutrient concentration and phytoplankton biomass in the aquatic ecosystem
experiences. To encapsulate the impact of pulse perturbations, the stochastic model (7.1) is
extended to the following form:
115

( )= − ( ) ( )−

⎧
⎪
⎪
⎪

( )=

( ) ( ) ( )
( ) ( )
( ) ( ) ( )

( ) ( ) ( )
( )

− ( ) ( )−

( ) ( )

+

( )

( ) ( ) ( )
( )

( )

( )=
− ( ) ( )− ( ) ( )
( ) ( )
⎨
) ( ),
⎪ ( ) = (1 +
⎪ ( ) = (1 +
) ( ),
⎪
⎩ ( ) = ( ),

( ),

+

( ) ( )

( ),

+

( ) ( )

( ).

⎫
⎪
⎬
⎪
⎭

≠

,

∈ ℕ,
(7.2)

=

,

∈ ℕ.

Note that model (7.2) shares a common biological meaning with model (7.1), and comprises a
sequence of real numbers, 0 <

<

assumed that 1 +

= 1, 2 and

> 0 for

<⋯<

< ⋯, lim
→

= +∞. For biological reasons, it is

= 1, 2, ⋯ . For

> 0 ( = 1, 2) , the impulsive

effects represent the process of species introductions. However,

< 0 ( = 1, 2) denote the

harvesting of species. In this research, only the former case is considered. Additionally, throughout
this research, it is always assumed that there exists a positive integer
(

) =

, = 1, 2,

∈

, and [0, ) ∩ { ,

∈

} = { , ,⋯,

such that

=

+ ,

}.

7.3. The main results
For the proposed model, some analytical results are presented in this chapter. Throughout this
chapter, (Ω, ℱ, {ℱ }

, ) is considered as a complete probability space with a filtration ℱ (

)

satisfying the usual normal conditions (i.e., it is right continuous and ℱ contains all -null sets).
To deduce the following results, some notations and auxiliary results are presented in Appendix U.

7.3.1 Existence and uniqueness of the global positive solution
In this subchapter, the existence and uniqueness of the global positive solution is investigated.
At first, the following model without impulses is presented:

116

⎧
⎪
⎪
⎨
⎪
⎪
⎩

( )=
( )=

( ) − ( )+ ∑
( )

− ()

+ ( ) ( )

( ),

( )

( )

( )

( )

( )

( )

+ ∑

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

− ( )− ( )

log 1 +

+ ( ) ( )
( )=

( )

log 1 +

( )

( )
( )

( )

(7.3)

( ),

− ( )− ( ) ( )

( ) ( )

+

( ),

with

where both

( ) and

( )= ∏

1+

∏

(1 +

),

( )= ∏

1+

∏

(1 +

).

( ) are periodic functions with period

(Zhang et al., 2017b). It is

assumed that the product is equal to unity if the number of factors is zero. Next, the following
lemma is presented.
Lemma 7.1. Let

( )=

( ) ( ), ( ) =

( ) ( ), and ( ) = 3( ).

(i) If ( ( ), ( ), ( )) is a solution of model (7.2), then ( ( ),

( ),

( )) is a solution of

model (7.3).
(ii) If ( ( ),

( ),

( )) is a solution of model (7.3), then ( ( ), ( ), ( )) is a solution of

model (7.2).
The proof of above lemma follows Zhang and Tan (2015), and hence is omitted.
Next, the following theorem is presented and the proof is given in Appendix V.
Theorem 7.1. For any initial values

(0), (0), (0) ∈

positive solution ( ( ), ( ), ( )) that is remain in

, model (7.2) exhibits a unique

with probability one.

The results from Theorem 7.1 indicate that the population size is remain nonnegative for
sufficiently large time .

7.3.2 Existence of positive -periodic solution
117

In this subchapter, the existence and uniqueness of positive periodic solutions of model (7.2)
is investigated.
Lemma 7.2. (Yu et al., 2019a). For any initial values
( ( ),

( ),
→

(0),

(0),

(0) ∈

, all the solutions

( )) of model (7.3) satisfy:
( ) < ∞,

. .,

Thus, there exist positive constants
( )≤

,

( ) < ∞,

→

,

, and

( )≤

,

. .,

( ) < ∞, . .

→

such that
( )≤

≥0 . .

Define
≜

1

≜

1

log 1 +

−〈 ( )+

( )〉 −

,

log 1 +

−〈 ( )+

( )〉 −

.

Now, one can have the following theorem.
Theorem 7.2. If

> 0 and

>0, then model (7.2) possesses a positive -periodic solution.

The proof of the above theorem is given in Appendix W. Biologically, Theorem 7.2 tells that
if the environmental noise is small,

> 0, and

>0, then model (7.2) exhibits a positive -

periodic solution, indicating that the phytoplankton blooms exhibit more or less periodicity.
7.4. Numerical simulation
Although some theoretical analyses of model (7.2) were conducted in the previous chapter,
obtaining detailed information on the combined actions of stochastic and impulse effects on
plankton dynamics is challenging due to the model’s complexity. Thus, this chapter presents some
numerical analyses to further explore the combined actions of stochastic and impulsive effects on
the biomass distributions of populations in the aquatic ecosystem. Unless otherwise mentioned in
118

Table 7.1. Biological explanations of parameters in model (7.1), and their numerical values used
for simulation results
Parameter Description

Unit

Value

Source

day-1

0.5

[0.2,1](a)

day-1

0.2

0.2(b)

day-1

0.5

0.6(b)

Natural death rate of phytoplankton

day-1

0.5

0.58(f)

Natural death rate of zooplankton

day-1

0.03

0.03(c)

Consumption rate of phytoplankton by
zooplankton

day-1

0.01

0.021(e)

/

0.2

Assumed

0.3

Assumed

0.01

0.01(d)

Nutrient concentration

/

Phytoplankton biomass

/

Zooplankton biomass

/

Maximal nutrient uptake rate by phytoplankton
Conversion rate of nutrient into the biomass of
phytoplankton
Conversion rate of phytoplankton biomass into
the biomass of zooplankton

Phytoplankton saturation constant
Strength of competition among the zooplankton
population

/

Washout rate for nutrient

day-1

/day

(a) Alijani et al. (2015); (b) Kartal et al. (2016); (c) Gao et al. (2008); (d) Tiwari et al. (2019);
(e) Das et al. (2018); (f) Gourley and Ruan (2003);
the text, the parameter values used for the numerical results are the same as in Table 7.1. The
impulsive perturbation and noise intensity are chosen as control parameters. It is well documented
that biological and environmental parameters are subject to fluctuation over time (Fan and Wang,
2000), and can induce more or less periodicity in population density, such as the seasonal blooms
of Microcystis spp. in Lake Taihu, China (Otten and Paerl, 2011). Thus, we assume that all the
parameters in model (7.2) are periodic and have a common period of 40 days. The parameter set
in the periodic environment is chosen as follows:
= 0.01 + 0.01sin (

/20),

= 0.5 + 0.01sin (

/20),

= 0.5 + 0.01sin (
= 0.5 + 0.01sin (
119

/20),
/20),

= 0.2 + 0.01sin (

/20),

= 0.03 + 0.01sin (

/20),

= 0.01 + 0.01sin (

/20),

= 0.2 + 0.01sin (

/20),

= 0.3 + 0.01sin (

/20).

7.4.1 Effects of impulsive control

Figure 7.2. The evolution of a single path of solutions for model (7.2) and the corresponding impulsive
model without random noise. The figures depict the paths of (a) nutrient, (b) phytoplankton, (c) zooplankton,
and the (d) phase diagram.

According to Theorem 7.2, model (7.2) exhibits a positive -periodic solution under certain
conditions. The noise intensity is chosen as
pulse intensity is set as

= 0.5,

= 0.02 + 0.01 sin(

/20), and the

= 0.1. The impulse frequency is selected as

= 5. Then, the

=

=

evolution of a single path of solutions for model (7.2) is presented in Figure 7.2. It is apparent from
the figure that the stochastic periodic solutions of model (7.2) fluctuate around the periodic
solutions of its corresponding deterministic model, but the overall trend is periodicity. The results
from Figure 7.2 indicate that the plankton density will exhibit periodicity, which is consistent with
the periodic bloom events observed in real aquatic ecosystems (Otten and Paerl, 2011).
120

Figure 7.3. Under

=

=

= 0.02 + 0.01 sin(

/20), the evolution of a single path of (a) nutrient,

(b) phytoplankton and (c) zooplankton in model (7.2) with respect to time for three different sets of pulse
intensity.

Figure 7.4. The evolutions of a single path of phytoplankton (first row) and zooplankton (second row) in
model (7.2) with respect to time for
0.01 sin(

/20), and

= 0.1, 0.4, 0.7, and 1. In this figure,

= 0.1 (first column),

= 0.3 (second column) and

=

=

= 0.02 +

= 0.5 (third column).

Next, the numerical analysis presents the effect of impulsive control on the distributions of
plankton populations. To do this, by fixing the noise intensity and the value of , Figure 7.3 shows
the evolution of a single path of nutrient concentration and plankton populations with respect to
three different sets of pulse intensity. The figure shows that for a low pulse intensity, i.e.,

=

= 0.01, the nutrient concentration becomes zero and the plankton populations tend to go extinct.
It is evident from the figure that when the pulse intensity increases to

= 0.5 and

= 0.1, the

nutrient concentration and the plankton populations remain in the aquatic system. However, when
121

increasing the pulse intensity to

= 1.5 and

= 1, the nutrient concentration and the plankton

populations show stronger oscillatory behaviors. The results from Figure 7.3 suggest that the pulse
intensity has the potential to alter plankton dynamics. For instance, heavy rainfall can lead to high
nutrient input into the water bodies due to river discharge or massive erosion, subsequently,
accelerating the growth of algae.

Figure 7.5. Under

=

=

= 0.02 + 0.01 sin(

/20),

= 0.5 and

= 0.1, the evolution of a

single path of (a) nutrient, (b) phytoplankton and (c) zooplankton in model (7.2) with respect to time for
three different values of .

Furthermore, the impact of pulse intensity on the plankton population is illustrated in Figure
7.4. Figures 7.4(a) & 7.4(b) show that if the nutrient pulse is equal to 0.1, both phytoplankton and
zooplankton populations disappear from the ecosystem, even if the pulse intensity of
phytoplankton is high. One possible reason behind this behavior of the ecosystem is that a small
nutrient pulse could not provide sufficient nutrients for the survival of the phytoplankton
population. Consequently, the aquatic system faces a lack of food to meet the requirements of the
zooplankton population, resulting in the extinction of both phytoplankton and zooplankton
populations. Additionally, it is clear that when the nutrient pulse intensity reaches 0.3, the
phytoplankton population becomes persistent, whereas the zooplankton population is doomed to
go extinct despite the high pulse intensity of the phytoplankton population (see Figures 7.4(c) &
7.4(d)). This may happen in the aquatic system as the nutrient concentration and phytoplankton
biomass are not enough to support the life of the zooplankton population. Furthermore, Figures
122

7.4(e) and 7.4(f) show that if the nutrient pulse intensity increases to 0.5, then both the
phytoplankton and zooplankton populations persist in the ecosystem, irrespective of the small
pulse intensity of the phytoplankton population. Thus, the results from Figure 7.4 indicate that the
survival, as well as the coexistence of plankton populations, are strongly correlated with the
nutrient pulse. Actually, in aquatic ecosystems, multiple external disturbances have been reported
to have the potential to increase phytoplankton abundance in the water bodies. For example, windinduced mixing may affect the migration of phytoplankton species, leading to an increment in the
abundance of phytoplankton at a particular location.
Apart from pulse intensity, pulse frequency also plays a significant role in the survival of
plankton populations (Liu et al., 2023). Figure 7.5 depicts the paths of nutrient concentration and
plankton populations for different values of . Obviously, when the pulse frequency is 1, i.e.,

=

1, the nutrient concentration does not vanish while the plankton populations go extinct. However,
as the pulse frequency increases to 5, the nutrient concentration and plankton populations persist
in the ecosystem. However, on further increasing the pulse frequency, the nutrient concentration
and plankton populations show stronger oscillatory behaviors. Biologically, a small nutrient pulse
frequency may not be able to provide sufficient nutrients for the survival of the phytoplankton
population, and the lack of phytoplankton reserves might not be sufficient to support the survival
of the zooplankton population in the aquatic ecosystem.

7.4.2 Effects of stochastic fluctuation
In real aquatic ecosystems, environmental stochasticity plays an important role in influencing
plankton dynamics (Yu et al., 2019a). Excessive noise intensity may threaten aquatic species,
leading to undesirable and potentially irrevocable changes in ecosystem functioning. For example,
a dramatic drop in temperature resulted in the complete disappearance of the Cylindrospermopsis
123

filaments (Borics et al., 2000). By fixing the pulse intensity as

= 0.5 and

= 0.1, Figure 7.6

shows the impact of environmental fluctuations on nutrient concentration and plankton populations
for

= 5 ,. Apparently, when the noise intensity is too low, i.e.,

0.01 sin(

=

=

= 0.02 +

/20), the plankton populations persist in the ecosystem. However, when the noise

intensity increases to

=

=

= 0.1 + 0.01 sin(

/20) , the nutrient concentration and

plankton populations exhibit a stronger oscillatory behavior, indicating a comparatively higher
chance of survival for the plankton populations. An interesting observation is that with an increase
in noise intensity to

=

=

= 0.5 + 0.01 sin(

/20), the nutrient concentration becomes

zero and the plankton populations tend to go extinct, whereas the deterministic model shows a
periodic solution. This implies that excessive noise intensity may accelerate the extinction of
plankton populations in the aquatic zone.

Figure 7.6. The evolution of a single path of (a) nutrient, (b) phytoplankton and (c) zooplankton in model
(7.2) with respect to time for

= 0.5 and

correspond to

= 0.02 + 0.01 sin(

=

=

=

=

= 0.5 + 0.01 sin(

= 0.1. In the figure, the red, blue and black lines respectively
/20) ,

/20).
124

=

=

= 0.1 + 0.01 sin( /20) and

The results from Figure 7.6 indicate that increments in noise intensity up to a certain limit can
enhance the amplitude of oscillations in plankton populations. However, if the noise intensity goes
beyond a certain range, it may threaten the entire ecosystem. In natural aquatic ecosystems,
considering the stochastic nature of turbulence, appropriate turbulent mixing can promote internal
nutrient release through sediment resuspension, contributing to the growth of plankton species.
However, strong turbulence can disrupt various elements of phytoplankton species’ life cycles,
such as the mitotic cycle and chromosome separation (Sengupta et al., 2017). Thus, controlling
environmental fluctuations can be a key factor in mitigating planktonic blooms.
7.5. Conclusion and discussion
Here, the dynamics of a stochastic nutrient-plankton model with impulsive control were
explored. Some theoretical and numerical analyses were presented to investigate how the plankton
populations respond to the combined actions of pulse perturbation and environmental fluctuation.
Analytically, the results indicated that model (7.2) exists a global positive solution, which suggests
that the population size will remain non-negative for all future time. In addition, some sufficient
conditions were obtained under which the model exhibits a positive -periodic solution, implying
that changes in plankton populations’ density will exhibit more or less periodicity. The results from
Theorem 7.2 indicated that both the pulse intensity and the noise intensity play important roles in
the existence of positive -periodic solution. Figure 7.2 showed that environmental noise can
change the biomass of nutrient and plankton, and induce fluctuations around the solutions of the
corresponding deterministic model. Figure 7.3 depicted that when the pulse intensity is small, the
plankton populations tend to go extinct. This happens as the low pulse intensity leads to insufficient
nutrient supply in the aquatic ecosystem. Furthermore, the results showed that increased pulse
intensity was capable of increasing the oscillatory tendency in the plankton populations, which
125

indicates that the pulse intensity can increase the maximum plankton population size. Thus, it can
be inferred that the pulse intensity can significantly affect the survival of plankton populations in
the aquatic habitat.
In natural aquatic ecosystems, some disturbances, such as wind-induced mixing, can lead to
an instantaneous increase in the biomass of phytoplankton. However, such phenomenon may not
be the primary driver of bloom events. Figure 7.4 showed a detailed view of how nutrient pulse
and phytoplankton pulse affect the evolution of plankton populations. It was observed that when
the nutrient pulse was small, both phytoplankton and zooplankton populations tended to go extinct,
even when a high pulse of the phytoplankton population was applied. One possible explanation is
that the growth of phytoplankton is directly correlated with the nutrient concentration, and the low
nutrient pulse may lead to nutrient deficiency in the aquatic ecosystem, ultimately causing
stagnation in the growth rate of phytoplankton species (Ittekkot et al., 1981). The low nutrient
pulse and a lack of phytoplankton in the aquatic reservoir might have impaired the growth, survival,
and possibly reproduction of the zooplankton species. The results also noted that with an increment
in the nutrient pulse, the phytoplankton population becomes persistent while the zooplankton
population goes extinct. This indicates that a high nutrient concentration in the aquatic reservoir
will allow the phytoplankton population to survive, but it will not be appropriate to support the
coexistence of both the phytoplankton and zooplankton species. Moreover, Figure 7.4 showed that,
if the nutrient pulse was 0.5, then both the phytoplankton and zooplankton populations become
persistent, even though the phytoplankton pulse was low. The results from Figure 7.4 suggest that
in comparison to the phytoplankton pulse, the nutrient pulse is the primary driver in controlling the
oscillations, survival, and extinction of both phytoplankton and zooplankton populations. Here, the
results are in agreement with the findings of Huang et al. (2018) and Paerl and Barnard (2020).
Owing to the fact that the number of pulses per period plays a significant role in the survival
126

of plankton populations (Liu et al., 2023), Figure 7.5 showed that when the number of pulses per
period is equal to 1, both the phytoplankton and zooplankton populations tend to go extinct in the
aquatic system. Biologically, a lower pulse number may result in insufficient nutrient concentration
in the aquatic ecosystem, which is unable to support the survival of plankton populations. But, with
an increase in the pulse frequency, the persistency of plankton populations increases. The results
also showed that the oscillatory tendency in the plankton populations also becomes stronger as the
pulse frequency increases. This implies that a high pulse number can increase the survival chances
of plankton populations in the aquatic system. The existing research has recognized the critical
role played by environmental fluctuations in regulating plankton dynamics (Yu et al., 2019a). The
simulation results demonstrated that random fluctuations have significant effects on the survival
of plankton populations in the aquatic habitat (see Figure 7.6). Figure 7.6 showed that if the noise
intensity was within a certain range, then environmental fluctuations are capable of enhancing the
oscillations of plankton populations. This indicates that appropriate noise control may increase the
survival chances of plankton populations in the aquatic system. However, excessive noise intensity
may result in the collapse of the entire ecosystem. For instance, a dramatic drop in temperature can
cause the complete disappearance of the Cylindrospermopsis filaments (Borics et al., 2000). The
results from Figure 7.6 suggested that environmental noise is another essential factor for
phytoplankton growth. Overall, the findings suggest that nutrient pulse and environmental
fluctuations play a crucial role in the survival of plankton populations. It is worth noting that the
objectives of this research are not the precision with which it predicts specific events within a
particular lake/ocean, but its contribution to the studies on how the coupling between
environmental stochasticity and impulsive effects influence the growth of plankton populations in
lakes/oceans. Moreover, our results may provide some insights into possible management
strategies to prevent algal blooms in aquatic ecosystems.
127

Chapter 8
CONCLUSION AND RECOMMENDATIONS
8.1. Synthesis and conclusion
Phytoplankton offer great potential for exploitation, serving as the base of the trophic chain
and playing a crucial role in stabilizing aquatic ecosystems. In recent years, considerable efforts,
such as experimental and mathematical studies, have been made to explore the dynamics of
phytoplankton growth. However, due to the nonlinear nature and complexity of aquatic ecosystems,
the underlying mechanisms that drive changes in phytoplankton dynamics have remained a
challenge. Thus, the current study developed five mathematical models to investigate the dynamics
of phytoplankton growth, including (i) delayed nutrient-plankton model with diffusion; (ii)
stochastic Leslie-Gower phytoplankton-zooplankton model with prey refuge; (iii) stochastic
nutrient-plankton model with seasonal fluctuation; (iv) stochastic nutrient-plankton model with
regime switching; and (v) stochastic nutrient-plankton model with impulsive control.
It has been recognized that delay effect exists in the phytoplankton growth process, such as
nutrient transformation and nutrient recycling related to decomposition. There is growing evidence
suggesting that spatial distribution also plays an important role in plankton dynamics, as the
abundance of plankton species changes not only in time but also in space. In Chapter 3, a delayed
nutrient-plankton model with diffusion was proposed to explore the effect of delay on the modeling
dynamics. The results indicated that the delay could destabilize the model via Hopf bifurcation
when the delay passed through the critical value. Furthermore, increasing delay could lead to the
emergence of stability switches instead of playing only a destabilizing effect in the model. More
importantly, numerical results revealed that chaotic behavior could be induced by delay, indicating
that delay-induced chaos can appear in a spatiotemporal plankton model. This work may provide
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insightful information about how delay affects the dynamics of spatiotemporal plankton growth
models. It is important to note that the emergence of chaos indicates the unpredictable feature of
the bloom phenomenon. Thus, the current study cannot predict algal bloom events but contributes
to the study of how delay affects phytoplankton growth.
In natural ecosystems, phytoplankton growth processes are inevitably influenced by
environmental fluctuations, which can be mathematically described by white noise. The noise
intensity is more difficult to measure in experimental studies, thus many researchers prefer to use
mathematical models to study the stochastic dynamics of aquatic ecosystems. Furthermore, recent
evidence supports that the refuge of phytoplankton is a common phenomenon in aquatic
ecosystems. For example, water column stratification can be used as a temporary shelter for
phytoplankton recovery. Thus, in Chapter 4, a stochastic Leslie-Gower nutrient-phytoplankton
model with prey refuge was proposed, in which the effects of random fluctuation and prey refuge
on the phytoplankton population were studied. The results showed that excessive noise intensity
could result in the extinction of plankton species. Theoretical analysis further indicated that the
refuge of phytoplankton played an important role in the coexistence of plankton species.
Furthermore, the numerical results revealed that the refuge of phytoplankton could significantly
change the mean and the variance of plankton biomass, indicating that prey refuge could enhance
the oscillation range of the phytoplankton population. The numerical results further showed that
too large or too small phytoplankton refuge was not conducive to plankton coexistence. The present
study provides insight into how phytoplankton refuge affects plankton interaction dynamics under
environmental fluctuations.
There is evidence that blooms occur periodically due to seasonal changes in the environment.
For example, Lake Taihu in China experiences phytoplankton blooms almost every year from May
to October. This periodicity is attributed to changes in nutrient supply, water temperature, and light
129

intensity, which all exhibit a periodic pattern and subsequently lead to periodic blooms. Although
extensive studies have focused on the effect of seasonal fluctuation on algal growth, the complexity
of the aquatic environment limits our understanding of seasonal phytoplankton blooms. In Chapter
5, a stochastic nutrient-plankton model with seasonal fluctuation was proposed, and the results
showed that seasonality and stochasticity may cause the periodic bloom of the phytoplankton
population. Additionally, the oscillation intensity of plankton biomass could be influenced by
nutrient input, and the overall trend of solutions presents obvious periodicity after nutrient input
went beyond a certain level. Importantly, the distribution of phytoplankton under constant and
periodic parameters does not show a notable difference, but seasonal fluctuation could
considerably alter the distribution of zooplankton. The simulation results further implied that the
distribution of zooplankton biomass was sensitive to changes in periodicity. The present study
provides insights into how seasonal fluctuations influence plankton dynamics under stochastic
environments.
In recent years, constant plankton mortality has been widely used in most aquatic ecosystem
models. However, during the decline phase of blooms, the decomposition of dead plankton
consumes a large amount of oxygen, resulting in a sharp decrease in dissolved oxygen. Thus,
phytoplankton mortality in the decline phase is much higher than that in the development phase.
In Chapter 6, colored noise was used to describe the shifting plankton mortality in the model and
a stochastic nutrient-plankton model with regime switching was proposed. The theoretical analysis
suggested that the plankton population could survive forever under certain conditions. Moreover,
sensitivity analysis showed that phytoplankton biomass was highly sensitive to the grazing rate by
zooplankton and least sensitive to the re-mineralization of dead biomass into nutrients. Numerical
results further revealed an important fact that the Markov chain could significantly enhance
plankton population oscillation. From an ecological point of view, regime-switching plankton
130

mortality is beneficial to the survival of plankton. The present study may not predict the algal
bloom phenomenon but provides an understanding of how regime-switching plankton mortality
influences plankton dynamics.
In the study of control and prevention of phytoplankton blooms, impulsive control has been
widely adopted in phytoplankton growth models. Specifically, growing evidence suggests that
proper impulsive control strategies can contribute to the survival of plankton populations.
Although extensive research has focused on the effects of impulsive control on phytoplankton
growth, little attention has been paid to the coupling between stochastic and impulse effects on
plankton dynamics. In Chapter 7, a stochastic nutrient-plankton model with impulsive control was
developed to investigate how random fluctuation and impulsive control influence phytoplankton
growth. Theoretical analysis obtained sufficient conditions for the existence of a positive

-

periodic solution. The numerical analysis revealed that nutrient pulse is the primary strategy for
controlling algal blooms, and noise disturbance played an important role in the survival of plankton
populations. Moreover, numerical analysis showed that excessive noise intensity could threaten
the entire ecosystem. The results may provide insightful information about the possible
management of excessive algal growth in aquatic systems.
Field observation and experimental studies offer a qualitative and quantitative analysis, while
mathematical models represent more of a mechanistic study. Notably, some dynamical behaviors,
such as bifurcation, cannot be obtained experimentally. Therefore, the use of phytoplankton growth
models has become a promising way to investigate bloom phenomena and has attracted much
attention. The present research provides a better understanding of the dynamics in the planktonic
community. For instance, complex dynamics can be induced by delay, indicating that
spatiotemporal chaos can appear in a spatiotemporal plankton model. White noise and colored
noise are suggested to play a significant role in planktonic communities, and the seasonal
131

fluctuation is considered an important factor affecting the periodic bloom events of phytoplankton.
Specifically, controlling nutrient input is considered as the primary strategy in preventing and
controlling algal blooms. It is worth mentioning that the proposed models may not accurately
mimic reality, but they can provide a better understanding of the dominant processes determining
the phytoplankton growth and offer some general suggestions for the control of bloom events.
8.2. Limitations and future research
Although the proposed models provided some understanding of the dynamics of
phytoplankton growth, they cannot explain all the mechanisms behind phytoplankton growth.
These models require further revision before they can be applied in the field. In fact, the real aquatic
ecosystems are complex. To reduce the complexity of models and computations, it is often
necessary to ignore certain aspects of natural aquatic ecosystems and focus only on the aspects
most relevant to the problem at hand. For example, during the model building process, I typically
assume that temperature and light are adequate. Therefore, the developed models may not fully
reflect the complexity of real aquatic ecosystem. Additionally, my research only focuses on the
interactions between nutrient concentration, phytoplankton population, and zooplankton
population. Thus, my research may only partially describe the nutrient-phytoplankton-zooplankton
interactions.
This study investigated delay-induced dynamics, such as periodic solution and chaos, and
provided insight into the oscillatory and chaotic behavior that is likely common in population
ecosystems. The emergence of chaos is beneficial in promoting phytoplankton biodiversity.
However, there is currently little empirical evidence to support the existence of chaos in aquatic
ecosystems. A possible explanation might be the limited access to long-term experimental data.
While the current studies demonstrate the existence of spatiotemporal chaos in phytoplankton
132

growth models, the mechanism for the formation of chaos in aquatic ecosystems require further
investigation. Additionally, previous studies have demonstrated that the distribution of plankton
can differ significantly not only in spatial but also in temporal scales, resulting in the emergence
of plankton patterns (such as stationary strips, spots, and strips-spots mixture patterns). Although
considerable efforts have been made to understand the principal features and spatiotemporal
variability of plankton species, they are still not well understood, especially delay-induced
spatiotemporal plankton patterns. Therefore, further research on delay-induced plankton pattern in
aquatic ecosystem is needed.
The dynamics of stochastic phytoplankton growth is another interesting problem in aquatic
ecosystems. The current study provided four different stochastic phytoplankton growth models and
investigated the plankton dynamics under various random disturbances. However, the complexity
of aquatic ecosystem limits the studies of stochastic phytoplankton growth dynamics. For instance,
aquatic ecosystems are not only affected by regime switching (e.g., plankton mortality, nutrient
supply) but are also influenced by abrupt disturbances (e.g., earthquakes, hurricanes). Although
extensive research has been carried out on the coupling effect of random fluctuations, it remains
unclear how these stochastic fluctuations affect planktonic ecosystems.
Another area of interest for further research is the impact of essential abiotic resources, such as
temperature and light intensity, on the structure and succession of phytoplankton. A considerable
amount of literature indicates that the temperature and light intensity are highly variable with
environmental fluctuations, resulting in changes in phytoplankton biomass. However, the proposed
dynamic models in this study have not considered these abiotic factors. To gain a better understanding
of how phytoplankton growth responds to temperature and light intensity, further research should
incorporate a phytoplankton growth model that takes these factors into account.
Additionally, experimental work is needed to provide validation for the results obtained in
133

current research, thereby enhancing the reliability and accuracy of the models. In the present
research, the models were developed relying on certain assumptions and simplifications, the
experiments can be used to validate the validity of these assumptions. By comparing model outputs
with experimental results, the scope and limitations of the models can be determined, leading to
model improvements and adjustments. Actually, models can be used for optimization, result
prediction, and decision-making, while experiments provide real-world validation and application
of the models. The integration of models and experiments is able to enhance the reliability,
accuracy, and applicability of research, providing a more comprehensive and in-depth
understanding of scientific inquiry. Hence, a combined experimental and modeling approach is
essential in future research.

134

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169

APPENDIX A. The proof of Theorem 3.1
Proof. As we have considered zero-flux boundary conditions, let 0 =

<

⁄ , ∈ ℕ = {0, 1, 2, … }) be the eigenvalues of the operator −∆ on

<

<⋯(

=

with the homogeneous

Neumann boundary condition. Following Dai et al. (2016), model (3.1) can be represented as an
abstract functional differential equation in the abstract space ([− , 0],
model (3.1) around the equilibrium

∗
( )

where

=

(

,

(

)= ( ,

(

∗

∗

,

∗

,

). The linear form of

) is given by:

= ∆ ( )+ (

(A1)

).

) and

,
,

) :

,

,

∈

([0,
∗)

−[( +
⎛
( )=⎜

∗

(− ) +

∗ ∗

(

],

∗
)

(

∗)

=
∗)

(0) −
∗

(

⎝

=

(0) + (

−

∗)

),

= 0,

= 0,

,

(0)]
∗
∗

(0)⎞.
⎟

(0)

⎠

Following Wu (1998), the characteristic equation corresponding to Eq. (A1) is rewritten as
follows:
−

− (

) = 0,

∈

(

),

≠ 0.

(A2)

Define:
,

=∑

∞

( )

,

,

(A3)

,

where

( ) composes of the eigenfunctions corresponding to the eigenvalues

Eq. (A3) into (A2), one can get:

170

. On substituting

−

∗

+

⎛
⎜

+
∗ ∗

∗

∗)

(

0

⎝

−(

∗)

−

∗

0

−
∗

(

)

(

∗)

∗

−

,

⎞
∗
⎟

,

=

,
,

−

,

,
,

⎠
= 0, 1, 2, ….

Thus, an equivalent equation of Eq. (A2) is obtained as follows:
+

+

)+(

+

+

+

+

(A4)

= 0, = 0, 1, 2, …,

where
= −(

+

=

−

=

−(
+(

) ,

+

,

=−

+

+

−

)

+

)

−(

,

=−

,

=−
+(

+

+

)

+

+

) ,
，

with
=− −
∗

=
For

,

=(

∗

∗
∗)

∗

(

,

−

= (
∗

,

∗

)
∗)

=−

∗
∗

,
.

= 0, Eq. (A4) reduces to the following form:
+

Since

∗

< 0, one can have

of Eq. (A5). If

> max

− ,

+(
+

+ ) +

+

> 0 in view of (

, then

> 0 and

has no positive roots, which yields Theorem 3.1.
This completes the proof.

171

+

= 0.

). Obviously,

(A5)
= 0 is not a solution

> 0 for any ∈ ℕ . Thus, Eq. (A5)

APPENDIX B. The proof of Lemma 3.1
Proof. For the considered nutrient-plankton model (3.1) with
(2003), we assume

(

> 0. Following Ruan and Wei

> 0) is a root of Eq. (A5). Putting

=

into Eq. (A5), one can get the

−
−

(

),

following two equations:
+

=
=−

(

+

,

+

=

−2

+

)−
( )−

(

(B1)

),

which yields
+

,

= 0,

(B2)

= 0, 1, 2, ….

where
,

Let

=

=

−2 ,

,

−

,

=

+

，

.

−

, then we define function ( ) as follows:
( )=

In view of (H1), we have

+

+

,

> 0, and denote

=

has no positive roots. However, if there exists

,

,

−3

,

(B3)

. Evidently for

∈ ℕ such that

< 0, Eq. (B3)

> 0, then the equation

′

( )=

0 has the following two real roots:
,

For

> 0 and

=

,

∆

,

,

=

,

∆

.

> 0, Eq. (B3) has two positive real roots if and only if

0.
This completes the proof.

172

0,

> 0 and (

0,

)<

APPENDIX C. The proof of Lemma 3.2
Proof. Denote two positive roots of Eq. (B3) by
(B2) will be

,

=

,

and

, , then the two positive roots of Eq.

, . From Eq. (B1), we obtain the critical value of time

=

,

and

,

delay as:
0,

=

0,

0

0,

0

0,

0

0

0

+2

0,

0

( = 0, 1, 2, ⋯ ,
Obviously, the right side of Eq. (C1) is an increasing function of
,

0,

. Let ( ) = ( ) +
0,

(C1)

= 1, 2).
. Thus, we have

=

( ) be the root of Eq. (A4) such that

= 0,

, ( = 0, 1, 2, ⋯ ,

=

0,

(C2)

= 1, 2).

On differentiating both sides of Eq. (A4) with respect to , we obtain:
=

(

− ,

)

(C3)

then
0,

′(

=

(C4)

),

where
∆ =

sin

0

+

cos

0

For convenience, we assume that
0 and

(

0,

) < 0. Hence,

( )

,

|

−

0,

<

0,

0,

cos
+

0,

sin

0,

.

, . Note that if Lemma 3.1 (ii) holds, then

> 0,

This completes the proof.

173

( )

|

0,

< 0,

= 0, 1, 2, ⋯.

(

0,

)>

APPENDIX D. The proof of Theorem 3.3
Proof. Let

= ( , , ),

=( ,

), and

,

∗

=

+ ,

∈ . We will track the direction

of Hopf bifurcation around the positive equilibrium and the stability of the bifurcating periodic
solutions by using normal formal theory and the center manifold theorem of differential equations
(Hassard et al., 1981).
In space , Eq. (A1) can be rewritten as
(

)+ ( ,

),

(D1)

( ) = ( ∗ + )(

(0) +

(−1)),

(D2)

= ∆ +
where

= ( ∗ + ) , and

) =( ∗+ )·

( ,

−
(−1)

∗

(0) +

( − )ℎ ∗ (ℎ +

∗)

(
∗

)

(0)

−

(0) + ⋯
∗

(0) + ( − )ℎ(ℎ +
0

=
Here,

=(

,

,

0
0

)

0
,

)

(0)
(0)

(0) + ⋯ ,
(0) + ⋯

0 0
0 0 ,
0 0

=

0

) ∈ . Hence,

∗

(0) − ℎ(ℎ +

0

= 0 is the Hopf bifurcation value of Eq. (D1).

On linearizing Eq. (D1) about (0, 0), we get
=

∆ +

(

(D2)

),

Recall that Eq. (A4) has a pair of purely imaginary roots

=

∗

, −

∗

with

= 0 and

other roots have negative real parts. The solution operator of Eq. (D2) is a semigroup whose
infinitesimal generator is given by
=

̇ ( ),
∆ (0) +

( ),

Hence, Eq. (D2) can be rewritten in the abstract form as,
174

∈ [− , 0),
∈ 0.

̇ =

+ ( ,

(D3)

),

where
( ,
Define

=

,

)=

∈ [−1,0),
= 0.

0,
( ,

),

= (0,

, 0) ,

, where

,

= ( , 0, 0) ,
⁄

=‖

⁄‖

, ‖

= (0, 0, ) ,

⁄ )‖ = ∫

(

.

⁄)

(

Denote by
⋅

=

+

+

,

⟨ ,

⟩=∫ [

=( ,

,

=( ,

) ∈ ([− , 0], ),

,

and

where

=(

For

,

=(

),

,
,

̄ +

,

̄ ]

̄ +

∈ .

), ,

) ∈ , we denote

,

,

=

,

,

,

,

.

,

Now, we define
=

,

∈ [−1, 0),

̇ ( ) ,
∫

( ,

)

( ) ,

−

(0) +

,

( ,

−( ∗ + ) ,
) = 0,
( ∗ + )( −

= 0.

Thus, we have
(

)=∫

( ,

)

( ),

where

Define

∗

= ([0, ],

) and a bilinear form (⋅ , ⋅) on
175

= −1
∈ (−1, 0)
= 0.

),
∗

×

as follows:

(D4)

( , ) = ∑∞,
where

=∑

∗

∈

Notice that ∫

,

(

=∑

= 0 for

,

) ∫

∈ ,

∈ ,

) = ∑∞

where (⋅ , ⋅) is the bilinear form defined on
=

(0)

Let ( )

,

)

of the operators

and

∗

∗

× , and satisfies the following equation:
( − )

∗

(0,

∑
∗

∞

( )

.

∈ (0, 1],

(0, )

∫

, and

)

as follows:

− ̇ ( ),

( )=

= (1,

.

,

,

(0) − ∫ ∫

Now, we define an adjoint operator
∗

∗

∈

≠ . Thus, we have
( ,

,

,

∗

( )

=

(− ) ,
( ,

= 0.
∗

, 1)

corresponding to the eigenvalues

and −

be the eigenfunctions
, respectively. Thus, one

can obtain the following equations:
−

+

⎛
⎜ −

−
∗

−

0

⎝

0
+

−

−

⎞
⎟

+

⎠

1
= 0,

and
+

∗

−

⎛
⎜
⎝

+

0
⎞
⎟

−

0

−

⎠

By direct calculation, we obtain
=
∗

=

⋅

176

,

=

,

=

⋅

;
.

= 0,
1

such that ( ∗ , ) = 1, ( ∗ , ̄ ) = 0, then we have

If we choose

=
This yields the decomposition

=

=

∈

+ ̄

+ ̄ +

⊕

by

∗

∗

.

with

=

+ ̄ ̄ | ∈

( ̄∗ ,

)=0

,

( ∗ ,

)=0 .

One can rewrite the solution of Eq. (D1) as follows:
=(
where

At

( )

=

∗

=

,

( )

,

,

( )

( ,

)=

+ ̄ ̄) ⋅

( , ̄ ),

+

(D5)

∈ . On the center manifold
( )

+

, we have

( ) ̄+

̄

( )

(D6)

+ ⋯.

satisfies the following relation:
( ) + ̄ ∗ (0) (0,

̇( ) =

),

.

(D7)

̄+

′′ ̄
̄ ̄

Next, we define
(0,

)|

′′

= (0, , ̄ ) =

+

′′
̄

+⋯.

For convenience, we rewrite Eq. (D7) in the following form:
+ ( , ̄ ),

̇=

(D8)

where
( , ̄) =

+

̄

̄+

(D9)

+ ⋯.

A simple computation will yield the following values:
=2

∗

−

̄ +

( − )ℎ ∗ (ℎ +
=

∗

∗

− (

)

∗

̄

+ ( ℎ ∗ (ℎ +

∗

)

− ) ̄

∗

]∫

,

+( − )ℎ(ℎ +
+ ̄ ) ̄ + ̄

− ℎ(ℎ +

∗

) (

( ̄

̄ + ̄

)
∗

+

∗

) + ( ℎ ∗ (ℎ +

)] + [( − )ℎ ∗ (ℎ +
177

− ℎ(ℎ +

∗

) 2

̄

∗

)

∗

)

̄ +

− )2

̄

∗

+( − )ℎ(ℎ +
∗

=2

) (

− ̄ ̄ + ̄

∗

( )

(−1)

+( ℎ ∗ (ℎ +
×

∗

( )

̄ (− )

+ ̄

∗

∗

( )

( )

×

(0) +

Now, we need to compute

∗

)
∗

(0) +

1
2

(0) ̄ +

1
2

)

(0) +

1
2

(−1) ̄ +

1
2

( )

( )

( )

( )

( )

+

( )

(0) +

( )

⋅ (2

( )

( )

( )

( )

̄ ̄ }∫

)

( )

∗

(0)

̄ ̄

(0)

+

(0)

+ (− ℎ(ℎ +

(0)

∫

+

)

,

(0) ̄

(0) + ̄

(0) ̄ + ̄

( ) and

( )

̄ +

∗

− ) ̄ − ℎ(ℎ +

̄ + ( − )ℎ(ℎ +

( )

+ ( − )ℎ ∗ (ℎ +

,

)

+

( )

̄

)]} ∫

+ ( ℎ ∗ (ℎ +

− ) 2

)

(0) +

∗

̄

+( − )ℎ ∗ (ℎ +
=2

̄ + ̄

∗

∗

) )

( )

(0)) + ( − )ℎ(ℎ +

( )

(0)

∗

)

.

∫

( ). From Eq. (D7), we have

̇ = ̇ − ̇

− ̇̄ ̄

=

−2
−2

{ ( , ̄ ) ( )}
{ ( , ̄ ) ( )}

=

+ ( , ̄,

+ ,

∈ [− , 0)
=0
(D10)

),

where
( , ̄,

)=

( )

+

( ) ̄+

( )=

−
−

( )
(0)

+ ̄
+ ̄

̄( )
̄ (0) +

−

( )

+ ̄

̄( )

−

(0)

+ ̄

̄ (0)

( )

̄

+ ⋯,

Further, we obtain

( )=

178

′′

,

∈ [− , 0),
= 0,
∈ [− , 0),

+

′′

̄,

= 0.

(D11)

After differentiating both sides of Eq. (D6) and comparing the coefficient with Eq. (D10), we
get
(
when

)

−2

( )=−

∈ [−1, 0). By the definition of

Denote

,

=∑

( )

∗

( )

∗

̄

−

and
∗

|

=−

= ∑∞

′

= 2

∗

̄( )

∗

∗

̄( )

∗

̄

−

. Thus,

=∑
−2

( )=−

(D12)

( ), ⋯

and Eq. (D12), we have

( )=−
( )=

( ),

̄,

∗

+

+

,
(D13)

.

satisfies the following relations:
′′

|

=−

,

,

,

= 1, 2, ⋯.

(D14)

where
′′

,

′′

,

̄ =∑

′

∗

−∫

=− ∫

∞

(0,

(0,
′′

)

̄,

̄,

,
′′

)

,

,

.

Thus, we have
∗

2
⎛
=⎜

−

∗

∗

2

0
−

⎝

−

−

⎝

⎛
=⎜

+

−

−

⎞
⎟

−

−

−

2

∗

+

′′

,

,

⎠

0
−

0

+

0

−

Thus, one can compute the value of

−

⎞
⎟

′′

̄,

.

⎠
. Finally, we will compute the values of the following
179

quantities:
(0) =
=−
=−
=2
The signs of

,

and

{

|

+

,

( )}
,
)

′(

{

{

( )}

′(

)

,

(0)},

will determine the properties of the Hopf bifurcation. Specifically,

determines the direction of the Hopf bifurcation, if
(subcritical);

| − |

− 2|

> 0(< 0), Hopf bifurcation is supercritical

determines the stability of the bifurcation periodic solutions, with the solutions

being stable (unstable) when

< 0(> 0);

determines the period of the bifurcating periodic

solutions, such that the period increases (decreases) when
This completes the proof.

180

> 0(< 0).

APPENDIX E. Auxiliar Lemmas
Lemma E.1 (Liu and Bai, 2016). Suppose that ( ) ∈ ( × [0, +∞),
(i) If there exist two positive constants
∑

( ) for all ≥ , where

≤ ⁄

( )

( )=0

. .,

(ii) If there exist three positive constants
∑

( ) for all ≥ , then

such that

( )≤

,

. .,

≥ 0;

and

such that

( )≥

≥ ⁄

( )

∫

→

( )

+

−

∫ ( )

+

a.s.

( ), ≥ 0, is a local martingale vanishing

at time zero. Then
( ) < +∞ ⟹

( )
→

=0

where
( )=∫
and 〈 ,

∫

< 0.

Lemma E.2 (see e.g. Liptser, 1980). Assuming that

→

−

, = 1, 2, ⋯ , , are constants, then

∫

→
→

and

).

〈 , 〉( )
(

〉( ) is Meyer’s angle bracket process.

181

)

,

≥ 0,

. .,

APPENDIX F. The proof of Lemma 4.1
Proof. We consider the following model
( )=

−

( )=

( )

−

−

−

(1− )

( )

(1− )

( )

( )

−

on ≥ 0 with initial value (0) = ln

(F1)

( )

+

( )

(1− )

( )

+

(0), (0) = ln

(0).

Obviously, the coefficients of model (F1) satisfy the local Lipschitz condition, then there
exists a unique local solution on [0,
( )=

( )

value ( (0),

,

( )=

( )

(0)) ∈ [0,

), where

is the explosion time. Therefore, one can see that

is the unique positive local solution to model (4.1) with any initial
). To show this solution is global, we need to verify

= +∞.

Considering the following auxiliary equations:
( )=

( )

−

( )

+

( )

( ),

(0) =

(0);

(F2)

( )=

( )

−

( )

+

( )

( ), (0) =

(0);

(F3)

( )=

( )

−

(

( )

) ( )

( )

+

( ),

(0) =

(0);

(F4)

According to the comparison theorem for stochastic equation (Peng and Zhu,2006), one can get
the following results for ∈ [0,

),

( )≤

( ),

( )≤

( ), a.s.

( )≤

(F5)

By Lemma 4.2 in Bao et al. (2011), Eq. (F2) has the explicit formula
( )=

)

{(
( )

( )}
)

{(

∫

( )}

,

(F6)

is the unique solution of Eq. (F2). Similarly,
( )=

)

{(
( )

{(

∫

182

( )}
)

( )}

,

(F7)

( )=

{(
( )

∫

(1− ) ( )

)

( )}
{(

)

( )}

is the unique solution of Eqs. (F3) and (F4), respectively. It therefore follows that
( ) are existent on ≥ 0, then we have

= +∞.

This completes the proof.

183

(F8)
( ), ( ), and

APPENDIX G. The proof of Lemma 4.2
Proof. Let

sufficiently large such that 0.5 exp{(

) } ≥ 1 for ≥ . Then when ≥ , by

−

Eq. (F6),
{(

( )=

( )

≤
≤
≤

(

)

( )}

∫

{(

{(

)

( )}

( ) ∫

{(

)

)

{(

( )}

)

(

)

|

( )}
{(

{(

∫

( )}
)

( )}

{(

)
( ) [

(

=

) }

)

exp

[

( )}
{(

Noting that
exp |

( ) − min

|[

( )] > 1.

Consequently, the following equation obtained:
∫

(

) ( )

exp{(

(

)

(

(

(

)

(

|

(

)

|

) (

)

× exp min
( )[exp{(

−

( )}
( )

|
)

)

) (

× exp −|

=

)

)

{(

≥∫

≥

( )}
{(

≥∫

=

) +

−

( )
( )}

|

( )

∫ exp{(

−

) +

( ) − min

( )

|

( )

( )}

exp |

| min

( )

exp{(

−

) }

) } − exp{(

−

) }],

184

( ) − max

) }

( ) − min

( )] .

( )−

|[

)

=

) }

( )

{(

≤

( )

]

( )]

where
(t) =

) (

(

exp |

)

| min

( ) − max

( ) + min

( ) .

Substituting the above inequality into Eq. (F8), we have
( )

≥ exp −(

)( − ) −

−

× [ (0) ( ) +
≥ exp{−(
×[

( )(exp{(

( )exp{(

=

≥ exp{(

−

−

) +

( )×

( ),

( ) = exp{(

−

) } − exp{(

−

−

) })]

( )}

) })]

−

( )+

) + 2(

( )

( )+

) } − exp{(

) −

−

−

) −

−

( )}
) −

( )}(1 − exp{−(

( ) exp − max

×
=:

−

( )exp{−(

−

( )(exp{(

) +(

−

( )−

( )+
−

( )}

)( − )})

( )

where

( )=

) +

( )}(1 − exp{−(

−

)( − )})，

( ) .

( ) exp − max

Hence,
ln ( ) ≤ −
Since

ln

( )−

ln

( ).

(G1)

, = 1,2 are bounded, we obtain
lim

→

( )

= 0 a.s., = 1, 2,

(G2)

which implies
lim

→

ln

( ) = 0,

lim

→

ln

( ) = 0, a.s.

Substituting the above identities and Eq. (G2) into Eq. (G1) leads to
185

lim sup

ln

→

ln ( ) ≤ 0, a.s.

( ) ≤ lim sup
→

Now we need to prove lim

( ) ≥ 0, a.s. Applying Ito’s formula to (F4) gives

ln

→

(G3)

ln

( )=

−

−

( )

+

( ).

( )=

ln

(0) +

−

−

∫

One can get that
ln

From Eq. (G3) we obtain that for arbitrary

> 0, there exist

− ≤

ln

( )

( )

.

+

(G4)

> 0 such that for ≥ ,

(0) ≤ ,

Then it follows from Eq. (G4) that

Let

ln

( )≤

−

+ −

∫

( )

+

ln

( )≥

−

− −

∫

( )

+

be sufficiently small such that

( )

(G5)

( )

(G6)

, ≥ ,
, ≥ .

−

− > 0, then according to Lemma I.1, Eqs. (G5) and

( )

≤ lim sup

(G6), we have
(

)

≤ lim inf
→

∫

( )

∫

→

(

≤

)

, a.s.

Due to the arbitrariness of , then
lim

→

Note that, lim ln
→

we

obtain

lim

ln

→

lim

→

∫

=

(

)

.

(G7)

( )⁄ = 0, and substitute Eq. (G7) into Eq. (G4), then

(0)⁄ = 0, lim
→

ln ( ) = 0

( )

a.s.

Thus

( ) = 0, a.s.

This completes the proof.

186

by

Eq.

(N6),

lim inf

→

ln

( )≥

APPENDIX H. The proof of Theorem 4.1
Proof. Applying Ito’s formula to model (4.1) gives
ln

( )=

−

−

ln

( )=

−

−

( )−

(

)

( )

(

)

( )

( )
(

)

( )

( ).

+

( )

( ),

+

It follows that
ln

( ) − ln

(0) = (

−

) −

∫

ln

( ) − ln

(0) = (

−

) −

∫

−

(

∫

( )
(

( ), (H1)

+

( )

)

(H2)

( ).

+

( )

)

( )

)
(

Now we prove (i). According to Eq. (H1)
ln
( )

Note that, lim
→

( )

≤
( )

= 0 a.s. and

−

(ii) If

, thus (i) implies lim

( )

( )

.

+

( ) = 0, a.s. Similarly, if

→

( ) = 0, a.s.

<

∫

, thus lim

<

(H2) we have lim
→

−

<

, from Eq.

( ) = 0, a.s. Then, the following inequalities obtained

→

for sufficiently large
( )

ln

( ) − ln

(0) ≤ (

−

) −

∫

ln

( ) − ln

(0) ≥ (

−

) −

( )∫

(

( )
(

(H4)

( ).

+

)

(H3)

( ),

+

)

Making use of Lemma E.1 to Eq. (H3) and (H4) respectively, we obtain
(

)(

(

) )

≤ lim inf
→

∫

Due to the arbitrariness of , lim
→

( )

≤ lim sup

∫

( )

(

)

, a.s.

→

∫

( )

=

The proof of (iii) is similar to (ii) and hence is omitted.
Now we prove (iv). By (H1) × − (H2) × (1 −
187

),

≤

(

)(

(

) )

,

ln

( )

(1 −

=
( )

( )

) ln
( )

∫

+ (

( )

) − (1 −

−

( )−

+

)(

−

(1 −

)

)−
(H5)

( )

Substituting Lemma 4.2 and Eq. (G3) into Eq. (H5), one can see that for sufficiently large ,
( )≤ + (

ln
When
)(
(

<1−
−

)

(

)

) − (1 −

−

)(

−

).

, then we can choose sufficiently small such that + (

) < 0, hence lim

( ) = 0, a.s. It therefore follows that lim

→

→

−

) − (1 −

∫

( )

=

, a.s.

Finally, let us prove (v). By Eq. (H2) we have
ln

( )−

(0) = (

ln

−

)−

∫

( )
(

)

( )

+

( )/ .

By virtue of Lemma 4.2 and Eq. (G3), one can see that
lim

( )

∫

→

(

)

, a.s.

=

( )

(H6)

It follows from Eq. (H1) that
ln

( )−

ln

(0) = (

)−

−

( )

(1 − ) ( )
+ (1 − ) ( )

−

( )/ .

+

(H7)

Substituting Eq. (G3) and (H6) into Eq. (H7), for sufficiently large the following inequalities
obtained:

Let

ln

( )≥(

−

)−

(

)(

)

ln

( )≤(

−

)−

(

)(

)

be sufficiently small such that (

−

)−

− −

∫

( )

+

( )/ ,

(H8)

+ −

∫

( )

+

( )/ .

(H9)

arbitrariness of , we obtain
188

(

)(

)

− > 0. By Lemma E.1 and the

lim

→

∫

( )

=

This completes the proof.

189

−

(

)(

)

, a.s.

APPENDIX I. The proof of Theorem 4.2
Proof. For the deterministic model corresponding to model (4.1), there is an interior equilibrium
∗

when

>

(1 −

(

=

)

∗

,

(

=

) ∗)

(

,

(I1)

).

From Ji et al. (2011), we know that a homogeneous Markov process ( ) that exists in

(

denotes Euclidean -space) can be described by the following stochastic equation
( )= ( )

+∑

( )

(I2)

( ).

The diffusion matrix is
( )=

( ) ,

( )=∑

Assuming that if there exists a bounded domain
the following properties, the Markov process

⊂

( )

( ).

with regular boundary , satisfying

( ) has a stationary distribution (⋅) and it is

ergodic.
(B. 1) In the domain

and some neighborhood thereof, the smallest eigenvalue of the diffusion

matrix ( ) is bounded away from zero.
(B.2) When

∈

\ , the mean time at which a path issuing from

and sup

< ∞ for every compact subset

reaches the set

is finite,

.

⊂

∈

Remark I.1. From the results in (Gard,1988; Strang, 1988), it is sufficient to prove
elliptical in , where

= ( )

+

∑,

( )

[

( ( )

)]

≥

∣

is uniformly

, that is, there exists a positive number

such

that
∣ ,

∈ ,

∈

,

which yields (B.1). To verify (B.2), we need to prove that there exists some neighborhood
a non-negative

-function such that for any

\ ,
190

is negative (Zhu and Yin, 2007).

and

Remark I.2. According to Eq. (I2), model (4.1) can be written as
( )

( )
=
( )

( )−

−
( )

−

(

(

)

( )

(
( )

)

( )

( )

+

( )

)

0

( )+

0

( ).

( )
Here the diffusion matrix is
( ,

0

)=

.

0

From above analysis, we know that there exists a solution
for any initial value

(0),

, and we write

(0) ∈

Then, we construct a nonnegative function :
( ,
where

)=

∗

−

∗

−

( ) and

=

of model (4.1)

( ) ∈

( ) as

and

respectively.

as follows:

→

+

∗

( ),

∗

−

∗

−

∗

≔

+

,

is a positive constant to be determined later. It follows from It ’s formula that

= 1−

∗

=(

−

∗)

−

=(

−

∗)

∗

= − (

−

= − (
+

∗

+

+

+(

( )( −

∗)

−

+

( )( −

)
−

∗)

∗)

(

(

)

(

)

(

) ∗

(

)

∗

(

)

∗(

(

) ∗ )(

−

−
∗)

(

)

∗

( ) +

+
(

)

(

)

−
)

(

∗ )(

)(
(

(

)

∗)

)

+

(1 − ) ( − ∗ )
(1 − )( − ∗ )( −
−
( + (1 − ) )
( + (1 − ) )
∗ ),

and
191

∗

( ) +

+

∗

+

∗)

+

1
2

∗

∗

= 1−
=(

−

∗)

=(

−

∗)

= (

−

)

(

)

(

∗)

)

(

) ∗(

(

+(
)
+
)

−

(

(

) (

∗ )(

(

)

(

∗)

)

)

∗)

∗

+

( )( −

+

∗

+

)

∗

( ) +

+

)

∗ )(

) ∗ )(

(

∗

( ) +

+

)

) ∗

(

(

(

(

∗

∗)

(

= (

∗

+

∗ ).

( )( −

+

∗)

Then we have
=

+

:=

+

(

−

∗)

=− (

−

∗)

+

( )+

(

) (

∗)

−

∗)

( ),

where

−
Let

(

)(

(
(

(

)

∗ )(

)(
(

(

)

∗)

)

)

(

∗

+

+

∗)

(

−(

)

)
∗

.

= , we have

=−
≤−

(

(

(

(
(

)

)

(

(

(

)
∗)

) (
(

)

∗)

) )(
)

−

)

(

−

(

∗)

(

)

(

)

(

∗)

(

)

)

∗

+

+

∗

+

∗

∗

+

.

Thus,
( + (1 −

) )

≤−

(

)

+( + (1 −
=−

(

(

−

∗)

∗

) )
)

−

−

∗

∗

+

− (

192

∗)

(

∗
(

) )

+

∗

−

(
+

(

(

−

)

entirely in
have

−

)

−

(

−

∗)

(

)

∗

+ (

< min
∗

∗

+

+ 1−

∶= −

then if

∗)

−

(
∗

−

∗

∗

+

∗

+

∗

− (

(

) )

∗

+

+ ,
∗

,

( ∗ ) , the ellipsoid

∗

(

−

∗)

to be a neighborhood of the ellipsoid with

⊆

(

) )
∗

− (

(

. We then choose

) )
∗

∗

) )

< − ( is a positive constant) for

there is

(

∗

+

+
−

+

lies

=0
=

, thus we

∈ \ , which yields condition (B.2). In addition,

> 0 such that
∑,

( ,

)

=

+

≥

∣

∣ all ( ,

)∈ , ∈

,

which leads to condition (B.1). Therefore, the stochastic model (4.1) has a stable stationary
distribution (⋅) and it is ergodic.
This completes the proof.

193

APPENDIX J. The proof of Lemma 5.1
Proof. Applying It ’s formula to model (5.2) yields that
( )

( )+

ln ( ) = − ∫

+ ∫
( )

where

( ) ( )

( )+

( )−

( )=∫

( ),

( )=∫

( ) = 0 a.s., for arbitrary 0 <

lim

→

⊂

with (

1+

( )

( ) ( )

(J1)

.

( ) ( )
( ) ( ) ( ) ( )

( )

< min 1,

) ≥ 1 − and a constant
1

( ) ( )

+

ln ( ) = − ∫
( )

( ) ( )

( )+

+

(J2)

.

( ) are two martingales. Obviously, when
, there exists a measurable set

+

= ( ) > 0 such that for

( ) ( )
( ) ( )+ ( ) ( )

( )

1
+ ln (0) < ,

∈

and >

≥ .

Combine this inequality with Eq. (J2), we have
ln ( ) =

ln (0) − ∫

≤ − ∫

( )+

( ) +

for ≥ . By Lemma E.1 (i), we have

→

( )−
( )

+

( ) ( )
( ) ( ) ( ) ( )

+

( )

( )

,

( ) = 0 a.s. and it follows that the conclusion holds

for model (5.2).
This completes the proof.

194

APPENDIX K. The proof of Theorem 5.1
Proof. From Eqs. (5.3) and (5.4), we have
( )

lim

→

Define ( ,

,

= 0 a.s., = 1, 2.

, ) = ( )[ ( ) + ( ) + ( )], then we have the following equation:

( , , , )= ( )

( )+ ( )+ ( ) +

( )( ( ) + ( ) + ( ))

= { ( ) ( ) ( ) − ( ) ( ) ( )}
( ) − ℎ( )

−
( )
=

( )

( ) ( )

( )

+ ( ) ( ) ( )

+ ( ) ( ) ( )

+ ( ) ( ) ( )

( )

( )−

( )+

( )−

( )( ( ) + ( ) + ( ))

( ) ( ) ( )− ( ) ( )−

( )+

( ) − ℎ( ) − ( )

( )+

( ) ( )

+ ( ) ( ) ( )

( )− ( )− ( )

( ) ( ) ( )

( )+ ( ) ( ) ( )

( )

( )−
( )+

( ).

Thus,
∫

( ) ( )

( )− ( )

=− ∫
( )

( )

( )+

( ) − ℎ( ) − ( )

+ ∫

( ) ( )

( )

( )
( )

+

( )

− ∫

( )+

( )−
(K1)

,

where
( ) = (0) − ( ) + ∫
+∫

( )

( ) ( )

From Eqs. (5.3), (5.4), we have lim

( )

( ) ( )

( )+∫

( )

( ) ( )

( )

( ).
( )

→

= 0 a.s.

Substituting Eq. (K1) into Eq. (J1) yields
( )

ln ( ) = ∫
∫

( ) ( ) ( )−
( )+

( )+

( )

( ) − ℎ( ) − ( )

( )
195

− ∫
( )

− ∫

( ) ( )
( ) ( )

( )+

( ) ( )

−

( )− ( )−

( )
where

( )

( )=

( )

( )

+

(K2)

,
( )

( ). One can get that lim

( )+

→

= 0 a.s., it then follows from the

periodicity of periodic functions and Eq. (K2), we have
( )

( ) ( ) ( )−

lim sup ln ( ) ≤ ∫

→

Hence, when ℜ < 0, we have lim

( )+

( ) = 0 a.s. Then lim

→

= ℜ a.s.

( )

( ) = 0 a.s. follows from Lemma

→

5.1.
Next, we prove (ii). From Eq. (K2), we have
( )

( ) ( ) ( )−

ln ( ) ≤ ∫

( )+

− ∫

( )+

( )

( ) − ℎ( ) − ( )

( )

( )

+

( )

,

(K3)

If ℜ > 0, then the following result is obtained from Liu and Bai (2016)
lim sup ∫

( )

→

≤

ℜ
a.s.
( )

(K4)

For Eq. (5.4), we obtain
( )

lim sup ln ( ) = − lim

∫

( )+

( )

≤ − lim

∫

( )+

( )−

→

→

→

( ) ( )

+ lim sup ∫

( ) ( )

→

( )

( ) ( )

=ℜ .

( )

(K5)

Obviously, one can get that lim

( ) = 0 almost surely if ℜ < 0. That is, for arbitrary 0 <

1 , there exist a set

with ℙ(

→

max

∫

( )

⊂

( )
( )

, ∫

( )

)≥ 1−

and a constant ℱ = ℱ( ) such that

< , for

∈

and > ℱ. By Eq. (K2), we have

the following equation for sufficiently large :
( )

ln ( ) ≥ ∫
− ∫

( ) ( ) ( )−
( )+

<

( )+

( ) − ℎ( ) − ( )
196

( ) −
( )

( )

+

( )

,

Due to the arbitrariness of

and Lemma E.1, we obtain
lim inf ∫

( )

→

≥

ℜ

a.s.

This completes the proof of Theorem 5.1 (ii).
Finally, we prove (iii). By Eq. (5.3), it follows that the left side of Eq. (K5) is nonpositive,
from Assumption 1, we have
lim sup ∫

( )

( )+

≤ lim ∫
→

→

+ a.s.

( )

(K6)

Then we take upper limit on both sides of Eq. (K2) and combine with Eq. (K6) yields
( )

lim sup ln ( ) ≥ lim ∫
→

→

( ) ( ) ( )−
( )

lim sup ∫
→

( )+

( )

− lim inf ∫

( )

→

≥ lim ∫

( ) ( ) ( )−

−

( )+

→

lim ∫
→

( )+

−

−

( )

( )

− lim inf

−

=

→

∫

( )

Applying Eq. (J1) again we have
ℜ

∫

( )

lim inf ∫

( )

≥

Obviously, lim inf ∫

( )

> 0 a.s. if ℜ >

→

→

This completes the proof.

197

( )

.

ℜ

−

. .

−

.

APPENDIX L. The proof of Theorem 5.2
Proof. Since model (5.2) exists a global positive solution, according to Theorem 3.8 in
Khasminskii (2011), we need to construct a
bounded closed set
max {(

(

) ,(

such that ℒ ≤ −1 on ℝ \

) },

∈{ , , }

=

,

.

=

) }, there exist a constant

construct the

− periodic function

Under

( , , , , )∈

and a

. Let

= min

min{

, ( − ℎ) , ( − ) } > max{(

> 0, satisfying

, ( − ℎ) , ( − ) , ( ) =

− ( ) (

=

) ,

+ 1) > 0. Hence, we

−function : ℝ → ℝ as follows:
( , , , )=

[− ( )( +
− ln

≡

+ ) − ln

( +

+
+

+

−

ln

+

+

( )]

+

)+(

+

) +

+ )

,

where
= ℜ max{2, },
=

sup

(

−

+

+

)+

(

( , , )∈ℝ

+ (

Γ +

) .

Here, Γ is defined as in Eq. (L1), and define
( )+

ℜ . Obviously,

( ) −

( , , , ) is also

lim

ℝ \

=

,

×

( )− ( )−

− periodic function on [0, +∞) and

−periodic satisfying
→ ,( , , )∈ℝ \

where

is a

= ( ) ( ) ( )−

,

×

,

inf ( , , , ) = +∞,

. We need to find a closed set

. By It ’s formula, we obtain
198

∈ ℝ such that ℒ ≤ −1 on

ℒ

=− ( )

( )( ( ) − ) + ℎ( ) − ( )
( )

( ) − ( )−

( )

( )

−

( )

− ( )−

( )

=[ ( ) ( )− ( )−

( )+

( )

( )] +

−

( ) ( ) ( )−

≤−

( )− ( )

ℜ +

( )

( ) +

( )

( )( +

( )

( )

( )

+

(t)

− ( )

+

( )

−[ ( ) + (ℎ( ) − ( )) ( )] −
−

+

( )

−

( )

( )− ( )

+

( )

( )+

( )

( )

+

( )− ( )−

( )
( )

( )

( )+

( )

.

+

Hence,
ℒ

≤−

ℜ +

.

+

Similarly, we have
ℒ

( )

=−
≤−

( )

+ ( ) − ℎ( ) − ( ) + ( ) +

+

) ,

+ (

+

( )

and
ℒ

=( +
+ (
≤( +
=

+ )

[ ( )( ( ) − ) + ℎ( ) + ( ) − ( ) − ( ) ]

+ 1)( +

+ ) [

+ )
( +

( )

− ( +
+ )

≤Γ −

( +

+ )

≤Γ −

(

+

+

+

+ ) + ( ) (

− ( ) (

−

( )

+

+ 1) ( +

).
199

( )

]

+ 1)( +
+ )

+ )

(t)

+ ) −

where
Γ =

( +

sup

+ )

(

−

+

) < ∞.

+

(L1)

( , , )∈ℝ

The above equations yield that
ℒ ( , , , )≤−
−

PZ + (

+

+

)+Γ +

≤ , ≤

≤ , ≤

ℜ +
1
2

(

+

) −
1
+ (
2

) .

Define a closed set as follows
= ( , , )∈ℝ : ≤
where

,

≤

> 0 satisfies the following conditions:
−

(L2)

+ Θ ≤ 1,

−

+ Θ ≤ −1,

0<

<

ℜ

0<

(L3)

,

(L4)

,

<

(L5)

and Θ , Θ are constants that are given in the expression (L6) and (L7), respectively.
Divide ℝ \

into six domains:

= {( , , ) ∈ ℝ : 0 <

< },

= {( , , ) ∈ ℝ : 0 <
= ( , , )∈ℝ :
Obviously,

=⋃

< },
>

1

= ( , , )∈ℝ :
= ( , , )∈ℝ :

,

< },
>

>

1

1

,

.

. Then, we need to prove that ℒ ( , , , ) ≤ −1 on each

1, 2, ⋯ ,6, which is equivalent to the result on
When ( , , , ) ∈

= {( , , ) ∈ ℝ : 0 <

× .

× , according to Eq. (L2) we have
200

× , =

ℜ

ℒ ( , , , )≤−
−

1
2

+(

+

(

+

+

) −

+

1
+ (
2

)+Γ +

≤−

+

≤−

+

≤ −1,

+(

+

) −

(

+

)

where
Θ =

sup

)+Γ +

+

( , , )∈ℝ

+ (

) .

(L6)

Similarly, when ( , , , ) ∈

or

×

× , using the definition of

, Eqs. (L4) and

(L5), we obtain
ℒ ≤−
When ( , , , ) ∈

× ,

×

ℜ ≤ −1.

or

× , using Eq. (L3), we have

ℒ ≤−

+ Θ ≤ −1,

where
Θ =

sup

(

+

+ (

) .

−

+

)+

+(

+

) +Γ +

( , , )∈ℝ

Obviously, for a sufficiently small

(L7)
such that

ℒ ( , , , ) ≤ −1, ∀( , , , ) ∈ (ℝ \

)× .

Then, by Theorem 3.8 in Khasminskii (2011), we obtain that model (5.2) has a nontrivial positive
−periodic solution.
This completes the proof.

201

APPENDIX M. The proof of Theorem 6.1
Proof. Clearly, the solution of model (6.2) will remain in ℝ for all ≥ 0 almost surely. For any
| (0)| < , define a stopping time as
= inf{ ≥ 0, | ( )| > }.
Then

↑ ∞ a.s. as

→ ∞. Define a

−function as follows:

( , , )=
where ( , , ) ∈ ℝ and
( , , )=
( )

+

( ) ( )+ℎ ( )

( ( ))
( )
( ( ))

+

( ( )) +

( ( ))

( )
As 0 <

+

,

∈ (0,1). By generalized It ’s formula, we have
+

( )

+

( )+

( )

+
+

( )+

( )
− ( ( ))

( )

+

+ ℎ− ̂

( )+

( )

−
(

+

( )+

( ( ))
+

( ) − ( ) −

+

( ( ))
( )
( ( ))

( ) ≤

( ( ))

+

)

−

( ( )) +

( ( ))

( )+

−

+

+

( )

( ),

( )

( )+

< 1, there exists a ( ) > 0 such that
( , , )≤

( )
( )

+

( )+

( )

( ).

We integrate both sides of the above expression to get
( ( ( ∧
where

), ( ∧

), ( ∧

is the stopping time. As

))) ≤

(0), (0), (0) +

∫

∧

→ +∞, we have

( ( ), ( ), ( ) ≤

(0), (0), (0) +

which gives
202

( ),

( )

,

( ( ( ), ( ), ( ))) ≤

(0), (0), (0) + ( ).

Note that
| ( )| = (

( )+

( )+

( )) ≤ 3 max

,

,

≤3

Thus,
| ( )| ≤ 3
Setting

(0), (0), (0) + ( ) < +∞.

= , then there exists a constant

> 0, such that

lim sup | ( )| ≤

→

Using Chebyshev’s inequality and taking

=

with 0 <

ℙ{| ( )| > } ≤

| ( )|

.
< 1, we obtain
,

which implies that
lim sup ℙ{| ( )| > } ≤

→

This completes the proof.

203

= .

+

+

,

APPENDIX N. The proof of Lemma 6.1
Proof. Define a

-function from ℝ to ℝ as,
( , , )=

.

( ) −

( )

+ ℎ ( ) −

+

( )

+

( )+

( )

( ).

By generalized It ’s formula, we have
( +

+ )= −

( ) −

( )

( )

( )

( )

+
( )

( )+

( )

( )

+

−

For any positive constant , it follows the differential operator ℒ acts on (1 + ) to give
ℒ(1 + ) = (1 + )
( ) −

( )

)

( )

+

( )

( )

+

= (1 + )
( )

( )

−

+

( ) −

( )

+

−

( ) −

( )

+ ℎ ( ) −

( )

+

( )

( ) −

( )

+ ℎ ( ) −

( )

+

( )

+

]+

[

+

+

( )

(1 + )

( )

( )

+

( )

+

(

+

( )
)

+

(1 +

( )

+

( )

.

= (1 + )
where
≤ℎ −

−ℎ

+

[

+

+

+

+

] < max{

,

,

} ,

,

}) −

−ℎ−

max{

,

+

+

As
[
we obtain
≤ (ℎ + max{

,

204

,

}

.

].

Thus, for a sufficiently small , we have
(1 + ) ) =

ℒ(

. Set

(1 + ) +

ℒ(1 + )

=

(1 + )

( (1 + ) +

)

≤

(1 + )

( −

)

+

≤

(1 + )

= (1 + )

. Obviously,

,

Where 0 <

≤

function for

− 2 > 0 while monotone decreasing function for 0 <

maximum of the function

− 2 ≤ 0 . Hence, the

takes the form

.

max 1,

Thus, ℒ(

is a monotone increasing

(1 + ) ) can be rewritten as
(1 + ) ) ≤

ℒ(

(1 + )

,

=

where

=

.

max 1,

Therefore,
(

(1 + ) ) ≤ 1 + (0)

(

− 1).

1+ ( )

≤

+

Hence, we have
lim sup ( ( ) ) ≤ lim sup

→

→

Note that,
205

.

( +

+ ) ≤3 (

+

+

) = 3 | ( )| ,

Thus, we get
lim sup

→

| ( )|

≤ 3 lim sup
→

This completes the proof.

206

(| ( )| ) ≤

= Π.

APPENDIX O. The proof of Theorem 6.2
Proof. According to Theorem 6.1, we have lim sup ℙ {| ( )| > } ≤ , which implies that
→

lim sup ℙ {| ( )| ≤ } ≥ 1 − .

→

By Lemma 6.1, we have lim sup | ( )|
→

≤ Π. Hence, for any given 0 <

, we have
ℙ{| ( )| < Κ} = ℙ | ( )| >

≤

| ( )|

which yields that
lim inf ℙ{| ( )| ≥ Κ} > 1 − .

→

This completes the proof.

207

= ,

< 1 and Κ =

APPENDIX P. The proof of Lemma 6.2
Proof. Eq. (6.4) can be rewritten as
= ,
where

∈ℝ ,

=(

,

,⋯,

=

Obviously,

∈

For each

×

≡

(1) −
−
⋮
−

⋯ −
⋯ −

−
(2) −
⋮
−

=

×

:

≤ 0, ≠

∈

×

(1) −
−
⋮
−

.

−
(2) −
⋮
−

⋯ −
⋯ −
⋮
⋯

=

( )+∑

≥

From Lemma 5.3 in Mao and Yuan (2006), we have
2.10 in Mao and Yuan (2006) that
=

.

⋮
( )−

.

, and the sum of each row is

( )−∑

equation

⋮
( )−

⋮
⋯

∈ , consider the leading principal sub-matrix

=

Then we have

) and

is a nonsingular

has a unique solution

( ), = 1, 2, ⋯ , .
> 0. It follows from Theorem

-matrix. That is, for any vector

= ( (1), (2), ⋯ , ( )) ≫ 0.

This completes the proof.

208

∈ ℝ the

APPENDIX Q. The proof of Lemma 6.3
Proof. Applying It ’s formula to model (6.2) yields that
( )

( ) +

( ) +

( )

+

ln ( ) = − ∫
+ ∫
( )

lim

→

⊂

( )

( )

( )
( )

( )

.

(Q1)
( )

( ) −

( ( )) ( )
( ) ( ) ( ( )) ( )

( )=∫

( )

( ) are two martingales. If

( ) = 0 a.s., then for arbitrary 0 <

< min 1, +

, there exist a measurable set

with (

= ( ) > 0 such that for

( ) +

ln ( ) = − ∫
where

( )

( )
( )

( )=∫

( ) ,

( )

) ≥ 1 − and a constant

1
1+

( ( )) ( )
( ) ( ) + ( ( )) ( )

1
+ ln (0) < ,

+

∈

.

(Q2)

and >
≥ .

Combine the above inequality with Eq. (Q2), we get
ln ( ) =
≤

ln (0) − ∫
− ∫

( ) +

( ) +

( ) −

( )

for ≥ . By Lemma E.1 (i), we have lim
→

+

( ( )) ( )
( ) ( ) ( ( )) ( )

+

( )

( )

,

( ) = 0 a.s., and it follows that the conclusion holds

for model (6.2).
This completes the proof.

209

APPENDIX R. The proof of Theorem 6.3
Proof. From Lemma I.2 we have
( )

lim

→

Define ( ,

,

,

= 0 a.s., = 1, 2.

) = ( )[ ( ) + ( ) + ( )]. By virtue of Lemma 6.2, we get

( , , , )
={ ( ) ( ) ( )−[ ( ) ( )−∑∈

( )] ( )}

−[( ( ) − ℎ( )) ( ) − ∑ ∈

( )] ( )

+ ( ) ( ) ( )

( )) ( ) − ∑ ∈

+ ( ) ( ) ( )

=

( )] ( )

( ) ( ) ( )− ( ) ( )−
( )− ( )− ( )

( ) ( ) ( )

( ) ( )

( )+

+ ( ) ( ) ( )

( )

( ) − [( ( ) −

( )

( ) − ℎ( ) − ( )

+ ( ) ( ) ( )

( ) ( )−

( )+ ( ) ( ) ( )

( )+
( )+

( ).

Consequently,
∫

( )

( )

( )

( ( ))

( )

( )

( ( ))

( )

( ) +

( ) −ℎ ( ) −

− ∫

( ) +

( ) −

+ ∫

( )

=− ∫

( )

( ) −
( )

,

( )

+

( )

( )+

(R1)

where
( ) = (0) − ( ) + ∫

( )

( )

( )

( )

( )+∫

∫

( )

( )

According to Eqs. (6.2) and (6.3), we have lim
→

( )

( )

( )

= 0 a.s. Substituting Eq. (R1) into Eq. (Q1)

yields
( )

ln ( ) = ∫

( )

( )

( ) −
210

( ).

( ) +

( ( ))

−

( ( )) ( )
( ) ( ) ( ( )) ( )

∫

where

( )=

( )

( )

( )

− ∫

( )

( )

( )

+

( ). Thus, lim

( )

( )+

( ) +

− ∫

→

( ) +

( ) −ℎ ( ) −
( ) −

( ) −

( )

,

(R2)

= 0 a.s. By the ergodic theorem of Markov chain

( ) and Eq. (R2), we obtain
( )

lim sup ln ( ) ≤ ∑ ∈

( ) ( ) ( )−

→

Thus, we have lim
→

( )+

( )

( ) = 0 a.s. if ℜ < 0. Consequently, lim

= ℜ a.s.

( ) = 0 a.s. follows from

→

Lemma 4.3.
Now, from Eq. (R2), we have
( )

ln ( ) ≤ ∫

( )

− ∫

( ) +

( )

( ) −

( ) +

( ) −ℎ ( ) −

( )

( )

( )

( )

+

( )

,

(R3)

If ℜ > 0, then we have following result from Liu and Bai (2016)
lim sup ∫

( )

≤

lim sup ln ( ) = − lim

∫

→

( ) ( )

∑ ∈

( )

( )

( )

=

ℜ

a.s.

(R4)

For Eq. (Q2), we obtain
( )

→

→

( ) +
( )

lim sup ∫

( )

→

∫

( ) +

lim sup ln ( ) ≤ − ∑ ∈

( )+

≤ − lim
→

( )

( )

+

( )
( )

( )

( ) −

( )
( )

.

Thus, one can get that
( )

→

211

( )−

( )
( )

= ℜ a.s.

(R5)

Obviously, lim
→

with ℙ(

⊂

( ) = 0 a.s. if ℜ < 0. That is, for arbitrary 0 <

) ≥ 1 − and a constant ℱ = ℱ( ) such that
max

for

∈

< 1, there exist a set

∫ ̌ ( )

, ∫

( )
( )

< ,

( )

and > ℱ. By Eq. (R2), for sufficiently large , we have
( )

ln ( ) ≥ ∫

( )

( ) +

∫

( )

( ) −

( ) −ℎ ( ) −

( ) +
( )

( ( )) −

( )

( )

+

Since is arbitrary, from Lemma 4 in Liu and Bai (2016), we obtain
lim inf ∫

→

( )

≥

∑ ∈

( ) ( )

This completes the proof.

212

( )

( )

( )

=

ℜ

a.s.

−
( )

,

APPENDIX S. The proof of Theorem 6.4
Proof. By virtue of Assumption 1 and the positive definiteness of the diffusion matrix of model
(6.2), and using Theorem 3.13 in Khasminskii (2011), it is obvious that we only need construct a
nonnegative function

and a bounded closed

∈

such that ℒ ≤ −1 for ( , , , ) ∈

× .
For simplicity, we introduce the following notations:
ℜ = ℜ − ∑ ∈

( ) ,

( )+

Denoted by
= min

∈{ , , } {(

=
For min

, ̂ − ℎ, −

−function : ℝ ×

( , , , )=

where

=(

,

,⋯,

.

=

+

+

> 0 such that

+ 1) > 0.

→ ℝ by
+ ) − ln

( +

+

(

−

[− ( )( +
− ln

≡

) },

= ,

> max{( ) , ( ) , ( ) }, there exists a constant
=

Define a

,

, ̂ − ℎ, −

−

ln

+

+(

+ | |)]

+ )

,

) is to be determined later and

> 0 is given by

= ℜ max{2, },
=

sup

−

(

+

+

)+

( , , )∈ℝ

+ ( ) .
213

(

+

)+(

+ ) +Γ +

with
Γ =

sup

( +

+ )

(

−

+

) < ∞.

+

( , , )∈ℝ

Obviously, ( , , ) exists a minimum point ( , , ̅ , )in ℝ × . Thus, we can define a
nonnegative function : ℝ ×

→ ℝ as follows:

( , , , ) = ( , , , ) − ( , , ̅ , ).
By It ’s formula, we get
ℒ

=− ( )

( )( ( ) − ) + ℎ( ) − ( )

∑∈

( )( +
( )
( )

( )

+ ) −
− ( )−

−[∑ ∈

( )

( ) − ( )−

( )

( )
( )

( )] +

( )

( )

+

( )

( )

( )

( )

( )

( ) ( ) ( )−

−

( ) −

− ( )

+∑ ∈

+∑∈

( )

( ) + ℎ( ) ( ) − ( ) ( )] −

( ) + ( )

( )

( ) +

−

( )

( )

( ) +

=[ ( ) ( )− ( )−∑∈

( )− ( )

+

− ∑∈

( )+

( )−

( )− ( )−

( )+

( )
≤

PZ + ∑ ∈
( ) ( ) ( )−

= −Ψ + ∑ ∈

− ∑∈

+

( )− ( )−
+

where Ψ = ( ) ( ) ( ) −
Note that

( )+

+

( )− ( ) − ( )+ ( )

( )+

−

( )

,
( )+

( ) −

[Ψ − ( Ψ) ] = 0 and ∑ ∈

( )+

( ) .

= 1, where Ψ = (Ψ , Ψ , ⋯ , Ψ ) ,

(1, 1, ⋯ ,1) ∈ ℝ . By Lemma A. 12 in Khasminskii (2011), the following equation:
= Ψ − ( Ψ) ,
214

=

has a solution

=(

,

) ∈ ℝ . Hence, one can get that

,⋯,

−Ψ + ∑ ∈

= −∑ ∈

Ψ =−

+

=−

ℜ

.

Thus,
≤ −Ψ + ∑ ∈

ℒ

+

ℜ

+

.

+

By a direct computation, we get
ℒ

( )

=−
≤−

ℒ

=( +

+ )

+ ) [

1)( +

≤( +

+ ( ) − ℎ( ) − ( ) + ( ) +

+

( )

+ ( )

+

[ ( )( ( ) − ) + ℎ( ) + ( ) − ( ) − ( ) ] + (
( )

+ )

( +

=

( )

+ )

≤Γ −

( +

+ )

≤Γ −

(

+

( )

+

+

( )

+

]

−

( +

+ ) +

−

−

(

(

+ 1) ( +

+ )

+ 1)( +
+ )

).

+

The above inequations yield that
ℒ ( , , , )≤−
−

ℜ

+

(

+

+(

+ ) −

+

)+Γ +

In order to validate that ℒ ( , , , ) ≤ −1 on some

+ ( ) .

× , we define a compact subset

by
= ( , , )∈ℝ : ≤
where

≤

> 0 satisfies the following conditions
215

1

, ≤

(S1)

1
≤ , ≤

≤

1

,

−

(S2)

+ Θ ≤ 1,

−

+ Θ ≤ −1,

(S3)

ℜ

(S4)

0<

<

0<

<

,

,

(S5)

Here, Θ and Θ are constants given explicitly in the expression (S6) and (S7), respectively.
We further divide ℝ \

into six domains as follows:

U = {(N, P, Z) ∈ ℝ : 0 < N < ϵ}, U = {(N, P, Z) ∈ ℝ : 0 < P < ϵ},
U = {(N, P, Z) ∈ ℝ : 0 < Z < ϵ}, U = (N, P, Z) ∈ ℝ : N >
U = (N, P, Z) ∈ ℝ : P >
Clearly,

1
1
, U = (N, P, Z) ∈ ℝ : Z > .
ϵ
ϵ

. Next, we prove that ℒ ( , , , ) ≤ −1 on each

=⋃

which is equivalent to the result on
For ( , , , ) ∈

1
,
ϵ

× , = 1, 2, ⋯ ,6,

× .

× , from Eq. (S1) we have
ℜ

ℒ ( , , , )≤−

−

1
2

≤−

+

(

+

+

≤−

+(

+ ) −

+

)+Γ +

+

1
+ ( )
2

≤ −1,

where
Θ =

+(

sup

+ ) −

(

+

+

)+Γ +

+

( , , )∈ℝ

(S6)

( ) .
Similarly, when ( , , , ) ∈

×

or

× , using the definition of
216

, Eqs. (S4) and

(S5) yields that
ℒ ≤−
For ( , , , ) ∈

× ,

×

or

1

ℜ ≤ −1.

4

× , using Eq. (S3), we have

ℒ ≤−

1
4

+ Θ ≤ −1,

+

+

)+

where
Θ =

sup

−

(

+(

+ ) +Γ +

( , , )∈ℝ

(S7)

+ ( ) .
Obviously, for a sufficiently small
ℒ ( , , , ) ≤ −1, ∀( , , , ) ∈ (ℝ \

)× .

In view of Theorem 3.13 in Zhu and Yin (2007), the stochastic process ( ( ),
( )) of model (6.2) is ergodic and it admits a unique stationary distribution in ℝ × .
This completes the proof.

217

( ),

( ),

APPENDIX T. The proof of Theorem 6.5
Proof. By Eq. (6.3), it follows that the left side of Eq. (R5) is nonpositive, and
( ) +

− lim ∫
→

( )

( )

+ lim sup ∫

( )

→

( )

( )

( )

≤ 0.

( )

From Assumption 2, we have
lim sup ∫

( )

( ) +

≤ lim ∫
→

→

+ a.s.,

( )

Further,
lim sup ∫

≤∑ ∈

( )

→

( )+

( ) a.s.

(T1)

Taking upper limit on both sides of Eq. (R2) and combine with Eq. (T1), we get
( )

( )

lim sup ln ( ) ≥ lim ∫
→

→

( )

lim sup ∫

( )

→

≥∑ ∈

( )

( ) −

( )

− lim sup ∫
→

( ) ( ) ( )−

− ∑ ∈

( ) +

( )+

( )+

( )

−

− lim inf ∫ ̌ ( )
→

( )

( ) − lim inf ∫ ̌ ( )

− .

→

Using Eq. (Q1), we get

lim inf
→

1

ℜ −
( )

≥

Obviously, lim inf ∫
→

∑ ∈

( )+
̌

( )

> 0 a.s. if ℜ >

1
2

( )
−
̌

218

ℜ
̌

−
̌

. .

. Thus, in view of Theorem 6.4,

phytoplankton is persistent in mean due to the ergodicity of model (6.2).
This completes the proof.

=

APPENDIX U. Auxiliary results
We assume that a product equals unity if the number of factors is zero. For a bounded and
continuous function ( ), we set
= sup ( ),

= inf ( ).

Moreover, if ( ) is integrable, we define
〈 〉= ∫

( )

,〈 〉 = ∫

.

( )

Now, to facilitate our further discussions, we present the following definitions and lemmas.
For an -dimensional stochastic differential equation,
( ) = ( ( ), )
with initial value ( ) =

+ ( ( ), )

( ),

(U1)

≥

, where ( ) is an -dimensional standard Brownian motion,

∈

the associated differential operator ℒ is defined by
+∑

ℒ=

( , )

+ ∑ ,

[

.

( , ) ( , )]

(U2)

Definition U.1 (Zhang and Tan, 2015). Consider the following stochastic-impulsive differential
equation:
( )=
, ( )
( )− ( )=
with the initial value (0) ∈

+
, ( )
( ), = ,

( ), ≠ ,
= 0,1,2, ⋯

. Then a stochastic process ( ) = (

> 0,

( ),

(U3)

( ), ⋯ ,

( )) ,

∈ [0, ∞) is said to be a solution of model (U3) if
(1)

( ) is ℱ adapted, and continuous on (0, ) and every interval ( ,
, ( ) ∈ℒ (

;

), ( , ( )) ∈ ℒ (

) , where each ℒ (

;

measurable ℱ -adapted process ( ) and satisfies ∫ ∣ ( ) ∣
(2) For any

,

∈ ℕ, (

) = lim
→

( ) exists and (
219

;

)∈
) is

,

.

∈

valued

< ∞ almost surely for every .

) = lim
→

( ) with probability one.

(3) For almost every ∈ (0, ), ( ) obeys the following integral equation:
( ) = (0) +
For all ∈ [ ,

( , ( ))

+

( , ( ))

( ).

∈ ℕ, ( ) obeys the following integral equation:

],

( )= (
Lemma U.1. For any

)+

( , ( ))

+

( , ( ))

( ).

> 0, the following inequality holds (Dalal et al., 2008):
≤ 2( + 1 − log ) − (4 − 2 log 2).

Proof The proof directly follows from Dalal et al. (2008).
Next, we present the definition of periodic Markovian process.
Definition U.2 (Zhao and Shao., 2021; Zhang et al., 2017b). A stochastic process ( ) =
( , )(−∞ < < +∞) is said to be periodic with period
numbers

,

,⋯,

, the joint distribution of random variables (

ℎ) is independent of ℎ, where ℎ =

,

Remark U.1. A Markov process ( ) is
is -periodic and the function
( , )=∫
for every

if for every finite sequence of
+ ℎ), (

+ ℎ), ..., (

+

= ±1, ±2, ⋯.
-periodic if and only if its transition probability function

( , ) = { ( ) ∈ } satisfies the following equation:
( ,

) ( , , + , )≡

( + , )

∈ ℬ (Khasminskii, 2011).

Lemma U.2 (Khasminskii, 2011). Let all the coefficients of the following It ’s differential
equation:
( ) = ( , ( ))

+ ( , ( ))

( ),

(U4)

are -periodic in , and satisfy the linear growing condition and the Lipschitz condition in every
cylinder

×

for

> 0 , where

function

= ( , ) which is twice continuously differentiable with respect to

= { ∶∥

220

∥ ≤ } . Further, assume that there exists a
and once

continuously differentiable with respect to in

×

, and

is periodic in and satisfies the

following conditions:
( , ) → +∞,

(U5)

→ +∞,

‖ ‖

ℒ ( , ) ≤ −1,

,

then there exists a solution of model (U4) which is -periodic Markovian process.

221

(U6)

APPENDIX V. The proof of Theorem 7.1
Obviously, for any fixed ≥ 0, there exists an integer

∈ {0, 1, 2, ⋯ } such that

≤ ≤

( + 1) .
As

=
+

and

+

=

(

Since [0, ∞) ∩ { ,

)

, we have

+

=

+

=⋯=

∈ ℕ} = { , , ⋯ ,

+

,

=

+

+( −1)

=⋯=

.

} , therefore, there exists a positive integer

∈

{1, 2, ⋯ , } such that
,
) ,

(

(

∈ [ , ( + 1) ),

,⋯,
) ,⋯,

(

)

∈ [( + 1) , + ).

Following Feng et al. (2021), we will show that model (7.3) has a unique globally positive
solution ( ( ),

( ),

( )).

Obviously, the coefficients of model (7.3) satisfy the local Lipschitz condition. Thus, there
exists a unique local solution of the model on [0,
this solution is global, we need to verify
of the

(0),

(0) and

), where

= ∞ a.s. Let

(0) lie in the interval [1/ ,

is the explosion time. To show that
> 1 be sufficiently large such that all

]. For any positive integer ( ≥

), we

define the stopping time as follows:
= inf{ ∈ [0,

]:

( ) ∉ (1/ , )

Now, we set inf ∅ = ∞ and let
Obviously, if

= ∞, then

function

∶

→

( ),

( ) =

( ),

≤

a.s.,

( ) ∉ (1/ , )

is increasing when
= ∞ and ( ( ),

( ) ∉ (1/ , )}.

→ ∞ . Also, let
( ),

( )) ∈

= lim

.

a.s. Define a

-

→

as follows:

( ) + 1 − log

( )+

( ) + 1 − log
222

( )+

( ) + 1 − log

( ).

For any

> 0, employing It ’s formula on ∈ [0,

( ),

( ),

( ) =ℒ

1
( )

+ 1−

+ 1−

∧ ], we get
( ) ( )

( ) ( )

( )

( )

( )+ 1−

( ) ( )

( )

( ),

where
ℒ = ( ( ) − 1) − ( ) + ∑
+( ( ) − 1)
+( ( ) − 1)

( )

log 1 +

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

+ ∑

− ()

( )

( )

( )

( )

− ( )− ( )

log 1 +

− ( )− ( ) ( ) +

( )

+

( )

( )
( )

( )

+

( )

+

( )

( ).

On manipulating the terms, we get
ℒ =− ( ) ( )+ ( )+
( )

−

( )

( )
( )

− ()

( )
( )

( )

( )

( )

+ ()

+

( ) ∑

( )

( )

( )

( )

( )

− ( ) ( )+ ( )− ( )
+

( )

( )

( )

( )
( )

( ) ∑

( )
( )

( )

− ()

( )

( )

( )

( )

+

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

− ( ) ( ) + ( ) ( )+ (

( )
( )

+ ()

( )

log 1 +

( )
( )

− ∑

log 1 +
( )

− ∑

log 1 +

log 1 +

( )
( )

( )

− ( ) ( )+ ( )

( )+

( )+

( )).

From the above, we get
ℒ ≤

+

+

1
+ [(
2

+

( )

1

) +(

) +(

) ]+

log 1 +

+

+

This yields,
223

( )

1

log 1 +

( )

+

( )

+

( )

.

ℒ ≤

1

+

log 1 +

+

( )+

+

+

+

1

( )

log 1 +

( ).

That is,
+2

∑

+ ∑

log 1 +

[ ( ) + 1 − log

( )]

+2

+

+

[ ( ) + 1 − log

( )]

ℒ ≤

log 1 +

[ ( ) + 1 − log

+

≤

+

( ),

( ),

+

+ [(

) +(

) +(
log 1 +

( )]

( ) ,

with
=

+

+

=2

∑

log 1 +

+

⋁ ∑

( ) ≤

+

( ),

) ],
⋁2

+

+

.

Thus, we have
( ),

( ),

( ),

+ ( )[ ( ) − 1]

( ) +

( )[ ( ) − 1]

( )[ ( ) − 1]

( )+

Integrating both sides of the above inequality from 0 to

∧

( )

( ),

and taking the expectation, we

obtain
∧

,

∧

,

∧

(0),

≤

(0),

(0) +

∧

( ),

+

( ),

( )

.

In view of the Gronwall’s inequality, we get
∧

,

∧

,

∧

≤
224

(0),

(0),

(0) +

.

Let Ω ( ) = {

∈Ω∶

( , ) equals either
,
(0),

( ) ≤ } . Then, for

=

,

≥

( , ) or

( , ) or

or 1/ . Also, we have

,

,

,

(0),

(0) +

≥ ( + 1 − log ) ∨

,

≥

( )

≥

≤

where Ω is the indicator function of Ω . Let

+ 1 + log

,

,

,

( ),
,

( ),

→ ∞, then lim {

≤ } = 0, which implies that

→

{

≤ } = 0. Since

{

= ∞} = 1 , which will ensure that model (7.3) admits a unique positive solution

( ( ),

( ),

Let

> 0 is arbitrary, we have {

,

≜

< ∞} = 0. Now, it suffices to show that

( )) for all ≥ 0 a.s.
( ) ( ),

( )=

( ) ( ) and

( )=

( ) . In view of Lemma 7.1,

( )=

( ( ), ( ), ( )) is the solution of model (7.2). Obviously,

( ), ( ), and ( ) are continuous

on the intervals (0, ) and ( ,

, we have

( )=

( ) ( )

( )

( ) ( ) − ( )−

=

( ) − ( )− ( )

( )
( ) ( ) ( )
( )+ ( ) ( )

+

( ) ( )

( ),

( ) ( )
( )

( ) ( )

( ) ( )

+

=

( ) ( )

( ) ( ) ( )
( ) ( )
− ( )−
( )+ ( ) ( )
( )+ ( ) ( )

( )

( )

+

( )

( )=

= ( )
,

+

=

+

For =

∈ ℕ. Also, for ≠

),

( )

( ) ( )

( ) ( )
−
( ) ( )

( )
( )

( )− ( )

∈ ℕ, we have
225

( )
( )

+

( ) ( )

( ),

( )

(

(

) = lim

( ) ( )=

= (1 +

)

→

) = lim

(

) (

( ) ( )= ∏

→

1+

(

1+

) = (1 +

)

) ( ),

1+

∏

1+

(

) = ( ).

(

)=

( ).

Similarly, we have
(

) = (1 +

) ( ),

Thus, one can conclude that model (7.2) has a solution ( ( ), ( ), ( )) for ≥ 0 with the initial
value ( (0), (0), (0)) ∈

.

Next, we show that the solution of model (7.2) is unique. For ∈ [0, ], model (7.2) can be
written as follows:
( )= − ( ) ( )−

⎧
⎪

( )=

⎨
⎪
⎩

( ) ( ) ( )

( )=

( ) ( )
( ) ( ) ( )
( )

( )

( ) ( ) ( )
( )

( ) ( )

+

( )

( ) ( ) ( )

− ( ) ( )−

( )

with the initial value ( (0), (0), (0)) = (

+

( ) ( )

( ),

+

( ) ( )

( ),

( )

− ( ) ( )− ( ) ( )
,

( ),
(V1)

,

) . The above model can be written as

+

( )

follows:
⎧
⎪
⎨
⎪
⎩

( )

( )= − ( )− ( )
( )=
( )=

( )

( )

( )
( )

−

( )

( )

( )

− ( )− ( )
( )

( )

( )

( )

( )

( )
( )

−

( )

− ( )− ( )

with the initial value ( (0), (0), (0)) = (log

( )

( ),
+

( )

( ),

+

( )

( ),

( )

−
, log

, log

(V2)

).

As the coefficients of model (V2) satisfy the local Lipschitz condition, so the model has a
unique solution ( (0), (0), (0)) by the theory of stochastic differential equations. Thus, in the
view of It ’s formula, ( ( ), ( ), ( )) = (

( ),
226

( ),

( )) is the unique solution of model

(V1).
For ∈ ( , ], model (7.2) can be transformed into the following form:
⎧
⎪
⎪
⎨
⎪
⎪
⎩

( )= − ( ) ( )−
( )=
( )=
(

), (

( ) ( ) ( )

− ( ) ( )−

( ) ( )

+

( )

( ) ( ) ( )

( ),
( ) ( )

( ),

− ( ) ( )− ( ) ( )

+

( ) ( )

( ),

) = (1 +

) (

( )

), (

( )

+

( ) ( )
( ) ( ) ( )
( )

( ) ( ) ( )

) (

( )

( )

), (1 +

), (

(V3)

) .

By using the same method, we can show that model (V3) has a unique solution for ∈ ( , ].
Thus, by the analogy, we obtain that model (7.2) has a unique solution with the initial value
( (0), (0), (0)) for each ∈ ( ,

].

This completes the proof.

227

APPENDIX W. The proof of Theorem 7.2
In view of Theorem 7.1, it suffices to prove that a periodic solution exists for the equivalent
model (7.3) without impulses. Thus, we only need to verify the conditions (U5) and (U6) of Lemma
U.2.
Define a

-function

∶ [0, +∞) ×

( )

( ,

,

,

)=

as follows:

→
( )

+ log

+

( )

+ log

+

+ log

=

+

+

,

where
( )=〈 ( )+

( )〉 − ( ) −

( ),

( )=〈 ( )+

( )〉 − ( ) −

( ),

( ) = −〈 ( ) +

( )〉 − ( ) −

( ).

(W1)

Owing to the periodicity of ( ), ( ), ( ), 1( ), 2( ), 3( ), 1( ), 2( ) and 3( )
are -periodic functions. We can show it by using the approach of Feng et al. (2021).
According to condition (U5) of Lemma U.2, we need to verify that
( ,

where

=

,

×

,

,

×

inf

)∈[ ,

,

,

)×(

\

)

( , ) → ∞,

→ ∞,

. Note that, this condition trivially holds as

1
1
+ log → +∞, . . → 0 , + log → +∞, . . → +∞.
2
Next, we prove condition (U6) of Lemma U.2. Using It formula, we get
ℒ

=−

( )

−

− ( )+ ∑
≤−

( )

−

( )− ( )+ ∑

log 1 +
( )− ( )−

( )

− ()

( )
( )

( )+ ∑
228

( )

log 1 +

− ()
( )

−
( )

( )

( )

( )

( )

( )

log 1 +

−

−

( )

+ ∑

ℒ

log 1 +

( )

=−

( )
( )

( )

−

(

)
( )

=−

( )

( )

+ ()
≤−

−

( )

,

( )

+ ()

( )

( )

( )

( )

( )

log 1 +

+ ()

− ( )−

( )

( )

( )

( )

( )− ( )−

( )+ ∑

+ ∑

log 1 +

+
( )

( )+ ( )

( )

( )

( )

( )

( )

( )

−

( )− ( )−

− ( )−

−
( )

− ( )−

( )

( )

( )

( ) −

( )−

log 1 +

− ( )− ( ) ( )−

− ( )− ( ) ( )−
( )

(

)

+

( )

.

In view of Eq. (W1), we obtain
ℒ =ℒ

+ℒ
( )

≤
−
−

−

− ( )−

( )

−

( )

−

+

( )

( )

− ( )−

log 1 +

( ) + ∑

−
( )

( )
( )

≤−

ℒ

( )
( )

( )

( )+ ∑

−

( )

( )

−

+ℒ
∑

( )

( )

−〈 ( )+

log 1 +
∑

( )−

−〈 ( )+

log 1 +

(〈 ( ) +

( )

( )〉 ) −
( )

−

( )〉 −

( )
(

229

)

−

+ ∑

( )〉 −
( )
(

)

log 1 +

−

( )

,

+ ∑

+

log 1 +

− ( )− ( )− ( )−

+

( )

−

+

( )

( )

.

−

Let
∗

= ∑

−

−

ℒ ≤

∗

log 1 +

+ ∑

log 1 +

−

−

−

−

+

+

+

.

Thus,
−

−

−

−

( )

( )

−
)

(

≜ ( ,

+

−

−

( )

( )

−

( )
(

(W2)

)

).

,

Obviously,
( ,

,

) → −∞, . .

→0

→0

( ,

,

) → −∞, . .

→ +∞

→ +∞

From (W2) and (W3), we take

=

(7.3) with

,

×

,

(0) > 0,

×

,

( ,

)∈

,

\ ,

. Therefore, the solution ( ) = ( ( ),

(0) > 0, and

(W3)

→ +∞.

sufficiently small such that

ℒ ≤ −1,
where

→0 ,

( ),

( )) of model

(0) > 0 is a positive -periodic Markov process.

The previous analyses yield that
( )=

( ) ( )= ∏

1+

∏

(1 +

)

( ),

( )=

( ) ( )= ∏

1+

∏

(1 +

)

( ),

( )=

( ),
230

where

( ) and

( ) are -periodic functions. Thus, ( ( ), ( ), ( )) is a positive -periodic

solution of model (7.2).
This completes the proof.

231