NONLINEAR DYNAMICS OF PLANKTON ECOSYSTEM WITH
IMPULSIVE CONTROL AND ENVIRONMENTAL
FLUCTUATIONS

by

He Liu
B.E., Xinyang College, 2010
M.E., Wenzhou University, 2014

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

UNIVERSITY OF NORTHERN BRITISH COLUMBIA
November 2021

© He Liu, 2021

ABSTRACT
It is well known that the density of plankton populations always increases and
decreases or keeps invariant for a long time, and the variation of plankton density is
an important factor influencing the real aquatic environments, why do these situations
occur? It is an interesting topic which has become the common interest for many
researchers. As the basis of the food webs in oceans, lakes, and reservoirs, plankton
plays a significant role in the material circulation and energy flow for real aquatic
ecosystems that have a great effect on the economic and social values. Planktonic
blooms can occur in some environments, however, and the direct or indirect adverse
effects of planktonic blooms on real aquatic ecosystems, such as water quality, water
landscape, aquaculture development, are sometimes catastrophic, and thus planktonic
blooms have become a challenging and intractable problem worldwide in recent years.
Therefore, to understand these effects so that some necessary measures can be taken,
it is important and meaningful to investigate the dynamic growth mechanism of
plankton and reveal the dynamics mechanisms of formation and disappearance of
planktonic blooms. To this end, based on the background of the ecological
environments in the subtropical lakes and reservoirs, this dissertation research takes
mainly the planktonic algae as the research objective to model the mechanisms of
plankton growth and evolution. In this dissertation, some theories related to
population dynamics, impulsive control dynamics, stochastic dynamics, as well as the
methods of dynamic modeling, dynamic analysis and experimental simulation, are
applied to reveal the effects of some key biological factors on the dynamics
mechanisms of the spatial-temporal distribution of plankton and the termination of
planktonic blooms, and to predict the dynamics evolutionary processes of plankton
growth. The main results are as follows:
Firstly, to discuss the prevention and control strategies on planktonic blooms, an
impulsive reaction-diffusion hybrid system was developed. On the one hand, the
dynamic analysis showed that impulsive control can significantly influence the
ii

dynamics of the system, including the ultimate boundedness, extinction, permanence,
and the existence and uniqueness of positive periodic solution of the system. On the
other hand, some experimental simulations were preformed to reveal that impulsive
control can lead to the extinction and permanence of population directly. More
precisely, the prey and intermediate predator populations can coexist at any time and
location of their inhabited domain, while the top predator population undergoes
extinction when the impulsive control parameter exceeds some a critical value, which
can provide some key arguments to control population survival by means of some
reaction-diffusion impulsive hybrid systems in the real life. Additionally, a
heterogeneous environment can affect the spatial distribution of plankton and change
the temporal-spatial oscillation of plankton distribution. All results are expected to be
helpful in the study of dynamic complex of ecosystems.
Secondly,

a

stochastic

phytoplankton-zooplankton

system

with

toxic

phytoplankton was proposed and the effects of environmental stochasticity and
toxin-producing phytoplankton (TPP) on the dynamics mechanisms of the termination
of planktonic blooms were discussed. The research illustrated that white noise can
aggravate the stochastic oscillation of plankton density and a high-level intensity of
white noise can accelerate the extinction of plankton and may be advantageous for the
disappearance of harmful phytoplankton, which imply that the white noise can help
control the biomass of plankton and provide a guide for the termination of planktonic
blooms. Additionally, some experimental simulations were carried out to reveal that
the increasing toxin liberation rate released by TPP can increase the survival chance
of phytoplankton population and reduce the biomass of zooplankton population, but
the combined effects of those two toxin liberation rates on the changes in plankton are
stronger than that of controlling any one of the two TPP. All results suggest that both
white noise and TPP can play an important role in controlling planktonic blooms.
Thirdly,

we

established

a

stochastic

phytoplankton-toxic

producing

phytoplankton-zooplankton system under regime switching and investigated how the
white noise, regime switching and TPP affect the dynamics mechanisms of planktonic
iii

blooms. The dynamical analysis indicated that both white noise and toxins released by
TPP are disadvantageous to the development of plankton and may increase the risk of
plankton extinction. Also, a series of experimental simulations were carried out to
verify the correctness of the dynamical analysis and further reveal the effects of the
white noise, regime switching and TPP on the dynamics mechanisms of the
termination of planktonic blooms. On the one hand, the numerical study revealed that
the system can switch from one state to another due to regime shift, and further
indicated that the regime switching can balance the different survival states of
plankton density and decrease the risk of plankton extinction when the density of
white noise are particularly weak. On the other hand, an increase in the toxin
liberation rate can increase the survival chance of phytoplankton but reduce the
biomass of zooplankton, which implies that the presence of toxic phytoplankton may
have a positive effect on the termination of planktonic blooms. These results may
provide some insightful understanding on the dynamics of phytoplankton-zooplankton
systems in randomly disturbed aquatic environments.
Finally, a stochastic non-autonomous phytoplankton-zooplankton system
involving TPP and impulsive perturbations was studied, where the white noise,
impulsive perturbations and TPP are incorporated into the system to simulate the
natural aquatic ecological phenomena. The dynamical analysis revealed some key
threshold conditions that ensure the existence and uniqueness of a global positive
solution, plankton extinction and persistence in the mean. In particular, we determined
if there is a positive periodic solution for the system when the toxin liberation rate
reaches a critical value. Some experimental simulations also revealed that both white
noise and impulsive control parameter can directly influence the plankton extinction
and persistence in the mean. Significantly, enhancing the toxin liberation rate released
by TPP increases the possibility of phytoplankton survival but reduces the
zooplankton biomass. All these results can improve our understanding of the
dynamics of complex of aquatic ecosystems in a fluctuating environment.

iv

CO-AUTHORSHIP
For all the chapters in this dissertation, I was the investigator including the
design of research, data collection and analysis, model building, modeling analysis
and computer simulation experiments. I wrote the original manuscripts and was
responsible for incorporating some comments, suggestions and feedbacks into the
revised manuscripts. Dr. Min Zhao and Dr. Jianbing Li supervised the research and
contributed to the research design, data analysis and the revision of the manuscripts.
Dr. Chuanjun Dai and Dr. Hengguo Yu contributed to the design and implementation
of experimental simulations and helped review and improve the manuscripts, so they
are included in authorship on all resulting publications. Dr. Qi Wang, Dr. Aimin Hao,
and Dr. Jun Kikuchi provided some useful suggestions in data collection and analysis,
and they contributed to some comments on the manuscripts, so they were included in
related publications.
Publication and authorships from this dissertation (Published or prepared for
submission):
He Liu, H.G. Yu, C.J. Dai, Q. Wang, J. Li, R.P. Agarwal, M. Zhao, 2019. Dynamic
analysis of a reaction-diffusion impulsive hybrid system. Nonlinear Analysis: Hybrid
System 33, 353-370. DOI: 10.1016/j.nahs.2019.03.001. (Chapter 3)
He Liu, C.J. Dai, H.G. Yu, Q. Guo, J. Li, A.M Hao, J. Kikuchi, M. Zhao, 2021.
Dynamics induced by environmental stochasticity in a phytoplankton-zooplankton
system with toxic phytoplankton. Mathematical Biosciences and Engineering 18(4),
4101-4126. DOI: 10.3934/mbe.2021206. (Chapter 4)
He Liu, C.J. Dai, H.G. Yu, Q. Guo, J. Li, A.M Hao, J. Kikuchi, M. Zhao, 2021.
Dynamics of a stochastic phytoplankton-toxic phytoplankton-zooplankton system
under regime switching. Submitted to Mathematical Methods in the Applied Sciences
on Feb 4, 2021 (Major revision on June 22, 2021), (Chapter 5)
He Liu, C.J. Dai, H.G. Yu, Q. Guo, J. Li, A.M Hao, J. Kikuchi, M. Zhao, 2021.
Dynamics of a stochastic non-autonomous phytoplankton-zooplankton system
involving toxin-producing phytoplankton and impulsive perturbations. Submitted to
Mathematics and Computers in Simulation on July 20, 2021 (Major revision on
October 29, 2021), (Chapter 6).
v

TABLE OF CONTENTS
ABSTRACT ......................................................................................................................... ii
CO-AUTHORSHIP............................................................................................................... v
TABLE OF CONTENTS ..................................................................................................... vi
LIST OF FIGURES.............................................................................................................. ix
ACKNOWLEDGEMENT................................................................................................... xii
Chapter 1 GENERAL INTRODUCTION.............................................................................. 1
1.1. Background ................................................................................................................ 1
1.2. Statement of the problem ............................................................................................ 5
1.2.1. Physical method................................................................................................... 7
1.2.2. Chemical method ............................................................................................... 10
1.2.3. Biological method.............................................................................................. 11
1.3. Objectives of the study ............................................................................................. 15
1.4. Organization of the dissertation................................................................................. 17
Chapter 2 LITERATURE REVIEW .................................................................................... 19
2.1. Plankton dynamics model ......................................................................................... 19
2.2. The research of nonlinear dynamics .......................................................................... 25
2.2.1. The dynamics of stochastic system ..................................................................... 26
2.2.2. The dynamics of impulsive control system ......................................................... 29
2.3. Summary of literature review.................................................................................... 31
Chapter 3 DYNAMIC ANALYSIS OF A REACTION-DIFFUSION IMPLUSIVE SYSTEM
........................................................................................................................................... 33
Abstract........................................................................................................................... 33
3.1. Introduction.............................................................................................................. 34
3.2. Dynamic analysis ..................................................................................................... 37
3.2.1. Preliminaries...................................................................................................... 38
3.2.2. Permanence of population .................................................................................. 41
3.2.3. Extinction of top predator population ................................................................. 48
3.2.4. Periodic oscillations of population density.......................................................... 50
vi

3.3. Experimental simulations.......................................................................................... 54
3.3.1 Impact of impulsive control on the dynamics of system (3-1)-(3-5) ..................... 55
3.3.2 Impact of environmental heterogeneity on the dynamics of system (3-1)-(3-5) .... 59
4.3. Conclusions.............................................................................................................. 61
Chapter 4 DYNAMICS INDUCED BY ENVIRONMENTAL STOCHASTICITY IN A
PHYTOPLANKTON-ZOOPLANKTON SYSTEM WITH TOXIC PHYTOPLANKTON .. 63
Abstract........................................................................................................................... 63
4.1. Introduction.............................................................................................................. 64
4.2. Dynamic analysis ..................................................................................................... 68
4.2.1. Preliminaries...................................................................................................... 69
4.2.2. Existence and uniqueness of global positive solutions ........................................ 70
4.2.3. Extinction and persistence induced by white noise.............................................. 73
4.2.4. The ergodic stationary distribution of plankton................................................... 76
4.3. Experimental simulations.......................................................................................... 83
4.3.1 Impact of white noise on the dynamics of system (4-2)........................................ 83
4.3.2 Impact of TPP on the dynamics of system (4-2)................................................... 88
4.4. Conclusions and discussion....................................................................................... 91
Chapter 5 DYNAMICS OF A STOCHASTIC PHYTOPLANKTON –TOXIC
PHYTOPLANKTON-ZOOPLANKTON SYSTEM UNDER REGIME SWITCHING ........ 95
Abstract........................................................................................................................... 95
5.1. Introduction.............................................................................................................. 95
5.2. Model formation....................................................................................................... 98
5.3. Dynamic analysis ................................................................................................... 101
5.3.1. Preliminaries.................................................................................................... 101
5.3.2. Existence and uniqueness of global positive solutions ...................................... 104
5.3.3. Extinction and persistence of plankton ............................................................. 106
5.3.4. Stationary distribution and ergodic property of plankton................................... 110
5.4. Experimental simulations........................................................................................ 114
5.4.1. Impact of regime switching on the dynamics of system (5-1)............................ 115
5.4.2. Impact of white noise on the dynamics of system (5-1)..................................... 120
5.4.3. Impact of TPP on the dynamics of system (5-1)................................................ 121
vii

5.5. Conclusions ............................................................................................................ 121
Chapter 6 DYNAMICS OF A STOCHASTIC NON-AUTONOMOUS
PHYTOPLANKTON-ZOOPLANKTON SYSTEM INVOLVING TOXIN-PRODUCING
PHYTOPLANKTON AND IMPULSIVE CONTROL ...................................................... 124
Abstract......................................................................................................................... 124
6.1. Introduction............................................................................................................ 124
6.2. Dynamic analysis ................................................................................................... 130
6.2.1. Preliminaries.................................................................................................... 130
6.2.2. Existence and uniqueness of global positive solutions ...................................... 132
6.2.3. Extinction and persistence induced by impulsive control and white noise ......... 133
6.2.4. Periodic oscillations of plankton density........................................................... 137
6.3. Experimental simulations........................................................................................ 142
6.3.1. Impact of white noise on the dynamics of system (6-3)..................................... 143
6.3.2. Impact of impulsive perturbations on the dynamics of system (6-3) .................. 145
6.3.3. Impact of TPP on the dynamics of system (6-3)................................................ 147
6.4. Conclusions ............................................................................................................ 148
Chapter 7 CONCLUSIONS AND RECOMMENDATIONS.............................................. 151
7.1. Conclusions ............................................................................................................ 151
7.2. Limitations and Future research .............................................................................. 153
References ........................................................................................................................ 156
Appendix A....................................................................................................................... 202
Appendix B....................................................................................................................... 203
Appendix C....................................................................................................................... 204

viii

LIST OF FIGURES
Fig. 1.1 Schematic diagram for the structure of the research……………………………..…........30
Fig. 3.1 Dynamic behaviors of species 1 , 2 and 3 of system (3-1)-(3-5) without impulsive
effects, here the initial conditions 1 0, = 2, 2 0, = 2, 3 0, = 6.85 for all ∈ .
(a): permanence of species 1 ; (b): permanence of species 2 ; (c): permanence of species
= 0……………56
3 ; (d): time series of
1 , 2 and
3 on the population dynamics with
Fig. 3.2 The effects of impulsive effects on the dynamic behaviors of species 1 , 2 and 3 of
system

(3-1)-(3-5)

1

with

3

,

1,

2,

2

= 0.9,

3

,

1 , 2,

, 1 , 2 , 3 = 0.95 for all
= 1,2, ⋯ , = 8 and the initial conditions
2, 2 0, = 2, 3 0, = 4.2 for all
∈ : (a): permanence of species
permanence of species 2 ; (c): permanence of species 3 ; (d): time series of

3

= 0.95,

1 0,

=
;
(b):
1
1 , 2 and

= 0…………………………………………………….57
3 on population dynamics with
Fig. 3.3 The effects of impulsive control parameter on the dynamic behaviors of species 1 , 2
and
of system (3-1)-(3-5), here
= 40, and the initial conditions
3
= 2, 2 0, = 3, 3 0, = 1 for all ∈ : (a): permanence of species 1 ; (b):
1 0,
permanence of species 2 ; (c): extinction of species 3 as → ∞; (d): time series of
= 0………………………………………...59
1 , 2 and
3 on population dynamics with

Fig. 3.4 The effect of impulsive control parameter (0 ≤ p ≤ 30) on the density of all species,
here the initial conditions
= 2, 2 0, = 2, 3 0, = 4.2 for all
∈
1 0,
……………………………………………………………………………………………...60
Fig. 3.5 The effect of impulse perturbation constants
species

3

with fixed

= 1,

3

,

1,

2,

3

= 1+

3

= 1,2,3,
0≤

∈

the density of

≤ 0.5 for all

= 1,2,

here ′2 , = 0.4 cos
+ 0.1 sin
+ 3, 3 , = 0.5 sin
sin
+ 2and the
initial conditions 1 0, = 2, 2 0, = 2, 3 0, = 1 for all ∈ ………………...60
Fig. 3.6 The effect of impulsive control parameter (0 ≤ p ≤ 30) on the density of all species,

here the initial conditions
= 2, 2 0, = 2, 3 0, = 4.2 for all
∈
1 0,
……………………………………………………………………………………….……60
Fig. 3.7 Numerical simulation of spatially non-homogeneous periodic solutions of the system
(3-1)-(3-5) with
= 8, here initial conditions 1 0, = 2, 2 0, = 2, 3 0, =
4.2 for all ∈ ……………………………………………………………………............61
Fig. 4.1 The effect of 2 on the stochastic dynamical behaviors of the system (4-2) with

change
1 = 0.1, 3 = 0.1,0 ≤ 2 ≤ 1.2. (a) The dynamical behaviors of species
2
from persistence in the mean to extinction in different areas of , and
for 0 ≤ 2 ≤
1.2. (b) The same path of species 2
for system (4-2) with respect to Fig. 4.1(a) for
2 = 0.1, 2 = 0.8, 2 = 1

and
on 1000,2000 and its corresponding deterministic
system (4-1)…………………………………………………………………………………85
Fig. 4.2 (a), (b), (c) The solution trajectories of system (4-2) and its corresponding deterministic
system (5-1). (d), (e), (f) The probability density function diagrams of 1 , 2 ( ) and
( ) for the system (4-2) with 1 = 2 = 3 = 0.1, and the red smoothed carves are
probability density functions for system (4-2)…………………………………………..….87

Fig. 4.3 Stochastic dynamical behaviors of system (4-2) with
ix

2 = 1.1

and its corresponding

deterministic counterparts on ∈ 0,1000 . (a) The persistence in the mean of species
( ) and extinction of species 2 ( ) of stochastic system (4-2). (b) The
1 ( ) and

persistence of deterministic system (4-1)…………………………………………………...87
Fig. 4.4 The effect of toxin rate
produced by population 1 ( ) on the stochastic dynamic
behaviors of the system (4-2) with 1 = 2 = 3 = 0.1,0 ≤ ≤ 1. (a)-(b) The persistence
in the mean of species 1 ( ) and 2 ( ); (c) The persistence in the mean of population
( ) for 0 ≤ ≤ 0.3846 and extinction for 0.3846 ≤ ≤ 1………………………….88
Fig. 4.5 The effect of one toxin rate
on the dynamics of system (4-2). (a), (b), (c) The

histograms of probability density functions for 1 , 2 ( ) and ( ) with different values
of . Here = 0.07 and σ1 = σ2 = σ3 = 0.1……………………………………………89
Fig. 4.6 The effect of one toxin rate γ on the dynamics of system (4-2). (a), (b), (c) The

histograms of probability density functions for P1 t , P2 (t) and Z(t) with different values
of γ. Here, γ = 0.39, δ = 0.07, and σ1 = σ2 = σ3 = 0.1. .................................................. 90
Fig. 4.7 Stochastic dynamical behaviors of system (4-2) and its corresponding deterministic

counterparts on ∈ 0,1000 . Here,
= 0.42, 1 = 2 = 3 = 0.1, = 0.07………....91
Fig. 4.8 The effect of toxin rate
produced by population The combined effects of two toxin
rates
and
on the dynamics of system (5-2). (a), (b), (c) The histograms of probability
density functions for 1 , 2 ( ) and ( ) with = 0.2, = 0.2, 1 = 2 = 3 = 0.1,
respectively…………………………………………………………………………………91
Fig. 5.1 (a), (b) and (c) denote the solution trajectories of 1 , 2
and
for system (5-1)

with
1 , 2 = 0.2,0.3
and
σ1 1 , σ1 2 = 1.5,1.5 , σ2 1 , σ2 2 =
1.3,1.4 , σ3 1 , σ3 2 = 0.7,0.8 , respectively.…………………………..…………117

Fig. 5.2 (a), (b) and (c) denote the solution trajectories of 1 , 2
for system (5-1)
and
with
1 , 2 = 0.2,0.3
and σ1 1 , σ1 2 = 0.1,0.05 , σ2 1 , σ2 2 =
0.1, .05 , σ3 1 , σ3 2 = 0.1,1.8 , respectively…...…………………………………117
Fig. 5.3 The effect of regime switching on the stochastic behaviors of zooplankton species
for system (5-1). (a) denotes the stochastic behaviors between extinction and persistence in
the mean zooplankton species
for system (5-1) with different values of
in different
areas of , ,
and other parameters as in Fig. 5.2; (b) denotes the solution trajectories of

zooplankton species
with respect to Fig. 5.3 (a) for = 1, = 15 and = 25,
respectively………………………………………………………………………………..118
Fig. 5.4 (a), (b) and (c) denote the solution trajectories of 1 , 2
and
for system (5-1)

with
1 , 2 = 0.2,0.3
and σ1 1 , σ1 2 = 0.1,0.05 , σ2 1 , σ2 2 =
0.1, .05 , σ3 1 , σ3 2 = 0.1,1.8 , respectively…..…………………………………118
Fig. 5.5 (a), (b) and (c) denote the solution trajectories of system (5-1) and its corresponding

deterministic counterparts, and (d), (e) and (f) denote the density function diagrams of
system
(5-1)
with
1 , 2 = 0.2,0.3
,
σ1 1 , σ1 2 = 0.1,0.05
,
σ2 1 , σ2 2 = 0.1, .05 ,
σ3 1 , σ3 2 = 0.1,0.05 in regime
= 1, = 2 ,

respectively……………..………………………………………………………………….119
Fig. 5.6 (a) denotes the movement of Markov chain in the state = 1,2 over time. (b) denotes
the probability density function (PDF) of ( )…………………………………………....119

Fig. 5.7 The effect of the toxin released rate
on the probability density function of system (5-1).
(a), (b) and (c) denote the histograms of probability density function for 1 , 2
and
of system (5-1) with
1 , 2 = 0.5,0.55 , respectively……………………..120
x

Fig. 6.1 The dynamics of system (6-2) with impulsive control parameter
= 2. (a)-(b): The
periodic solutions of phytoplankton population
and zooplankton population
of
deterministic system (6-2), respectively………………………………..............................143

Fig. 6.2 The stochastic dynamics of system (6-3) with

=

1

2

= 0.005 + 0.005 sin

40

.

(a)-(b): The periodic solutions of phytoplankton population
and zooplankton
population
of system (6-3), respectively……………………………………………144
Fig. 6.3 The stochastic dynamics behaviors of system (6-3) with

1

= 1.8 + 0.005 sin

40

and

its corresponding deterministic counterparts. (a)-(b): Blue curves are the extinction of
phytoplankton population
and zooplankton population
of stochastic system
(6-3), respectively, and red curves represent the persistence of phytoplankton and
zooplankton populations for its corresponding deterministic counterparts, respectively….146
Fig. 6.4 The stochastic dynamic behaviors of system (6-3) with = 40 and its corresponding

deterministic counterparts. (a)-(b): the persistence in the mean of phytoplankton population
and the extinction of zooplankton population
of stochastic system (6-3),
respectively; (c)-(d): the persistence of phytoplankton population
and the extinction of

zooplankton population
of its corresponding deterministic counterparts,
respectively………………………………………………………………………………...146
Fig. 6.5 The effect of TPP on the stochastic dynamic behaviors of system (6-3) with
γ t = m + 0.06 sin

πt
40

. (a) The persistence in the mean of population

persistence in the mean and extinction of population

( ) ; (b) The

( )……………………………......148

Fig. 6.6 The effect of TPP on the stochastic dynamic behaviors of system (6-3). Panels (a) and (b)
denote the sample path of phytoplankton ( ) and zooplankton ( ) of the stochastic
system (6-3) and the periodic solutions of its corresponding deterministic system (6-2) with
different
0.06 sin

, respectively.

40

(red curves) and

= 0.06 + 0.06 sin

40

= 0.5 + 0.06 sin

(green curves),

40

(cyan curves), and their

corresponding deterministic counterparts (blue, black and magenta curves). Here

xi

= 0.18 +
= 2..148

ACKNOWLEDGEMENT
First of all, I would like to express my great gratitude and appreciation for my
supervisor Dr. Jianbing Li and co-supervisor Dr. Min Zhao whose guidance, support,
help and encouragement have been invaluable throughout my research study, and they
are the leaders of my scientific research. Without their help, support and advice, I
would not have been able to complete my research. Further, I would like to thank my
supervisory committee members: Dr. Ron Thring and Dr. Youmin Tang for their
insightful comments, guidance, and discussions for thesis draft. Also, I would like to
thank my external examiner Dr. Zoe Li for her useful comments and suggestions for
the final thesis.
Special thanks go to Dr. Chuanjun Dai and Dr. Hengguo Yu for sharing their
expertise, experience, and they continuously provided help, encouragement, guidance,
and discussions. They were always willing to help me with my research. And also, I
would like to express my thanks to Dr. Geoff Payne and Dr. Roger Wheate for their
supports, understanding and help, and I would also like to express my thanks to Dr.
Bill McGill, Dr. David Connell, Dr. Ken Otter, and Dr. Peter Jackson as my course
supervisors in my first year of PhD at UNBC. They always encouraged me to take a
broader viewpoint in my later research, which greatly helped me to deal with my
research problems from multiple perspectives.
Special thanks also go to Dr. Mingjiang Wu, Dr. Xiufeng Yan, Dr. Zengling Ma,
Dr. Xiangyong Zheng and Dr. Yubao Li and all staffs from the College of Life and
Environmental Science, Wenzhou University, for their unconditional supports,
understanding and help.
My gratitude also extends to the graduate or non-graduate students at UNBC for
sharing their help, knowledge, skills, experience and thoughts. They are Dr. Yuan
Tian, Dr. Jingjing Shi, Ya Zhang, Qing Guo and all the team group members from Dr.
Li‟s laboratory. I would also like to extend my gratitude to the team group members:
Jingjing Su, Lijun Wang, Xinxin Li, Fangfang Chen, Kaidong Niu, Wenchang Chen,
xii

Juan Ye, Huanyi Liu, Shengyu Huang, Heng Yang, Qiulin Huang, and Lingzhen
Zheng, who all come from Wenzhou University for their friendship and understanding.
They are always supportive and inspiring. The friendship from my friends has helped
me to gather my strength to pass through many hard times during my PhD study.
I would like to dedicate this work to my families for their unconditional supports,
endless understanding, companionship and love. Special thanks go to my wife, who
has always concerned and supported me silently and took good care of my family,
especially for my daughter, Mofan Liu, thus letting me concentrate on my research.
This research was financially supported by the National Key Research and
Development Program of China (Grant No. 2018YFE0103700) and the National
Natural Science Foundation of China (Grant No. 61871293, No. 61901303, No.
31370381, and No. 31570364).

xiii

Chapter 1 GENERAL INTRODUCTION
1.1. Background
Plankton populations are floating organisms of many different phyla that live in
the pelagic of oceans, in freshwater lakes, or in larger rivers, estuaries and reservoirs,
and they are greatly influenced by water movements (Sommer, 1994, 1996).
Phytoplankton are the microorganisms and commonly unicellular and microscopic in
size, requiring some essential elements (such as light, carbon dioxide and a range of
inorganic and organic nutrients) to photosynthetically produce biomass. Zooplankton
population includes the animals in the plankton community living on phytoplankton.
As the foundation of the entire aquatic food webs, plankton populations play an
important role in the material circulation and energy transportation for real aquatic
ecosystems, and they are great of global significance for climate regulation and
biogeochemical cycling (Odum, 1971; Simo, 2001). On the one hand, phytoplankton
are of fundamental importance in supporting the primary productivity of the real
aquatic ecosystems (Behrenfeld and Falkowski, 1997; Field et al., 1998; Hoppe et al.,
2002), which contribute to the climate regulation by absorbing carbon dioxide from
surrounding environments (Duinker and Wefer, 1994) and provide the materials and
energy for all life in aquatic environments (Kawecka and Eloranta, 1994). On the
other hand, zooplankton can control the quantity and spatial-temporal distribution of
phytoplankton by feeding on them (Martin, 1970; Lampert and Taylor, 1985; Lampert
et al., 1986; Wyatt and Horwood, 1973; Levin and Segel, 1976; Griffin et al., 2001).
Meanwhile, plankton populations are the main food resources for high-trophic
organisms such as fish and other larger animals, and their distribution and biomass
variation can directly affect the status of fishery and other aquaculture (Platt et al.,
2003), which play a critical role in the structures and functioning of aquatic
ecosystems (Akhurst et al., 2017). Thus, plankton play a significant role in economic
and social values for aquatic ecosystems.
1

Due to the rapid urbanization, industrialization, intensifying agricultural
production and the increase of the global population (United Nations Population
Division, 2008), a large amount of industrial and agricultural wastewater, domestic
sewage and farming practices are discharged into bays, rivers, lakes and oceans,
enriching high level of nutrients and minerals in these water bodies, and finally
resulting in water eutrophication (Webster et al., 2001). Actually, the process of
eutrophication consists of a continuous increase in the contribution of essential
biological nutrients, including nitrogen and phosphorus (organic loading), until it
exceeds the capacity of the water bodies (i.e. the capacity of lakes, rivers or oceans to
purify themselves), which may contribute to the changes of structures and functions in
these water bodies. Actually, these changes strongly depend on the use of fertilizers,
untreated industrial wastewater, aquaculture runoff and discharge, and the reduction
of self-purification capacity of water bodies. Under the conditions of eutrophication,
the rates of organic production for real aquatic ecosystems exceed those of
consumption (Qin et al. 2013), causing the massive accumulation of organic matters
in water bodies that may seriously destroy the ecological balance of aquatic
ecosystems and greatly promote the occurrence of HABs (Jager et al., 2017). In
addition, the eutrophication process can be greatly accelerated due to the effects of
some factors, for example, a slow current velocity, the poor self-purification capacity
(Heisler et al., 2008), a suitable temperature (for example, 25℃ or above) (O‟Neil et
al., 2013), an adequate light exposure (Sheiwastav et al., 2017), and point-source
discharges and non-point loadings of limiting nutrients. Based on these, the
Organization for Economic Cooperation and Development (OECD) that was led by
18 member countries to monitor and carry out the research of the trophic status and
eutrophication in about 150 lakes, proposed the standard criteria of quantifiable
indicators for eutrophication in 1982, that is, the average concentration of total
phosphorus (TP) >0.035 mg/L, the average concentration of Chlorophyll-a >0.008
mg/L, and the average water transparency <3 m (OECD, 1982). The ecologist
Jorgensen pointed out that the excessive growth of algae is a key process in the
2

formation of eutrophication (Jorgensen, 1983), and a number of studies showed that
the water bodies in the areas where algal blooms occur have been seriously polluted,
and the levels of nutrients (i.e. nitrogen and phosphorus) have greatly exceeded the
standard criteria from OECD. According to these thresholds, many lakes in China
have experienced different degrees of eutrophication, which have become one of the
most important ecological and environmental problems facing China‟s freshwater
lakes (Qin et al., 2013).
Eutrophication is an enrichment of essential nutrients in water bodies, which can
lead to changes in the structures and functions of aquatic ecosystems. For example,
increasing production of aquatic plants (Rast and Thornton, 1996), consuming large
amount of dissolved oxygen (Mu et al., 2017), deteriorating water quality (Western,
2001; Capuzzo et al., 2015), reducing the biomass of the harvestable fish and shellfish
(Smith, 2003; AI Gheilani et al., 2011), increasing the biomass of consumer species
(Smith, 2003), and ultimately affecting biodiversity (Nasri et al., 2008). Thus, water
eutrophication has become an urgent environmental issue worldwide in recent years
(Nyenje et al., 2010; Liu et al., 2011). Indeed, it was reported that over 75% of closed
water bodies in the world are associated with eutrophication (Freedman, 2002).
According to the survey of the state of the World‟s Lakes, a project promoted by the
International Lake Environmental Committee (ILEC), eutrophication affects 54% of
Asian lakes, 53% of lakes in Europe, 48% of lakes in North America, 41% of lakes in
South America, and 28% of lakes in Africa in varying degrees. Due to the degradation
of water quality caused by eutrophication, many limnological studies on lakes, rivers,
and streams have been emphasized in recent years (Saxena, et al., 1988). Especially,
some freshwater lakes in USA, such as 17 lakes of Western Washington, Washington
(Welch and Crooke, 1987), Lake Okeechobee, Florida (the average water depth of 2.7
m) (Schelske, 1989), Lake Apopka, Florida (the average water depth of 4.7 m)
(Coveney et al., 2002), and Lake City Park, Louisia (Ruleya and Rusch, 2002), have
seriously suffered from eutrophication, and the estimated cost of damage caused by
entrophication in USA alone was approximately $2.2 billion annually for tourism
3

restoration (Dodds et al., 2009). In China, 25% of drinking sources are lake water or
reservoir houses, but more than half of lakes have become eutrophic and
hypereutrophic (Yang, 2009). For example, Lake Taihu (the average water depth of 2
m), the third largest freshwater in China, experienced the toxic cyanobacterial blooms
in the summer of 2007, leading to the crisis of drinking water supplies for 2 million
people in Wuxi City, Jiangsu Province (Qin et al., 2010). According to the work of
Yang et al. (2008), approximately 30 billion ton of polluted water have been
discharged directly into the lakes, which may cause all the urban lakes and the most of
the medium-sized lakes at the urban-rural fringe areas in China facing the problem of
water eutrophication by 2030. Other cases of lake eutrophication such as Lake
Kastoria in Greece (Koussouris et al., 1991), Lake Biwa in Japan (Yamashiki et al.,
2003), and Lake Bellandur in India (Chandrashekar et al. 2003), have been reported
globally in recent years. The main reason for the deterioration of water quality in
these cases is primarily associated with high nutrient enrichment derived
from anthropogenic activities.
Dating back to the 1960s and 1970s, many scientists and researchers have linked
the issue of algal blooms to nutrient enrichment resulting from human activities such
as agriculture, industry, and sewage disposal (Schindler, 1974; Hallegraeff, 1993), and
an obvious and problematic symptom of eutrophication is the increase of harmful
phytoplankton biomass resulting in the occurrence of algal blooms (Tang et al., 2010).
More precisely, when phytoplankton blooms occur at the surface or at specific depths
in the water column, the number of a certain dominant phytoplankton species rapidly
increases or almost equally rapidly decreases in aquatic ecosystems and then returns
to its original low level under some conditions (Beltrami and Carroll, 1994). The
dominant species of algal blooms are different during harmful algal blooms (HABs)
(Smayda, 1997) due to their living aquatic environments. In this respect, HABs
mainly include different types of taxa, such as diatom (e.g. Alexandrium),
dinoflagellate (e.g. Pfiesteria), and cyanobacteria (or blue-green algae), among which
cyanobacteria are of special importance because of their potential impact on drinking
4

water sources or recreational water bodies (Carmichael and Wayne, 2001; Yang et al.,
2012). Additionally, algal blooms are accompanied by toxin productions from HAB
species (Paprdimitriou et al., 2012), such as cyanobacterial toxins, which can
contaminate seafood and fish or kill other higher trophic level organisms (Penaloza et
al., 1990 and Tencalla et al., 1994), and even directly or indirectly threaten human
health through the food chain of aquatic ecosystems (Francis, 1878; Hallegraeff, 1993;
Codd and Bell, 1996). Actually, the events of the death of marine animals or even
human being caused by toxic algae have been reported globally with an increasing
frequency (Naiman et al., 1995) since the first case of animal death caused by toxic
cyanobacteria in 1878 (Francis, 1878). For example, many poisoning episodes caused
by algal toxins occurred in the United States, Australia, France, Zimbabwe, and other
countries (Falconer et al., 1983; Gugger et al., 2005; Ndebele and Magadze, 2006;
Nasri et al., 2008). Significantly, HABs have dramatic consequences for the
ecological balance of aquatic ecosystems, drinking water resources, economic losses,
fisheries and tourism (Capenter et al., 1998; Wells et al. 2015; Yang et al., 2012).
Based on these huge effects of algal blooms, it is urgent and important to find an
effective way in the aquatic ecology for controlling and preventing HABs.
1.2. Statement of the problem
HABs have become one of the most severe problems in real aquatic ecosystems
worldwide recently, and there have been reported globally with an increasing
frequency and intensity over the last few decades (Hallegraeff, 1993). The outbreaks
of algal blooms have and continue to pose a serious threat to the balance and
stability of aquatic ecosystems, which not only cause profound and deleterious
effects on water quality (Capuzzo et al., 2015), water landscape (Mitra and Flynn,
2006), and aquaculture development (Uye, 1986; Wyatt and Horwood, 1973; Levin
and Segel, 1976), but also pose a serious health hazard to animals and human beings
(Nasri et al., 2008; Falconer et al., 1983; Gugger et al., 2005; Qin et al., 2010).
Moreover, excessive growth of algae can be caused by nutrients enrichment, and
5

especially phosphorus which is the primary nutrient necessary for abundant algae
and aquatic plants growth (Paerl et al., 2001; Mainstone and Parr, 2002). Once a
water body is eutrophicated, it will lose its primary functions and structures and
subsequently affect the sustainable development of economy and society. For a
successful water restoration and the plan of eutrophication mitigation, the basic aim
is to identify the sources of pollution and their subsequent remediation. Thus, many
municipal authorities have passed some relevant legislations and laws to mitigate
and limit the point sources and non-point sources loading of nutrients, such as
prohibiting the usage of phosphate-containing laundry detergents (Kundu et al.,
2015), optimizing agricultural land use and best policies (Alvarez et al., 2017; Dai et
al., 2018), restoring animal and plant communities in local ecosystems (Zhang et al.,
2009), and adapting domestic wastewater treatment and waste classification (Huang
et al., 2017). But eutrophication and algal blooms are still prevalent on the water
surface worldwide (Smith et al., 1999; Smith and Schindler, 2009) and some
refinements are still going on due to the ever-increased complexity and specific
problems that require wider experience. Therefore, it is important and urgent for
water resource managers to understand how to minimize the intensity and frequency
of algal blooms (Paerl and Paul, 2012), and the prevention and control approaches of
algal blooms have become an important ecological topic in the protection of aquatic
environments around the world. Actually, many researchers have investigated and
developed technologies and methods for the prevention and controlling of algal
blooms since the 1960s, and the purposes of these strategies are to kill harmful algae
organisms or limit their growth (Schindler, 2006; Carpenter, 2008; Lewis and
Wurtsbaugh, 2008; Sengco, 2009), but the algal blooms control or suppression
activities are controversial and challenging. Generally, the most common types for
controlling and suppressing HABs include physical, chemical, and biological
technologies (Anderson, 2009).

6

1.2.1. Physical method
Among the algal control methods, physical method is one of the most applicable
techniques in freshwater lakes. This method is a technical method to control algae
pollution through physical engineering measures, such as ultrasonic algae removal,
flocculants addition, shading algae removal, and other related technologies.
Ultrasonic algae removal is an important research direction of algae removal
technology, and ultrasonic technology has been applied for monitoring algae growth
for water and wastewater treatment (Mahvi, 2009; Wang et al., 2019). Actually,
ultrasound is a sound wave with high frequencies above the limit of human hearing
(22 kHz) that can control algae growth at special frequency, and the commonly used
frequencies for industrial cleaning are those between 20 and 50 kHz (Suslick and
Price, 1999). In addition, the process of ultrasound is to generate a sound wave layer
in the top layer of a water body, and then the generated sound wave layer can directly
destroy the buoyancy of algae cells, causing these cells to sink and eventually die.
Generally, the ultrasound technology is advantageous to monitor the real-time water
quality, and predict and control algal blooms. In past decades, a lot of works have
been done by researchers and some excellent results have been achieved. For example,
the work of Tang et al. (2003) showed that a high frequency of ultrasound has a great
influence on the density of cyanobacteria cells at the early stage of its growth, and
further concluded that the growth rate of cyanobacteria cells after irradiation can be
reduced to 38.9% of the control group in a short time, while dispersive ultrasound
irradiation is an effective method to inhibit the rapid growth of cyanobacteria cells.
Joyce et al. (2010) studied the effect of ultrasonic frequency on the growth of
Microcystis aeruginosa in an experiment, and the best results were achieved when the
ultrasonic frequency was set at 580 kHz. Significantly, ultrasonic algae removal has
been proven the most effective, environmentally and friendly technology to control
algal blooms in lakes, reservoirs, and ponds, which provide a long-term solution for a
healthy ecosystem and is harmless to humans, plants, and other aquatic life. However,
most modern ultrasonic devices strongly rely on transducers which are composed of
7

piezoelectric materials, and it is difficult to implement this technique successfully to
remove algae from large areas of water bodies due to the high power and determining
the specific ultrasonic frequency (NEIWPCC, 2015).
One promising control strategy may be the rapid sedimentation of HABs through
flocculation with clay (Shirota, 1989; Anderson, 1997; Sengco, 2001; Sengco and
Anderson, 2004). The mitigation of HABs using clay was first proposed in the 1970s
(Shirota, 1989), and the addition of clays particles (such as nontoxic and inexpensive)
or any flocculants can help to carry bloom to the bottom of sediments, which is the
oldest and most widely used approach to control HABs (Anderson, 1997; Pierce et al.,
2004; Sengco and Anderson, 2004). When sprinkled on the surface of water bodies
during HABs, these tiny but dense clay particles will „flocculate‟ or combine with
other particles in water bodies, including HABs cells. The process of flocculation
formation is to remove these cells through sedimentation, and the rapid sedimentation
of algae cells may be caused by the flocculation (Jackson and Lochmann, 1993). And
also, clays have been investigated in some countries as a means of removing harmful
algae from water column (Shirota, 1989; Yu et al., 1994; Sengco and Anderson, 2004;
Lee et al., 2008; Song et al., 2010). For example, South Korea, where a fish-farming
industry worth hundreds of millions of dollars is seriously threatened by HABs, this
control method has been proved to be a flexible and economically and socially
feasible method for water treatment, and thus great progress has been made (Na et al.,
1996; Lee et al., 2008; Liu et al., 2010). However, one of the limitations of this
strategy may need more clay consumption (Sengco, 2001; Yu et al., 2004), resulting in
excessive sediment siltation and heavy dredging work. Additionally, when algae are
removed from underwater by the clay coagulant, they cannot be prevented from
floating in shallow rivers and lakes, and the release of toxic substances from
decomposition can cause secondary pollution (Lee et al., 2008; Orizar et al., 2013).
It is well known that the growth of algae is closely related to the photosynthesis
by the light, and thus light is one of the important factors restricting algae growth and
diversity (Marra and Heineman, 1982; Nanninga and Tyrrell, 1996). For example, the
8

work of Shen (2002) showed that the growth rate of algae increases with the
increasing of light intensity, and found that the best situation of algal growth is at the
luminescence of 4000 lux. Also, the strong or weak light was unfavorable for
phytoplankton growth, and the function together with suitable temperature, light
intensity and ample sunlight encouraged algal blooms under the same water quality
and hydrodynamic conditions (Cao et al., 2011). However, light-shielding can
significantly inhibit the photosynthetic rate of algae and promote algae extinction due
to the consumption of dissolved oxygen (Kirk, 1994). For example, Chen et al. (2009)
concluded that the algae biomass can reduce rapidly under the light-shading condition,
and also, the reduction efficiency is increased when the light-shading is accompanied
by aeration, indicating the feasibility of reducing harmful algae by light-shading plus
aeration. The work of Ye et al. (2007) illustrated that the concentration of
Chlorophyll-a, turbidity and chemical oxygen demand decrease significantly after 9
days of light-shading, and their removal rates reach at 80.1%, 68.0% and 93.8%,
respectively, which verifies the feasibility of the light-shading method, and further
indicates that the shading of sunlight may still be a viable measure to control algal
blooms in natural water purification systems. Generally, it seems to be ineffective,
expensive and impractical to apply this strategy to the large scale and complex
ecosystems for the proper algal control (Edmondon, 1970), and also the light-shading
efficiency strongly depends on the temporal and spatial differences of different lakes
and objectives of algal control.
Other physical technologies such as membrane filtration (Castaing et al., 2010;
Liang et al., 2008), air flotation (Teixeira et al., 2010) and electrochemical
technologies (Rodrigo et al., 2010; Hasan et al., 2016; Chaplin, 2019), have also been
widely used around the world. For example, electrochemical technologies, the
methods

combining both

physical chemistry and electronic science, can

simultaneously treat multiple classes of contaminants with high removal efficiency
through the production of chemicals at the electrode surfaces with low power and
energy demands, which have proved to be a clean, flexible and environmentally
9

friendly way for wastewater treatment (Giwa et al., 2019; Chaplin, 2019). However,
the development of the destructive electrochemical technologies requires the
manufacture of non-toxic, low-cost, and high surface area electrodes, which have long
service life and can be operated without harmful toxic by-products (Tang et al., 2019;
Chaplin, 2019).
1.2.2. Chemical method
Chemical treatment method involves treating wastewater with various chemical
additives that inhibit or be lethal to HAB organisms, including chemical agent method,
electrochemical method and photochemical degradation method. The former is the
most commonly used method, while the latter two methods are less reported. The
chemical agents, such as metals (Magdaleno et al., 2014), algaecides and herbicides
(Nagai et al., 2016), can control the propagation of algal blooms due to its adverse
effects on the other organisms and the expedited release of Microcystins (EPA of
China, 2000). In fact, chemical control was attempted in 1957 against the Florida red
tide organisms using copper sulfate delivered with crop dusting airplanes in small
lakes and ponds (Rounsefell and Evans, 1958). After that many related works have
been done. For example, the work of Kaya et al. (2005) found that the combination of
lysine and malonic acid could selectively control toxic Microcystis blooms and induce
the growth of Macrophytes. Li et al. (2007) carried out an experiment to study on
the emergency treatment of toxic cyanobacteria blooms in basalt lake, and the results
shown that algaecides may remove cyanobacteria temporally, but it cannot resolve the
core problem of lake eutrophication. However, the most types of chemicals are not
species-specific which may have a potential negative impact on ecological balance.
For example, the use of algaecides, such as copper sulfate, copper chelates, chemical
Endothall, has been proved to be an effective way to control and prevent HABs in
small-sized lakes and ponds temporally (Boyd and Tucker, 1998). If the amount of
required copper sulfate may not be prepared carefully by considering the alkalinity or
acidity of the reservoir, the copper sulfate will become toxic to fish and other large
10

aquatic animals (Anderson, 2009). Although the use of chemicals is the most common
and versatile management strategy for controlling nuisance aquatic plant populations,
this method is neither practical nor advised in large ecosystems, or any waters to be
used for fishing, drinking and other animal and human use. Also, most chemicals are
too expensive and need frequent dosing, and they may not resolve the primary cause
of the problems, or even may pose risks to human being, livestock and wildlife since
these chemical treatments are often associated with the changes of PH value or
salinity (Murphy et al., 1999; Hullebusch et al., 2002), in addition to causing damage
to non-target aquatic organisms that may reappear after treatment, and thus it is
difficult to find environmentally friendly and acceptable chemical way to help to
control particular harmful algae (Anderson, 2009).
1.2.3. Biological method
The biological method is a way to increase the grazing pressure on toxic
phytoplankton or to reduce recycling of nutrients by using the principle of the food
chains of aquatic ecosystems and the relationship between organisms, including
ecological floating island technology, aquatic plants competition technology,
filter-feeding fish control technology and harmful phytoplankton control technology.
Ecological floating island, a biotical-ecological restoration technology, refers to
hanging biological fillers on the fixed support to make microorganisms, protozoa and
small zooplankton floating on the surface of the fillers, so as to improve the
wastewater purification by increasing the biomass per unit volume. Ecological
floating island technology has drawn increasing attentions in biotical-ecological
restoration due to its convenient operation, low environmental risks and effective
treatment (for example, pollutant purification, ecosystem restoration and landscape
improvement). The existing evidence showed that the effects of water spinach on
nitrogen and phosphorus removal, and chemical oxygen consumption are great of
significance for wastewater treatment and can improve water quality of eutrophication
(Wang, 1997). In addition, Wu et al. (2010) proposed a biopond-wetland system for
11

controlling cyanobacterial blooms in a pond at Kunming City, Western China, and
shown that when the hydraulic load of the biopond-wetland system is 500 m3 d on
non-rainy days, the efficiencies of the overall average nutrients removal rate are 83%
(Chl-a), 57% (total nitrogen), 70% (total phosphorus) and 66% (Ammonia),
respectively, indicating that the biopond-wetland system can significantly control the
outbreaks of toxic cyanobacterial blooms. However, it is difficult to control the
hydraulic retention time and the hydraulic loading rate, as well as the selection of
floating island plants for ecological floating island due to the local climate change and
the surrounding environments.
The aquatic plant competition method is to use the competition between aquatic
plants and harmful algae for light, nutrients, oxygen, and release allele-chemicals into
water environments to inhibit the growths of harmful algae. The work of Fang et al.
(2007) showed that the growth of algae can be inhibited by the cultivation of
beneficial algae or higher aquatic plants in water bodies. The integrated ecological
floating bed simultaneously used plants (for example, Ipomoea aquatica), freshwater
mussels (for example, Clam fluminea) and biofilm carriers (in artificial and semi-soft
assembly) to perform better than the other two floating beds, one of which is
composed of freshwater mussels and organisms, while the other is made of plants and
biofilm carriers (Li et al. 2010). However, the decomposition of aquatic plants after
apoptosis reduces the concentration of dissolved oxygen in the water bodies, resulting
in emitting a foul smell and accelerating the deposition and inundation of the water
bodies.
Macrophytes or algae are important organisms in the aquatic ecosystems, and
they are the primary food sources for herbivore fishes and other large aquatic
organisms. Thus, the excessive growth of algae can be controlled by adding
filter-feeding aquatic organisms, such as silver carp, nile tilapia and bighead carp.
Actually, the basic premise is that secondary consumers are removed through the
addition of tertiary consumers, which allows for the dominance of large-bodied,
generalist grazers to control phytoplankton population. For example, the study of
12

Infante and Riehl (1984) indicated that nile tilapia can swallow large amounts of algae
and zooplankton by the alteration of a food-chain to restore the health of the
ecosystem, resulting in the rapid decrease in the biomass of plankton populations.
Starling (1993) carried out a mesocosm experiment on a medium controlled
ecosystem in a tropical reservoir in Brazil, and the results showed that silver carp can
successfully control Microcystis blooms. Godlewsk and Swierzowski (2003) also
found that aquatic animal community can effectively inhibit the growth of algae in the
lake, which can improve water quality for the long-term, but the effect of algae
control is mainly related to the selection of aquatic species, distribution and pollution
degree of the water bodies. However, when toxic phytoplankton or harmful algal
blooms have filtered as food resource by fishes, their toxins may accumulate in
shellfish and the high level of toxins can be lethal to humans or other consumers.
Significantly, the effect of filter-feeding fish on water quality is typically short-lived
and most obvious in small and easily managed aquatic ecosystems.
It is widely recognized that toxin-producing phytoplankton are a group of
phytoplankton that have the ability to produce toxic chemicals into the aquatic
environments during harmful algal blooms, which contribute to the negative effects
on the economic and ecological values, and even pose a health hazard to animals and
human beings (Hallam et al., 1983). For example, Noctiluca scintillans (Macartney),
dinofagellates Alexandrium acatenella (Whedon et Kofoid) Balech, diatoms Nitzschia
pungens f. multiseries Hasle, etc., are harmful to planktonic organisms (Halllegraeff,
1993). Thus, many researchers have paid much attention to study the effect of TPP on
the termination of planktonic blooms from their field works and modeling analysis
over the past decades (Chattopadhayay, et al., 2002; Chattopadhayay, et al., 2004;
Sarkar et al., 2005; Sarkar and Chattopadhayay, 2003). For example, taking the
species Noctiluca scintillans (harmful phytoplankton), Chaetoceros sp. (harmful
phytoplankton) and Paracalanus sp. (zooplankton) as the research objective, and the
monitoring of plankton populations were carried out from March, 1999 to January,
2001 in the Northwest coast of Bay of Bengal by Sarkar et al. (2005), the results from
13

the biomass distribution observed in their filed works displayed that the presence of
harmful phytoplankton leads to the persistence of all species through the termination
of planktonic blooms and can be used as a controlling agent for the stability of marine
ecosystem (Sarkar et al., 2005). Although some excellent and interesting results have
been obtained, the reason for the occurrence of planktonic blooms and their possible
control mechanisms is still unclear currently. Thus, the progress of such important
areas urgently requires special attention both from experimental and mathematical
ecologists, which may greatly stimulate researchers to further explore the possible
control mechanisms of planktonic blooms.
As the basis of food chains in oceans, lakes, and reservoirs, plankton populations
are the natural and essential components of healthy aquatic ecosystems, which can
provide energy and materials to aquatic life. In most water bodies, there is a fine
balance between adequate irradiance and nutrient supply that determines the rate of
production of phytoplankton biomass, or primary productivity (Harris and Piccinin,
1977; Reynolds, 1984; Cloern, 1999). However, unnecessary and excessive growth
of phytoplankton may have a harmful impact on the aquatic ecosystems under some
suitable conditions, and their density levels can significantly affect ecological
balance by disrupting natural balance and reducing water quality. Thus, to control
the occurrence of algal blooms and regulate the balance of aquatic ecosystems, some
common methods technically, such as physical, chemical, and biological strategies,
have been widely applied to control harmful algae growth and inhibit bloom
formation. However, these algal blooms treatment methods all have their own pros
and cons currently. To be more precise, chemical control and physical control may
have the advantage of efficiency, but the practical observations show that these two
control methods consume much resources (i.e., labor-intensive and high costs), and
may cause secondary pollution. The effectiveness of biological control strategy may
not be as fast as that of the former two control strategies, but some of biological
control method are successful and environmentally friendly. Significantly, their
control durability may be better, as these control methods are implemented based on
14

the principle of ecological balance. Although all these three methods have made
great progress in controlling algal blooms, the process by which harmful algal
blooms occur is still not completely understood, and understanding of dynamics
mechanisms of changes in plankton populations becomes much more significant.
Therefore, the development of the effective methods and technologies for controlling
and preventing aquatic algal blooms is needed.
1.3. Objectives of the study
In recent years, biologists, as well as ecologists, have paid increasing attention to
explain the influence of some factors on the formation and disappearance of algal
blooms using a number of experiments (De Baar et al., 1995; Beman et al., 2005), and
some hypotheses have been established. For example, Critical Depth Hypotheses
(Sverdrup, 1953) and Dilution-Recoupling Hypotheses (Behrenfeld, 2010). The
results from experiments indicated that many factors affecting the dynamics of
plankton growth are bound to exist, for example, iron supply (Boyd et al., 2000; Coale
et al., 1996) and light (Stomp et al., 2004), but they can only qualitatively explain the
dynamics of plankton growth. Due to the complexity and openness of real aquatic
ecosystems and the limitations of the current technology, which contribute to the
difficulty to understand the mechanisms of plankton growth only depend on
experimental and field observations, and the key to solve the issue of algal blooms is
to understand the dynamics mechanisms of algae growth from dynamical analysis. In
this respect, mathematical modeling of plankton can provide the quantitative insight
into the dynamics of plankton growth. Moreover, the study of algal blooms is to
establish dynamics models for the evolutionary processes of blooms formation and
dissipation according to the dynamics mechanisms of population and water physical
environments, and thus mathematical modeling is now a classical way to study the
planktonic blooms (Truscott and Brindley, 1994a). Generally, dynamics modeling can
be used to analyze and solve the solutions of the models, understand the parameter
characteristics of the dynamics systems, and finally simulate the models by the means
15

of computer technology.
Based on the background of the ecological environments in the subtropical lakes
and reservoirs, the dissertation research mainly takes the planktonic algae as the
research objective, aiming to model the dynamics mechanisms of plankton growth
and evolution. In this dissertation, some theories related to population dynamics,
impulsive control dynamics and stochastic dynamics, as well as the methods of
dynamics modeling, dynamics analysis and experimental simulation, are applied to
reveal the dynamics mechanisms of the spatial-temporal distribution of plankton and
predict the dynamic evolutionary processes of plankton growth. Significantly, the
objectives of the dissertation are to study some nonlinear problems of population
dynamics, such as the dynamics mechanisms of the termination of planktonic blooms,
the dynamics mechanisms of spatial-temporal evolutionary processes of plankton
growth and the dynamics mechanisms of the dynamic prevention and control of
planktonic blooms. The specific objectives include:
Ø Establishing an impulsive reaction-diffusion hybrid system and analyzing the
dynamics of the proposed system. Several simulation experimental factors,
for example, impulsive control and the heterogeneity and homogeneity of
environments, are taken into account to investigate the effects of these factors
on the survival of species, the dynamics spatial-temporal distribution of
species, and the dynamics evolutionary mechanisms of species by means of
the experimental simulation technology.
Ø Considering a stochastic phytoplankton-zooplankton system with toxic
phytoplankton. Environmental stochasticity (for example, white noise) was
firstly chosen as a control parameter, aiming to explore the dynamical
behaviors of plankton under the environmental fluctuations. And then the
toxin liberation rate released by TPP was examined to reveal the dynamics
mechanisms of the formation and disappearance of planktonic blooms, and
finally the combined effects of two liberation rates on the process
performance of planktonic blooms were further discussed using the
16

experimental simulations.
Ø Building a stochastic phytoplankton-toxic phytoplankton-zooplankton system
under regime switching, and devoting our attention to address the issues
though studying the dynamics of the system. To be more precise, studying
the long-time behaviors of phytoplankton-toxic phytoplankton-zooplankton
system in a fluctuating environment and revealing the effects of white noise,
regime switching and TPP on the dynamics mechanisms of the termination of
planktonic blooms.
Ø Developing a stochastic non-autonomous phytoplankton-zooplankton system
involving toxin-producing phytoplankton and impulsive perturbations, and
investigating the effects of these factors on the dynamics of plankton model.
More precisely, we aim to study how do environmental noise and impulsive
control influence the survival of plankton populations and what influence in
the peak of the cyclic outbreaks of planktonic blooms in an impulsive
perturbations and fluctuating aquatic environments.
1.4. Organization of the dissertation
The dissertation is organized as follows: Chapter 2 presented some literature
review. In chapters 3-6, four different mathematical and biological models of plankton
dynamics were proposed, respectively, aiming to study some key biological factors
affecting the growth of plankton populations in aquatic ecosystems and reveal the
dynamics mechanisms of the termination of planktonic blooms (see Fig. 1.1). They
mainly include an impulsive reaction-diffusion hybrid system (in chapter 3), a
stochastic phytoplankton-zooplankton system with toxin-producing phytoplankton (in
chapter 4), a stochastic phytoplankton-toxic phytoplankton-zooplankton system under
regime

switching

(in

chapter

5),

and

a

stochastic

non-autonomous

phytoplankton-zooplankton system involving toxin-producing phytoplankton and
impulsive perturbations (in chapter 6). The conclusion of this research and the
recommendations for future research were provided in chapter 7.
17

Study on the
modeling
mechanisms of
plankton growth
and evolution
Taking impulsive
perturbations white
noise, regime
switching, and TPP
into new plankton
dynamics models

An impulsive
hybrid reactiondiffusion hybrid
system

A stochastic
phytoplanktontoxic
phytoplanktonzooplankton
system with
regime switching

A stochastic
phytoplanktonzooplankton
system with two
toxin-producing
phytoplankton

A stochastic
phytoplanktonzooplankton
system with
impulsive
perturbations

Using modeling analysis
and experimental
simulation technologies

Revealing the dynamics
mechanisms of the
termination of
planktonic blooms
Fig. 1.1. Schematic diagram for the structure of the research.

18

Chapter 2 LITERATURE REVIEW
2.1. Plankton dynamics model
The dynamics of plankton studies mainly the dynamics mechanisms of plankton
and the effects of some internal and external factors on the mechanisms in real-world
aquatic environments. Dating back 100 years ago, many scientists and researchers
have investigated the dynamics of plankton models, and the plankton research have
regularly combined with field observations, laboratory experiments, as well as
mathematical modeling from the beginning. Due to the strong positive correlations
between zooplankton and fish abundance, fisheries and climate change have greatly
stimulated the common interest of researchers in studying the dynamics of plankton
populations. Actually, since the pioneering work of Fleming (1939), many
mathematical models have been developed and explored extensively for the
phytoplankton productivity (Ivlev 1945; Riley, 1946; Odum, 1956). Subsequently, a
collection of the most frequency used models was presented by Behrenfeld and
Falkowski (1997), which could be of great help in estimating the productivity of
marine ecosystems. All these works have greatly lead to the rapid development of
plankton dynamics models.
More precisely, the first attempt to study plankton dynamics models by using a
simple ordinary differential equation can be traced back to the work of Fleming
(1939), who demonstrated that zooplankton grazing plays a crucial role in the
controlling of phytoplankton blooms and opened the chapter of the study of the
dynamics of plankton models. Subsequently, Ivlev (1945) introduced the functional
response to more realistically describe the interactions between phytoplankton and
zooplankton populations. And then some other functional responses, such as
Holling-type response terms (Holling, 1959, 1973), Mayzaud and Poulet (1978), have
been widely considered to study the dynamics of plankton models (Steele and
Henderohn, 1981, 1992; Malchow, 1993; Truscott and Brindley, 1994a; Scheffer,
2004; Garvie and Ttenchea, 2007). Other approaches the researchers are trying to
19

construct the data-fitted functions (Riley, 1963) and apply Lotka-Volterra model to
describe the interactions of prey-predator for phytoplankton-zooplankton (Dubois,
1975; Mimura and Murray, 1978; Segel and Jackson, 1972; Gierer and Meinhardt;
1972; Levin and Segel, 1976). In this respect, the understanding of plankton dynamics
models has been greatly improved in recently years, which provide a deeper insight
into solving some ecological problems (Huo and Li, 2004; Mukhopadhyay and
Bhattacharyya, 2008; Zhang et al., 2014; Priyadarshi et al., 2017; Yu et al., 2018). In
addition, due to the rapid development of population model and the idea of its
construction, which have greatly lead to the establishment and development of
plankton dynamics models. Thus, the following mainly introduces the related research
progress of population model.
The study of population dynamics looks back over two centuries of history in the
mathematical and ecological sciences, and the earliest and simplest population model
is well-known Malthus population model. Actually, Thomas Robert Malthus
published his "Principle of Population" thesis in 1798, and proposed the Malthus‟s
theory that population is always growing in a geometric progression and advocated
the implementation of population control (Malthus, 1798). It is assumed that the per
capita growth rate of change in the number of individuals is a constant

and derived

the following model of the variation of population:

with an initial value

0 =

Noted that the trajectory of

0 , where

( )=

=

(2-1)
is the number of population at the time .

0 exp

is exponential. Such an exponential

trajectory is of considerable theoretical significance, as it represents the long-term
behaviors of age-structured population affected by constant mortality and fertility
rates (Keyfitz, 1977). But the essence of this model is that each individual generates a
constant number of offspring, regardless of the crowding (i.e. environmental carrying
capacity) and the availability of food supply, which is not in line with the reality. In
fact, the influence of external factors on the population growth is not considered in the
Malthus model, which makes population growth always a constant. When the typical
20

size of the population is small, the model can make a more accurate prediction, while
it cannot reflect the real situation for a large human population base. In other words,
the exponential population growth is incompatible with linear growth of food
resources.
Considering that unchecked infinite exponential growth is patently unrealistic
and the growth of population must be regulated by some biological factors, for
example, the availability of food supply and the living environments of population.
Thus, Verhulst (1838) introduced the density constraint to the basis of Malthus
population model and initially derived the logistic growth model, where the per capita
rate of change decreases linearly with the population size. Afterwards, Pearl applied
the logistic growth equation, which has been derived independently by Lotka (1925),
to model the population growth in US (Pearl, 1920). The equation can be presented as
follows:

where

(

=

1−

(2-2)

> 0) is the environmental carrying capacity of population size, which

means the maximum number of population that the natural environment can
accommodate. Also, Verhulst assumed that the growth rate of population follows the
law of logistic growth

1−

. This implies that the population growth slows down

as the number of population increases, indicting the environmental resources can
significantly affect the growth of population. Other models, such as the Gompertz
growth

=

ln

, can also show many of the same properties, but the logistic

growth equation is arguably the best-known and most widely applied rate equation for
the population growth and population invasion (Mendez et al., 2014).
It should be noted that both Eq. (2-1) and Eq. (2-2) only discussed the single
population models, but species in nature are not independent of each other, for
example, predation, competition, and parasitism. The first attempt to establish the
multi-population prey-predator model by the work of Lotka and Volterra (Lotka, 1925;
Volterra, 1926), who presented the Lotka-Volterra competition model that describes
21

the population densities of prey

and predator
=

−

=

1

1

−

,

.
(2-3)

.

The construction of the model is based on the following assumptions: the growth rate
−1

of the prey population density , given as
of the predator density

and greater than zero when the predator density is zero.

Conversely, the predator growth rate
but be negative when

, should be a decreasing function

−1

should increase with the prey count,

= 0. In addition, the parameters , 1 , , and

1

describe the

prey reproduction rate, predation rate, predator death rate, and conversion efficiency
of prey into predator, respectively, and all the parameters are positive. This set of
coupled ordinary differential equations gives rise to characteristic, undamped,
non-linear oscillations. Generally, the classical Lotka-Volterra model can be given as
the following form:
=

+

=

+

1

1

+

+

,

2

.

2

(2-4)

However, the relationship between predator and prey described by classical
Lotka-Volterra model is rather realistic, with the most obvious flaw being the
unchecked growth of the prey in the absence of predator. In the 1930s, consequently,
Gause proposed a generalized mathematical model in which the rate parameters
effectively become response function of the respective species, allowing more
realistic control of populations compared to the original Lotka-Volterra model (Gause,
1934; Royama, 1971). The Gause type predator-prey model is introduced as follows:
=
where

=−

−

+

is the natural death rate of predator and

prey into predator. Function

,

.

(2-5)

is the conversion efficiency of

denotes the growth rate function of the prey and

( ) represents the predation response function. The selections of functional
22

response are of great significance to describe the predator-prey phenomenon in
different environments and many scholars have done a lot of works to construct a
more accurate functional response function by field observations and laboratory
experiments, for example, Michaelis-Menten uptake dynamics (Dugdale, 1967)
provide a more realistic reflection of nutrient uptake dynamics (Huppert et al., 2002).
In this dissertation, we mainly consider Holling-II functional response into the model.
Actually, Michaelis and Menten (1913) proposed the following functional response
when studying the saturation models of enzyme kinetics:

where

=

+

is the maximum growth rate of population and

(2-6)
( > 0) denotes the

half-saturation constant. Actually, Holling obtained this function through the

laboratory experiment (Holling, 1959) and named the function as Holling-II
functional response. Fortunately, we apply the functional response to analyze the
dynamics of phytoplankton-zooplankton system in this dissertation. Other functional
responses such as Crowley-Martin type (1989), Beddington-DeAngelis type
(Beddington, 1975; DeAngelis and Goldstein, 1975) are also considered in the
following discussion.
Plankton models can be regarded as a specific application of predator-prey
models, and its mathematical structure relies strongly on the generalization or
improvement of the mentioned population systems. Motivated by the above works,
the construction of plankton models is based on the following assumptions: the
natural growth rate of phytoplankton follows the law of logistic growth; the capture of
phytoplankton by zooplankton needs to be described by an appropriate functional
response function and consider the natural death of zooplankton populations.
Moreover, the study of plankton ecology is an important significance of the survival
of our earth, but the occurrences of planktonic blooms have been reported globally
with an increasing frequency over the past decades (Hallegraeff, 1993), and TPP are
among the contributors in these blooms (Hallegraeff, 1993; Philips et al., 2004;
Hallam and Luna, 1984). Therefore, considerable scientific attention towards harmful
23

phytoplankton when studying phytoplankton-zooplankton systems has been paid
(Estep et al., 1990; Huntley et al., 1986; Buskey and Hyatt, 1995; Wyatt and Horwood,
1973). For example, the field observation (Estep et al., 1990) and laboratory
experiment (Huntley et al., 1986) shown that the toxicity may be as a strong mediator
in the zooplankton feeding rate. Thus, it is necessary to consider the effect of the
toxins produced by harmful phytoplankton on zooplankton when studying
phytoplankton-zooplankton systems. Based on the field observation and model
analysis, Chattopadhyay et al. (2002) proposed the following nonlinear coupled
ordinary differential equations:
=

=

1−

−

−

,

−

(2-7)

.

Compared with Eq. (2-5), system (2-7) added an additional term −

describes the effect of phytoplankton toxins, where
produced by TPP and

that

denotes the toxin liberation rate

represents the distribution of toxic substances.

Interestingly, the authors investigated the existence and local stability of positive
equilibria and the existence of Hopf-bifurcation of the system by considering different
combinations of functional response

and

, and concluded that TPP may be

used as a biological way to control the planktonic blooms (Chattopadhyay et al.,
2002).
Based on (2-7), the dynamical behaviors of plankton systems with harmful
phytoplankton, such as stability, bifurcation and chaos, have been explored
extensively in recent years (Saha and Bandyopadhyay, 2009; Sarkar and
Chattopadhayay, 2003; Luo, 2013; Jang, 2014; Roy et al., 2006; Upadhyay and
Chattopadhyay, 2005; Chaudhuri et al., 2013). For example, the work of Saha and
Bandyopadhyay

(2009)

shown

some

dynamical

properties

of

the

toxic

phytoplankton-zooplankton system without time delay, and further investigated the
existence of local Hopf-bifurcation induced by time delay, and studied the existence
of stability switching phenomena and the stability of Hopf bifurcation and the
direction of the bifurcating periodic solution. Sarkar and Chattopadhayay (2003)
24

introduced the environmental fluctuations into the phytoplankton-zooplankton system
with TPP, and the results shown that TPP and the controlling of the rapidity of
environmental fluctuations are key factors influencing the termination of planktonic
blooms. In this dissertation, we apply schematic few-species models to demonstrate
that phytoplankton-zooplankton interactions can give rise to the dynamics
mechanisms of plankton systems and predict the dynamic evolution process of
plankton growth.
2.2. The research of nonlinear dynamics
The commonality of the above mentioned studies is to investigate the dynamics
of the nonlinear problems in the plankton dynamics models. Generally, the research of
nonlinear dynamics system includes: the study of nonlinear dynamics of nonlinear
phenomena in dynamic systems and the establishment a nonlinear dynamic model
respect to the practical problems. Due to the development of mathematics and
computer technology, an increasing attention has been paid to the study of nonlinear
dynamic systems. Especially, at the end of the 19 th century, Poincaré proposed the
concept of dynamics and established a qualitative theory (the theory that directly
studies the behaviors of solutions without using the solutions of the equations) when
studying celestial mechanics, which laid the foundation for the study of nonlinear
dynamics (Poincaré, 1890). Almost at the same time, Lyapunov published his doctoral
dissertation "General Theory of Motion Stability" between the year of 1882 and 1892,
which greatly promoted the development of nonlinear dynamics (Lyapunov, 1966).
Moreover, Lorentz was the first to discover the strange attractor (or called Lorentz
attractor) and described it as the famous "Butterfly Effect" (Lorentz, 1963) in 1963,
which opened the chapter of the study of chaotic dynamics. The discovery of chaos
has become one of the most important achievements in the field of nonlinear
dynamics (Lorenz, 1993). In addition, Prigogine (1969) put forward the theory of
dissipative structure in 1969, which had a great influence on physics, chemistry,
biology and other disciplines, and pushed the study of nonlinear dynamics to a new
25

high level. Here, we do not full cover the discussions of all the branches of nonlinear
science. Instead, we concentrate on the brief description of the dynamic methods of
nonlinear dynamics in related to this dissertation.
2.2.1. The dynamics of stochastic system
An important problem in nonlinear and statistical physics is to under the
underlying laws for various phenomena in dynamical systems forced by random
environmental fluctuations. Many phenomena in nature are positively or negatively
affected by environmental fluctuations, which can be called the stochastic phenomena.
For example, among the modern industrial process, the social economy and other
fields of a variety of dynamic systems, almost all of them are subjected to various
random factors. When the intensity of random disturbance is small or the accuracy of
the system is not high, the system can be generally regarded as a deterministic one.
On the contrary, the interaction between nonlinearity and stochasticity of dynamical
systems can give rise to unexpected phenomena which have no analogue in the
deterministic case (Moss and MeClintock, 2007; Anishchenko et al., 2007). The
problem is fundamental because it involves the interplay between the modern
qualitative theory of deterministic dynamical systems and the stochastic analysis. For
example, some properties of deterministic systems may be lost by using deterministic
methods to describe some stochastic systems, and it is difficult to achieve the
expected results by means of the analysis and control methods of the deterministic
system theory for studying stochastic systems. Thus, it is necessary and important to
consider the effects of stochastic fluctuations on the systems and take the study of the
dynamic systems as stochastic systems in these cases (Mao, 1997a). Actually,
stochastic fluctuations are usually referred to as random noise, and the solutions of the
equations with random parameters (or stochastic dynamical systems) are called
stochastic processes. Generally, deterministic systems can be described by ordinary
differential equations, while the stochastic systems can be presented by stochastic
differential equations (Mao, 1995, 1997a).
26

Since Kiyoshi Ito initiated stochastic analysis and introduced stochastic integral
and stochastic differential equations in 1942, the stability theory of stochastic systems
has been developed rapidly. In the middle of the last century, the study of stochastic
dynamics was widely applied to the natural sciences, engineering and social sciences,
as well as in the study of nonlinear stochastic problems in physics, chemistry, biology
and medicine. Some related theories, such as stochastic stability, stochastic optimal
control, stochastic limit cycle, stochastic bifurcation, and stochastic chaos, have been
established by mathematicians, physicists and mechanics. Subsequently, many
monographs, anthologies and textbooks on stochastic dynamics have been published
in the world (Gihman and Skorohod, 1979; Khasminskii, 1980; Meyn and Tweedie,
1983; Mao, 1994, 1997b; Arnold, 1998; Chueshov, 2002). For example, Mao (1997b)
introduced stochastic differential equation and its application theory in detail.
Chueshov (2002) introduced the theory of stochastic differential dynamics system and
its application. All these results have further lead to the development of the theory of
dynamics of stochastic system.
In the real world, there exist many types of random noise and all species have
evolved in the presence of environmental disturbance. The work of Robert et al. (1998)
showed that the environmental fluctuations can be divided into the following groups:
biologically based disturbance of populations (for example, overharvesting, invasion,
disease, and their interactions) and physically based disturbance such as storm,
volcanic eruption and forest fire. Generally, the Gaussian noises (Brownian motion or
Wiener process) can describe the continuous and random amplitude at any time that
often occurs in reality, which satisfies the Gaussian distribution function based on
biological disturbance as a whole (Kingman and Feller, 1972). If the governing
variables are or can be approximated as discrete, one can employ the well-established
theory of Markov chains (Cox and Miller, 1977; Norris, 1998). There is now
compelling evidence that the nature is usually assumed to respond to gradual change
in a smooth way, however, many studies on lakes, coral reefs, oceans, forests and arid
lands have shown that the smooth change can be interrupted by sudden drastic
27

switches to a contrasting state (Scheffer et al., 2001). To describe such phenomenon,
continuous discrete state Markov chains can be used to simulate random populations
switching from one state to another or more (or regime switching) (Mao and Yuan,
2006). For example, Sun et al. (2018) revealed that the death rate of newborn Little
Yellow Croaker (Larimichthys polyactis) are 0.53 and 0.84 at 22℃ and 16℃,
respectively. At the same time, some biological experiments and observational
evidences illustrated that many populations have scale-free characteristics in spatial
distribution, which may be strongly related to the discontinuous Levy noise in the
environments (Levy, 1925; Sato, 1999) (for example, micro-plankton (Reynolds, 2008)
and bees (Stout and Goulson, 2001)). Therefore, many researchers have applied levy
noise to describe the sudden and catastrophic interference based on physical
principles (Cognata et al., 2010; Applebaum, 2009). In recent years, ecological
models with random disturbances have attracted the interest of many researchers, and
some excellent results have been achieved. However, most of these models are based
on the classical Lotka-Volterrs models or the epidemic chamber models, and there are
few literatures to study the aquatic ecological populations, especially plankton
populations. Moreover, accumulating evidence showed that the environmental
fluctuations can play an important role in real ecosystems (Carpenter et al., 2011;
Meng et al., 2014), and the survival of plankton populations can be significantly
affected by environmental stochasticity (Melbourne and Hastings, 2008). All the
results could provide us a deeper understanding for the realistic aquatic ecosystems
when incorporating environmental fluctuations into the systems. Here the question
may arises: can environmental fluctuations such as white noise and regime switching
are a cause of the termination of planktonic blooms? Therefore, in this dissertation,
we mainly apply the theories of the dynamics of stochastic systems to study the
temporal and spatial evolution of plankton distribution for plankton ecosystems and
further reveal the distribution mechanism of plankton populations in fluctuating
aquatic environments.

28

2.2.2. The dynamics of impulsive control system
In the real world, the states of systems will change suddenly because of the
influence of some factors, such as drought, flooding, hunting, planting. Due to these
changes of relatively short time interval at some fixed times, it is usually regarded as
instantaneous discontinuous phenomena, which can be represented by impulsive
effects that may be suitable for such phenomena (Lakshmikantham et al., 1989;
Bainov and Simeonov, 1993). For example, during the outbreaks of aquatic algae, the
implementation of mechanical measures, such as algae removal, will result in a
short-term decrease in the density of harmful algae. In the production of fishery
resources, the launch of young fish and the harvest of adult fish will cause
instantaneous changes in fish density. Also, in agricultural production, people usually
control pests by spraying insecticides or releasing natural enemies regularly, which
lead to the rapid changes in the number of pests and natural enemies in a short period
of time. These transients or almost instantaneous behaviors can indeed cause sudden
and abrupt changes in the state of species, which cannot be described by continuous
systems. Therefore, the greatest interest for people is what the response of systems to
experiencing such sudden changes in the real world, and a relatively effective method
to study such problems is to use the impulsive dynamics systems. The most prominent
feature of impulsive dynamics systems is that it can fully consider the impact of
instantaneous mutation on the system states, and profoundly and accurately reflect the
law of action of things, which seems to be more realistic (Jannash and Mateles, 1984;
Jin et al., 2004).
Impulsive dynamics system is a description of the evolution process of sudden
changes of the states of systems under instantaneous disturbance using impulsive
differential equations. Actually, since the pioneering work of Minman and Myshiks
(Mi‟man and Myshkis, 1960), impulsive dynamics systems have been the common
interest among researchers, who are committed to its theoretical researches
(Simeonov and Bainov, 1988, 1989; Bainov and Covachev, 1994; Samoilenko and
Perestyuk, 1995; Chen, 2011, 2013). Especially, the applications of Lyapunov
29

function and Razumikhin technique have given a good explanation of the properties
of the solutions of impulsive systems, which greatly promoted the development of
theory and practical applications. For example, the treatment of some diseases
(Lakmeche and Arino, 2000; Panetta, 1996), pulse immune control and therapy
(Donofrio, 2002a, 2002b; Shulgin et al., 1998, 2000), Microbial culture (Agur et al.,
1993), population optimal control (Zhang et al., 2003; Mailleret and Grongard, 2008),
pest control and management (Lu et al., 2003; Jatav and Dhar, 2014) and ecological
environment management (Martins et al., 1997; Huppert et al., 2002; Liu et al., 2003;
Dai et al., 2013; Wang et al., 2014). Significantly, the research on the impulsive
dynamics systems is attractive and challenging because of the combined with the
characteristics of continuous and discrete systems, but it is not simply a superposition
of the two systems, but a new system. Additionally, the biggest difference among
them is the impulsive dynamics system can describe some complex problems that
cannot be described separately by the continuous or discrete dynamics systems.
Generally, impulsive dynamics system can be divided into the following systems:
impulsive dynamics system in which impulse occurs at a fixed time (periodic
impulsive dynamics system), impulsive dynamics system in which impulse occurs at
variable time (time-varying impulsive dynamics system) and impulsive dynamics
system depending on the state of system (state-dependent impulsive dynamics system)
(Lakshmikantham et al., 1989; Bainov and Simeonov, 1993). In the past decades, the
periodic impulsive dynamics systems have been widely explored and its theoretical
systems have been improved (Liu and Chen, 2003; Chen et al., 2009; Liu et al., 2006;
Wang et al., 2009; Wang et al., 2008; Yu et al., 2011; Xiang et al., 2009). In these
existing literatures, they mainly studied the impulsive dynamics system in which the
impulse occurs at a fixed time in the biological system, proved the existence and
boundedness of the semi-trivial periodic solution of the system, and obtained the local
and global asymptotic stability of the semi-trivial periodic solution, which are the key
conditions ensuring the permanence of the biological populations. Significantly, the
results from experimental simulations showed that chaos can occur in the process of
30

long-time dynamic behaviors, and further discussed how to control the development
of biological populations so that we can make use of biological resources and thus
obtain greater economic benefits. As another important branch of impulsive control
dynamics system, state-dependent impulsive control dynamics system can simulate
the advantages of the controlling depending on the state of population. Recently, the
studies on state-dependent impulsive differential equations have become the common
interest among many researches (Tang and Cheke, 2005; Jiang et al., 2007; Dai and
Zhao, 2012; He et al., 2015), and some properties of periodic solution, such as the
existence of periodic solution, stability and periodicity, have been investigated (Nie et
al., 2009; Dai et al., 2012). In addition, some existing literatures revealed that the
existence, stability and bifurcation of periodic solutions of state-dependent impulsive
differential equations using successive functions (Guo et al., 2014; Wei and Chen,
2014; Zhang et al., 2014; Ji et al., 2015; Xiao et al., 2015), indicating the easier way
to obtain the economic benefits and practical value by taking the states of populations
as the impulsive control parameter. For example, in the management of
disaster-related problems such as pest outbreak, algal blooms and red tide, their
control implementations strongly depend on the states of species. In recent years, with
the frequent occurrence of algae blooms, people usually rely on their own experience
to control them, but the corresponding treatments are not satisfactory and may lead to
secondary contamination. Here the question may arises: does the impulsive control
affect the dynamics of plankton systems and further what does role the impulsive
control play in the formation and disappearance of planktonic blooms? Therefore, in
this dissertation, we will establish the related plankton dynamics systems under the
impulsive effects, and try to study the effect of the impulsive control on the dynamics
mechanisms of the termination of planktonic blooms and predicting the dynamic
evolution process of plankton growth.
2.3. Summary of literature review
Based on the dynamic models of differential equations, especially for the
31

relationship between phytoplankton and zooplankton populations, the influence of
some different key factors on the dynamics mechanisms of the termination of
planktonic blooms can be revealed, which can provide a theoretical basis for
exploring the dynamics mechanisms of the formation and disappearance of planktonic
blooms. Although many studies on the dynamics of plankton systems have made great
progress, the dynamics mechanisms of plankton growth are still not completely clear
recently, especially for the detailed dynamics mechanisms of planktonic blooms. In
recent years, some biological factors, for example nutrition (Huppert et al., 2002),
water temperature (Zhao et al., 2020), time delay (Rehim and Imran, 2012), flow rate
(Dai et al., 2015), predation pressure (Wyatt and Horwood, 1973; Uye, 1986), and
harmful phytoplankton (Banerjee and Venturino, 2011), have significantly affected the
dynamics mechanisms of planktonic blooms. However, other biological factors, for
example population diffusion, impulsive control, environmental fluctuations and TPP,
can also affect the dynamics of plankton growth, but they are rarely considered into
the existing literatures. Here some questions may arise: what role does the
reaction-diffusion effect play in the dynamics of spatial and temporal evolutionary
process of plankton distribution, and how the impulsive control affect the survival of
plankton? Also, how the impulsive control and white noise influence on the dynamics
of plankton, especially the extinction and persistence in the mean of plankton in a
fluctuating environment and what influence the peaks of the cyclic outbreaks of
planktonic blooms under man-made factors and fluctuating environment? Moreover,
what are the potential mechanisms that contribute to the extinction of plankton in an
evolutionary setting when plankton is subjected to a white noise? Furthermore,
predicting how the prolonged coexistence of plankton under the effects of white noise
and regime switching and studying what are the mechanisms that affect the peaks of
the outbreaks of planktonic blooms in a fluctuating environment? In a word, these
interesting problems in the dynamics mechanisms of the formation and disappearance
of the planktonic blooms can be explored in the dissertation, and it is still necessary
and important to study the nonlinear problems of planktonic blooms.
32

Chapter 3 DYNAMIC ANALYSIS OF A REACTION-DIFFUSION
IMPLUSIVE SYSTEM1
Abstract
In this paper, a predator-prey system with Crowley-Martin functional response,
which is described by a couple of reaction-diffusion equations with impulsive, is
studied analytically and numerically. The aim of this research is to analyze how the
impulsive effect influences on dynamic of the system. Dynamics of the system,
including the ultimate boundedness, permanence and extinction, are investigated
firstly under impulsive effects. Significantly, it is found that there exists a unique
positive periodic solution that is globally asymptotically stable when impulsive effects
reach some critical state. Additionally, a series of nu merical simulations are carried
out to further study dynamics of the system, which are consistent with the analytical
results.

Keywords: Crowley-Martin functional response, Reaction-diffusion equation,
Permanence, Periodic solution, Impulsive effect

1

This work has been published as Liu, H., Yu, H.G., Dai, C.J., Wang, Q., Li, J.B.,
Agarwal, R.P., Zhao, M., 2019, Dynamic analysis of a reaction-diffusion impulsive
hybrid system. Nonlinear Analysis: Hybrid Systems 33, 353-370. DOI:
10.1016/j.nahs.2019.03.001.
2
This work has been published as Liu, H.,33Dai, C.J., Yu, H.G., Guo, Q., Li, J.B., Hao,

3.1. Introduction
In real world, the spatial distribution of species seems to be heterogeneity in the
bounded domain, and each species has a nature tendency to spread to areas where the
population concentration is much smaller. Moreover, stabilization of predator-prey
dynamics can be achieved through spatial heterogeneity, whose stabilizing effect of
partial isolation of habitat patches is very important (Scheffer, 1998; Akhmet et al.,
2006). Then, the reaction-diffusion equations may be a very useful tool to describe the
spatial dispersal of species. Actually, since the pioneering work of Turing (1952),
diffusion phenomena have been observed as causes of the spontaneous emergence of
ordered structures, called patterns, in many non-equilibrium situations (Pearson, 1993;
Bois et al., 2011; Liu et al., 2012; Hillerislambers et al., 2001). In recent decades, the
studies on reaction-diffusion equations have made great progress, by which a great
variety of systems with diffusion process are widely explored, and many excellent
results have been obtained (Melkemi et al., 2005; Pang and Wang, 2003; Peng and
Wang, 2005; Hu et al., 2014; Dai et al., 2015; Dai and Zhao, 2014). Those works
mainly focus on continuous dynamic systems with diffusions. However, there exist
impulsive phenomena in some diffusion processes, which cause studies on the
dynamics of the system with diffusion and impulsive to be one of the significant
research interests currently.
In reality, practical populations undergo inherent discontinuity of many natural
and man-made factors, which lead to rapid population decrease or increase over a
fixed time (for example, fire, drought, hunting, harvesting breeding, etc.). Systems
with such kinds of discontinuous changes refer to impulsive differential equations,
which have attracted the interest of many researches in the past decades since they
provided a natural description of observed evolutionary behavior of certain real-world
problems (Samoilenko and Perestyuk, 1995; Zavalishchin and Sesekin, 1997). For
example, many biological phenomena involving thresholds, bursting rhythm models
in medicine and biology, optimal control models in economics, pharmacokinetics, and
frequency modulation models can exhibit impulsive effects, and then, impulsive
34

effects have been extensively investigated (Nieto and 'Regan, 2009; Dai et al., 2012c;
Jatav and Dhar, 2014; Chakrabory et al., 2015; Zhao et al., 2012; Wang et al., 2014).
Especially, the dynamic of reaction-diffusion system with impulsive have been the
common area of interest among many researchers (Akhmet et al., 2006; Wang et al.,
2010; Liu et al., 2011, 2014; Zhong and Liu, 2010; Struk and Tkachenko, 2002).
Akhmet et al. (2006) have investigated the dynamics of an impulsive ratio-dependent
predator-prey system with diffusion, and they obtained some conditions for the
permanence of the predator-prey system and for the existence of a unique globally
stable periodic solution. Liu et al. generalized the impulsive reaction-diffusion system
to three populations and obtained a series of interesting results (Liu et al., 2011, 2014;
Zhong and Liu, 2010). All these works have lead to the development of impulsive
reaction-diffusion systems.
In addition, functional response plays an important role in the population
dynamic due to its characterization for the interaction between predator and prey.
Holling (1959) introduced the concept of the functional response, which describes the
asymptotic relationship between prey removal rate per predator and the density of
prey. There exist many functional responses, such as Leslie-Gower, Holling type II,
ratio-dependent, and so on. One well-known example is the Crowley-Martin
functional response (Crowley and Martin, 1989), where it is assumed that
predator-feeding rate decreases with higher predator density even when prey density
is high. At present, the system with Crowley-Martin functional response have been
explored extensively (Upadhyay and Naji, 2009; Jazar 2013; Dong et al., 2013; Shi et
al., 2011; Meng et al., 2011; Liu et al., 2013; Dong and Zhang, 2015), such as a
delayed Chemostat model (Dong et al., 2013), a stage-structured predator-prey model
(Shi et al., 2011), a stochastic predator-prey model (Liu et al., 2013), a diffusive
predator-prey system (Dong and Zhang, 2015), and so on, and many interesting
results have been shown. Obviously, it is interesting and important to study the
dynamics of the prey-predator systems with Crowley-Martin functional response.
Additionally, impulsive effect exists in some diffusion processes in ecosystems.
35

However, the impulsive diffusion prey-predator system with Crowley-Martin
functional response has rarely been studied, and dynamics of the impulsive
prey-predator system with Crowley-Martin functional response under diffusion is still
not very clear currently. Hence, the study of the dynamics on the impulsive reaction
diffusion prey-predator system with Crowley-Martin functional response is the aim of
this paper, especially, the influence of the impulsive effect on the dynamics of the
prey-predator systems with Crowley-Martin functional response under diffusion.
Motivated by above works, a impulsive diffusion prey-predator system with
Crowley-Martin functional response, which describes the interaction among a
top-predator, an intermediate predator and a prey, is presented, as follows:
∂

1

∂

2

∂t

∂t

∂

=

1∆ 1 +

=

2∆ 2 +

1+ 3 ,

3

∂t

and

where

=

+

,

=

= 0,

1 and

predator;

1

2

2

2+ 4

3∆ 3 −

,

1

3

,

2

,

,

,

,

−
2 3

3+ 3

−
,

2

1

,

4

3 + 1+

,

,

1

3

−

1

,

2
2 3

,

,

2+ 4

,

+
′

2

,

,

2

1

1+ 1 ,

1+ 2 ,

1+ 1 ,

1+ 2 ,

,

2 3

3+ 3

,

,

4

1 and

2+ 1

1 2

2+ 1

,

2

,

,

2 3

,

= 1,2, ⋯

= 1,2,3

2

1 2

′1 ,

,

3

,

2

,

1 2

,

1 2

(3-1)
−

(3-2)
(3-3)
(3-4)

(3-5)

are the respective intrinsic growth rates of prey and an intermediate
2 are the respective coefficients of intra-species competition of

prey and intermediate predator;

1 and

2 are the respective maximal predator per

capital consumption rates, i.e., the maximum number of prey that can be eaten by an
intermediate predator and maximum number of intermediate predators can be eaten
by top predators in each time and space unit;

′1 and

′2 are the respective

conversion of prey to inter-mediate predator and intermediate predator to top predator;
3 is the death rate of the top predator;

1,

36

3 and

2 , 4 are the handling time and the

magnitude of interference among predators, respectively;

= 1,2

is the

2

2
1 +

respective environmental carrying capacity of the prey and intermediate predator.
,

⋯+

, , ′
2

2

= 1,2 ,

3 and

= 1,2,3,4 are positive functions.

is the Laplace operator;

1, 2

and

3 are

=

positive diffusion

coefficients and reflect non-homogeneous population dispersion; Neumann boundary
conditions (3-5) characterize the absence of migration;
derivative,

=

∪

is the outward normal

.

In addition, it is assumed that prey and predator populations are confined to a
fixed bounded space domain

⊂

with smooth boundary and are non-uniformly

distributed in the domain. Furthermore, they are subjected to short-term external
influences at fixed time
< ⋯ with lim →∞

, where is a sequence of real numbers 0 = 0 < 1 < ⋯ <

= +∞.

The rest of this paper is organized as follows: Section 3.2 presents the basic

assumptions and useful theorems firstly, and then, we apply the upper and lower
solution method and comparison principle of impulsive differential equations to
obtain the conditions for ultimate boundedness of the solutions, the permanence of the
system, and the extinction of the top predator, and finally the existence of a unique
periodic solution by establishing of an appropriate auxiliary function. A series of
numerical simulations are carried out to further study the dynamics of the system in
Section 3.3. In Section 3.4, we summarize the results and present our conclusions.
3.2. Dynamic analysis
The population survival is an interesting topic in ecology, and the uniform
persistence, from an ecological viewpoint, means all the species can coexist at any
time and any location of their inhabited domain. Therefore, in this section, we firstly
investigate the ultimate boundedness, permanence and extinction of the system, and
then the existence, uniqueness and globally asymptotic stability of positive periodic
solutions are studied from dynamic analysis.

37

3.2.1. Preliminaries
Let

,

and

be the set of all integers, positive integers and real numbers,

respectively, and denote R + = 0,+∞ . Throughout this paper, we always assume that:
,

(C1) the functions

and

3

,

,

,

,

,

1,

2,

,

,

3 ,

= 1,2,3,

∈

arguments and positive-valued;
(C3) the functions
,

= 1,2 ,

and ;
,

(C2) the functions

3

,

are bounded positive-valued functions on

differentiable in

and

, ′

are periodic in
∈

(C4) there exists a number
(C5) the sequences

satisfy

≥ 1 and , 1 , 2 , 3 .

,

,

,

, ′

,

= 1,2 ,

+

=

+

, such that
,

1,

= 1,2,3,4

, continuously

are continuously differentiable in all

with a period

+

×

,

2,

3

> 0;

=

,

for all
1,

2,

3

,

= 1,2,3,4
≥ 1;

for all

= 1,2,3 ,

In order to analyze the dynamics of an ecological system, it is necessary and
important to consider system with the periodicity conditions (C3)-(C5), which might
be quite naturally exposed (for example, seasonal effects of weather, food supply,
mating habits, etc.) (Cushing, 1977)
In the following, we introduce the following notations:

=
and denote

=

,
,

=

: ∈
: ∈

+×

−1 ,
−1 ,

,

=

+×

, ∈
, ∈

38

(3-6)
,
,

∈
∈

,
,

=∪ ∈
=∪ ∈

(3-7)
(3-8)

ì
ü
æ - ö
1,2
1,2
i
j
t
,
x
Î
C
å
,
j
t
,
x
Î
C
ï
( ) ( ) t ,x ( k ) ( ) t , x ç å k ÷ , k Î N ï
ï
ï
è
ø
_
ï
ï.
X = íj：G ® R (ii ) lim- j (t , x ) = j (t k , x ) exists
ý
t ®t k
ï
ï
ï
ï
(iii) lim+ j ( t , x ) = j (t k+ , x ) exists.
t ®t k
ï
ï
î
þ

A vector function

,

1

,

2

,

,

,

3

∈ Ξ × Ξ × Ξ is called a solution
∈

, and (3-4) for

= inf ,

,

of system (3-1)-(3-5) if it satisfies (3-1)-(3-3) on , (3-5) for
∈

every

.
,

For a continuous function
= sup ,

,

.

(3-9)

, we define

and

Definition 3.2.1.1. The solutions of system (3-1)-(3-5) are said to be ultimately
bounded if there exist positive constants
solution

1

, ,

10 ,

20 ,

∈

and ≥ .

30

=

exists a moment of time
= 1,2,3,

for all

,

2

10 ,

, ,

20 ,

10 ,

30

20 ,

′

30

= 1,2,3
,

3

such that

such that for every

, ,

10 ,

, ,

20 ,

10 ,

30

20 ,

, there
30

≤

′

Definition 3.2.1.2. The system (3-1)-(3-5) is called permanent if there exist positive
constants

and

initial functions

10

a moment of time
= 1,2,3,

for all

∈

′
=

= 1,2,3 such that for every solution with non-negative

,

20 (

) and

and

≥ .

10 ,

20 ,

30

30 (

) that are not identically zero, there exists

such that

≤

, ,

10 ,

20 ,

30

≤

′

Now, we discuss the existence of the solutions in system (3 -1)-(3-5). According
to the method of upper and lower solution method for quasi-monotone systems (Pao,
1992), it can be verified that, for continuously differentiable initial functions
0

:

→

+ and

0

≢ 0, = 1,2,3, there exists a classical solution of system

(3-1)-(3-3) and (3-5), if it is of class

2

in

,

∈

, of class

1

in

,

∈

, of

1

in , > 0, and satisfies the system, then it is the classical one of system (3 -1)-(3-3)
and (3-5).

By the existence of solutions of system (3-1)-(3-3) and (3-5), we can verify the
39

∈ 0, 1 , the solutions

existence of solutions for the system (3-1)-(3-5). Actually, if

of the system (3-1)-(3-5) are well-defined as classical solutions of system (3-1)-(3-3)
and
1

(3-5).
+
1 ,

2

Impulsive
+
2 ,

3

+
3

condition

(3-4)

implies

that

the

functions

are continuously differentiable in x , and satisfy boundary

conditions (3-5). Hence, assuming

+
1 ,

1

functions, we can continue the solution on

+
2 ,

2

1, 2 .

+
3

3

as a new initial

Proceeding in this way, the

solution of the system (3-1)-(3-5), for all > 0, can be constructed.

Based on the biological interpretation, we just take the non-negative solutions

into account for the system (3-1)-(3-5), and we firstly discuss and guarantee the
3

non-negative and positive quadrants of
(3-1)-(3-5) in the following.
Suppose
3

, ,

10 ,

condition
deduce

10

that
20 ,

30

∂ 1
= 1∆ 1 + 1
∂t

−

≤

−

≥0 ≢0 ,

20

1

,

1+ 1 ,

−

1

,

1

1

,

1+ 1 ,

−

1

,

10 ,

20 ,

30

≥ 0 ≢ 0 , obviously, we can

1

,

system (3-1), respectively, where

and
1

,

1

40

,

1 2

2+ 1

1−

1

1

1+ 2 ,

∂ 1
− 1∆ 1 − 1
≥
∂t

We can simply verify that

1

1+ 2 ,

∂ 1
− 1∆ 1 − 1
∂t

∂ 1
= 1∆ 1 + 1
∂t

following equations:

, ,

30

,

, ,

2

10 ,

20 ,

30

,

is a solution of system (3-1)-(3-5) with the initial

≥0 ≢0 ,

and

1

are positively invariant for system

,

1−

1

,

1 2

1 −

2+ 1

1 1 .

,

1

2

,

,

2

,

1 2

2

,

1 2

are lower and upper solutions of

and

1

,

are solutions of the

∂

1

−

1∆ 1 −

1

1 − 1

∂

1

−

1∆ 1 −

1

1

∂t

and

∂t

Due to
1

,

10

> 0 and

Then, we have

positive function
positiveness of
function

1
1 .

1

1

1

1

1

1 −

= 0,

2

1 0,

=

10

,

(3-10)

1 0,

=

10

.

(3-11)

− 1 1 = 0,

≥0 ≢0 ,

20

,

∈ 0, 1 because

,

,

> 0 for

> 0 for

≥0 ≢0 ,

≥ 0 ≢ 0 , we can get

30

∈ 0, 1 using Lemma A2 (see Appendix A).
,

1

is bounded below by

. Obviously, we can repeat the same argument to prove the

,

∈

for

1, 2

because of the positiveness of the

Similarly, we can obtain that

explanation of species

,

2

> 0 and

1

,

3

,

> 0 for

∈ 0, +∞ . The

> 0 are very similar, which is

omitted here. Thus, we can obtain that the non-negative and positive quadrants of
are positively invariant for system (3-1)-(3-5) if the conditions (C1)-(C5) hold.

3

3.2.2. Permanence of population
Based on the analysis in the previous subsection, we will discuss and guarantee
the ultimate boundedness of solutions for the system (3-1)-(3-5) firstly, and then
investigate the permanence of the system. The following analysis shows that
impulsive control can influence the permanence of the system directly.
Firstly, we discuss the ultimate boundedness of population
seen from Eq. (3-1) that
∂ 1
= 1∆ 1 + 1
∂t

−

Let

1( ,

,

1

,

1+ 1 ,

1

,

1+ 2

∂ 1
− 1∆ 1 − 1
≥
∂t

10 ) be a solution of

Then, we have

−

∂

1

∂t

−

1∆ 1 −

1

41

1

1

,

,

1 −

1

−

1 2

2+ 1

1 1 .

1 1

,

= 0,

,

1

2

,

. It is readily

1 2

(3-12)

∂ 1
− 1∆ 1 − 1
∂t

0=

1 −

≥

1 1

∂ 1
− 1∆ 1 − 1
∂t

According to Lemma A1 (see Appendix A), we have
1( ,

1

), if

the solution
depend on

10

1( ,

1

= max ∈

1( ,

ordinary differential equation
∂ 1
− 1
∂t

Hence
1

+

, ,

10 ,

20 ,

=

≤

1

, ,

1

1 1 .

10 ,

20 ,

30

1

1

1 −

C

, ,

,

10 ,
1

1

1

1 1

= 0,

20 ,

30

1

,

≤

does not

) is the solution of the following
1 0,

1
1

30

. From the uniqueness theorem,

) of (3-12) with initial conditions independent of

> 0. Therefore,

for

≤

10

1 −

,

.

1

,

1

=

1

2

,

,

.

,

,

3

C

Since the corresponding impulsive differential equations
∂

1

∂t

1

=

+

1

=

1

1

−

1 1

1( 1( ,

(3-13)

))
1

are ultimately bounded (from Lemma B (see Appendix B)), thus, we get ultimate
boundedness of solutions of Eq. (3-1) and Eq. (3-4), i.e., there exists a positive
′1 such that

constant

,

1

≤

′1 , starting with some time t1 .

Secondly, we analyze the ultimate boundedness of population
population

2 , from Eq. (3-2), we have

∂ 2
= 2∆ 2 + 2
∂t

+
−

≥

2

,

1+ 1 ,
1+ 3 ,

−

2

,

2

′1 ,
1+ 2 ,
2 ,
2+ 4 ,

∂ 2
− 2∆ 2 − 2
∂t
42

2 −

1 2

2+ 1
2 3

3+ 3

2 2+

,
,

2

′1
1

4

.

,
,

2

1 2
2 3

,

. For

Since there exists a positive-valued function
2

∈

if

population

,

2 ≤

,

1 > 0,

3 >0

2

and

with some time t2 .

Finally, for the top predator population

∂ 3
= 3∆ 3 − 3 ,
∂t

≥

and it follows that

3+

1+ 3 ,

∈

3 , when
2+ 4

′

2

∂ 3
′2 ′2
− 3∆ 3 + 3 3 −
,
∂t
4
3

, ,

10 ,

20 ,

solution of the initial value problem

30

≤

( ,

1,

2,

3) ≤

≤

′2 , starting

, the same analysis with the

′2 such that

1 , there exists a positive constant

2

such that

3

,

,

≥ t2 , we can obtain that

,

,

2

2 3

3+ 3

3

,

, where

4

,

2 3

3

,

3

is a

∂ 3
′ 2 ′2
=− 3 3+
∂t
4

with
3 0,

=

3

The linear periodic impulsive equation
∂

3

=− 3 3+

∂t

has the general solution

3

3

+

=

piece-wise continuous function, and

(Samoilenko
−

as

3 +

and

0

+

3

( ,

′

3

2

4

.
′

2

(3-14)

( ) , where

is a constant and

= exp − 3 +

Perestyuk,

=1 ln sup ( , 1 , 2 , 3 )

=

3

3

1995).

1,

2,

0<

< ln

3

Obviously,

0

is a

-periodic

if

the

condition

.

3 ) < 0 holds, then we have

→0

→ ∞. Thus, all solutions of Eq. (3-14) are ultimately bounded. That is, all

solutions of Eq. (3-3) and Eq. (3-4) are also ultimately bounded.

To sum up, we can conclude: if the conditions (C1)-(C5) hold, and
43

(i)

there
1

(ii)

( ,

1,

exists
2,

a

positive-valued

function

1(

) if

,

2 ≤

,

3) ≤

∈

,

there exists a positive-valued function

(iii)

∈

if

2

,

the inequality

3

holds, where

= sup ( , 1 , 2 , 3 )

1 > 0,

−

3 +

3

1,

( ,

1 ≤
2

2 > 0,

=1

2,

3

3 > 0 and
2

such that

3 > 0 and

ln

such

1

∈

;

( ,

1,

that
∈

2,

;
3) ≤

<0

3 ). Then all solutions of the system

(3-1)-(3-5) with non-negative initial conditions are ultimately bounded.

Next, according to the ultimate boundedness of solutions for system (3-1)-(3-5),
we will analyze the permanence of the system in the following, and further illustrate
how the population survival under the impulsive effects.
Since the assumption that there exists a positive-valued function
1

that

( ,

1,

2,

3) ≤

if

1

′

can deduce that there exist

∈

,

1 ≤

= 1,2,3

,

2 > 0,

3 >0

≤

such that

and

20

= 1,2,3,

≥0 ≢0 ,

interval

∈

, ,

10 ,

1

≥

≥ 0 ≢ 0 , then

30

and > 0, for some small

20 ,

,

30

,

we

, ,

2

20 ,

,

−

30

10 ,

the
,

initial

∂ 1
= 1∆ 1 + 1
∂t

−

≤

1

1+ 1 ,

1

,

1+ 2

∂ 1
− 1∆ 1 − 1
∂t
44

20 ,

30

10

, we

≥0 ≢

> 0 for all

, ,

3

10 ,

conditions

20 ,

30

,

2

zero. Without loss of generality, we assume that min ∈Ω
all = 1,2,3. The inequality:

∈

> 0, considering the solution on the

get

10 ,

, ,

such

′ , starting with a

particular time. Lemma A2 (see Appendix A) implies that if
0 ,

1

1

1

,

,

1−

1 2

2+ 1
1

1−

1

2

have

separated from
=

0

,

1 2

>0 for

is valid, so

0=

∂ 1
− 1∆ 1 − 1
∂t

1−

1

1−

1

≤

2

∂ 1
− 1∆ 1 − 1
∂t

= 1,

Now, according to Lemma A1 (see Appendix A) for

1

1−

1

1−

, ,

10 ,

20 ,

2,

3 .

1

.

2

≥

30

for ∈ 0, 1 . Applying the last inequality for = 1 , together with Eq.

1

(3-3), we have
1

+
1,

,

10 ,

Thus, the solution

1

20 ,

≥

30

, ,

1

10 ,

20 ,

1∆ 1 −

1

inf ∈Ω ,

1
1

,

1,

1,

2,

3

1, 2, 3 ∈

is bounded from below by a solution of

30

periodic logistic equation with impulse:
∂

1

∂t

1

Obviously,
1 −

1

2

if

=

+

the

=

1 − 1

inf ∈Ω ,

1

1 =

condition

> 0 holds, where

=

1,

1

1−

2

1, 2, 3 ∈

1

=1 ln inf ∈ ,
2,

3 :0 <

,

1, 2, 3 ∈

(3-15)

1

,

1,

2,

3

+

<

′ , = 1,2,3 , and by

10 ,

20 ,

30

≥

,

2

,

1 2

Theorem 2.1 (Smith, 1999), then Eq. (3-15) has a unique piece-wise continuous and
∗
1

strictly positive periodic solution
∗
1
1

as
, ,

10 ,

20 ,

30

≥

For the population

∗
1

for ≥ 1 .

2 . Using the inequality

∂ 2
− 2∆ 2 − 2
∂t

−
+

≤
If the condition

4

, ,

1

→ ∞ . Therefore, there exists a positive constant

0=

2

. Thus,

> 0 holds, where

2

1+ 1 ,
1+ 3 ,

,

−

2

′1 ,
1+ 2 ,
2 ,
2+ 4 ,

∂ 2
− 2∆ 2 − 2
∂t

=

2 =

1,

=1 ln inf ∈ ,
2,

,

3 :0 <

45

2−

2

1 2

2+ 1
2 3

3+ 3
2

4

−

1, 2, 3 ∈

<

2

,

2

2

,

.

,

4

1,

∗
1

1

1

→

such that

2 3

2,

3

+

2 −

′ , = 1,2,3 , and following the

same analysis as for
2

, ,

10 ,

20 ,

1 ,

≥

30

∗
2

for ≥ 2.

,

3−

For the top predator population

∗
2 , thus,

∂ 3
−
∂t

0=

3∆ 3 +

≤
+

3

∂ 3
−
∂t

Hence,

′2
3 +

3

, ,

10 ,

and ∈ 0, 1 , then
3

,

3

≥

′

=

3
3 = inf ∈Ω ,

+

3

2

=

−

3

3
3 3
3

1, 2, 3 ∈

exp
3

3

3

Therefore, if

≥ 33
3

exp
3
≤

,

2 3

3+ 3 ,

3

3

20 ,

′

1

3

2

3

′

3+

,

2,

−

1

′

3

2

3 , for some

2

3 −

1,

3 −

−

46

3

=

10 ,

20 ,

4

,

30

≥

2 3

,

,

3

3

∗ 2
2
′
3+ 4 3+ 2 3 4 2 3
2 ∗
2
3
2+ 3 4 3+ 3
4 3

3

3 +

3 −

3 >0

≥

30

and
+
1,

2

2+ 4 ,

′2

−

3 0,

is the solution of

where

′

, ,

2

such that

2
∗ 2
3+ 4
3 + ′2 3 4 2 3
.
2 ∗+
2
3
2
3 4 3+ 3
4 3

where

3

≥ 2, since

3 . When

1+ 3 ,

3∆ 3 +

3

3

∗
2 > 0

there exists a positive constant

. If

′

2

3

+ 4

3

≤
3+

3
′

2 ∗+
2
3 4

(3-16)

for some

2 3

3 +

4

∗
2 3
2
3
4

∗
′
2 + 4 3+ 2 3 4 2 3
2 ∗+
2
3 + 3
2
3 4 3+ 3
4
′

and

∈ 0, , and

3 >0

3

3

.

3

,

≥

3

exp
3

=

3 =
1,

2,

such that

For
3

=1

3 :0 <
′

ln 33 +

0
3 ∈
3

=1 ln inf ∈ ,

3

2

−

such that
3

,

1,

30

≥

0
3 .

σ∗3 = inf

3

, 0,

, 0,

,

30

=

,

3

time ∈

,

3

, 0,

3

, then

, 2 , then

30

: 0 ∈ 0,

,

30 ∈

, 0,

30

≥ σ∗3 .

: 0 ∈ 0,

,

30

enough to consider

30
3

3

, ,

3

,2 ,

3 2 ,

−

>0 ,

3

where

3

,

, ,

<

for all

3

30 ≥

30

, ≥2

with

30

∈ 0,

3 ,

∈

0, 2

.

0
3,

3 ,

∈

0, 2

≥ σ∗3 ,

from the definition of

, ,

,

3

0
3,

≥

3

> 0.
3

3

≥
.

as the solution of Eq. (3-16)

with

≥

=

such that

30

30

30

30 ∈

3

3 >0

30 > 0, there exists a time

30 , and consider a positive number

and consider a solution
0, 2

3

, 0,

Denote

Let

∈

3

3

of (3-16) with

30

3

= inf

′2

+

3

.

3

∗
′
2 + 4 3+ 2 3 4 2 3
2 ∗+
2
3 + 3
2
3 4 3+ 3
4

0
3 , with the additional condition

3

3

′

Then, for all ≥ 2 , one can obtain that,

all

2,

0, 3 , there exists a positive integer

≥

−

′ , = 1,2,3 , we can take sufficiently small

3 −

Hence, for every solution

with

3

1, 2, 3 ∈

<

3

2

∗
′
2 + 4 3+ 2 3 4 2 3
2 ∗+
2
3 + 3
2
3 4 3+ 3
4
′

−
If

=1

′

ln 33 +

≥

30 ≥

. If
. If

3

3

, ,

, ,

30

30

<

>

3

for
3

at

. Therefore, it is

. By construction, these

solutions are bounded from below by positive constant σ∗3 for

∈ 2 ,3

.

Proceeding in this way, it can be proved that the solutions of the system are bounded
from below for ≥ 3 .

Therefore, we can conclude that, if the conditions (C1)-(C5) hold, and further
47

suppose that
(i)

the solutions of system (3-1)-(3-5) are ultimately bounded, i.e., there exist
positive
1

exists

, ,

10 ,

=

20 ,

10 ,

30

20 ,

,

= 1,2,3, x ∈ Ω and

(ii)

′

constants

1 > 0,

the conditions

2

30

= 1,2,3

, ,

10 ,

such

20 ,

,

30

> 0 such that

≥ ;

2 > 0 and

that

for

, ,

10 ,

3

, ,

10 ,

20 ,

every
20 ,

30

30

≤

solution
,

there

′ for all

3 > 0 hold.

= 1,2,3 such that any solution of system

Then, there exist positive constants

(3-1)-(3-5) with non-negative initial functions not identically equal to zero satisfies
1

,

,

2

,

,

3

,

∈

=

1,

2,

3 :

<

<

starting with a particular time, i.e., system (3 -1)-(3-5) is permanent.

′ , = 1,2,3

According to above analysis, we can easily induce the following results:
Remark 3.2.2.1 If there are no impulses in problems (3-1)-(3-5), then conditions
> 0 ( = 1,2,3) can be taken as the following form:
1 −−

1

2

> 0,

2 −

2

4

> 0,

′

3

2

−

which are sufficient to establish permanence for the system.

3

> 0,

3.2.3. Extinction of top predator population
Are there certain populations in the ecosystem that tend to become extinct after a
certain time? The extinction of some species is often desirable, such as the extinction
of pests, while some species are undesirable. For example, in order to protect the
diversity of species, we should take protection measures for some endangered species.
Based on the previous analysis, it is interesting to find that the top predator will
undergo extinction under some a condition, and the result shows that if the condition
4 =

=1 ln sup

1, 2, 3

3

,

1,

2,

3

+

′

2

3

−

3

< 0 holds, then the top

predator tends to extinction, indicating the impulsive control can significantly affect
48

the dynamics of system (3-1)-(3-5). Now we give a specific analysis.
Suppose
3

,

3

is a fixed positive constant such that

3

3

is the solution of initial value problem
d

∂ 3
= 3∆ 3 + 3 ,
∂t

3−

′

=

0,

3

From the inequality

≥

3

−

′

3−

3

=

3

1+ 3 ,

∂ 3
− 3∆ 3 +
∂t

2

3

2+ 4
2

2

3,

3

,

,

2 3

3+ 3

and Lemma A1 (see Appendix A), we can deduce that, for
3

, ,

10 ,

10 ,

20 ,

30

and
3

+
1,

,

20 ,

≤

3

30

≤

,

3

3

and

30

3

3

′

≥

,

3

sup

,

4

,

,

1,

2 3

≤ 1

3

1, 2, 3

2,

3

if the impulsive condition (3-4) holds. Proceeding in this way, we conclude that every
solution of Eq. (3-3) and Eq. (3-4) is bounded from above by the corresponding
solution of the linear impulses equation
3
3

=

+

′

=

2

3

3

−

3

sup

3

1, 2, 3

3

,

1,

2,

3

Applying the Lemma B (see Appendix B), all solutions of the last equation tend to
zero as

→ ∞ if the inequality

4 <0

holds.

Obviously, we can obtain the following results if the system does not experience
the impulsive perturbations:
Remark 3.2.3.1 Suppose that the conditions (C1)-(C3) hold, then the top predator
3

,

in system (3-1)-(3-3), (3-5) is extinctive if
′

2

3

−

3 < 0.

49

3.2.4. Periodic oscillations of population density
In the previous subsection 3.2.2, we have obtained the result that the system is
permanent. Thus, in this subsection, we will further study the existence of periodic
solutions in system (3-1)-(3-5) by constructing a proper auxiliary function in the
following, which implies that all the species can coexist at a stable state for a long
time.
In order to study the existence of periodic solutions in the system (3-1)-(3-5), we
firstly suppose that

,

1

,

2

,

,

,

3

and

1

are two solutions of system (3-1)-(3-5) bounded by constants

,

,

2

and

,

,

3

,

from below

and above, respectively. Consider the function
=
with derivative

=2
where

1 = 2
2 =2

3 =2

3
=1
Ω

Ω

1−

−

2−

−

1

=1

2,

1

1 −

=1 Ω

1

1,

Ω

3

1 −

Ω

3

∆

1−

1 −

1 1

1 1 2

3

−

2

2 −

2 2

2 2 3

1,

1

2,

3

2

= 1 + 2 + 3 + 4,

,
−

1 1 2

1+ 1 1+ 2 2+ 1 2 1 2

1+ 1 1+ 2 2+ 1 2 1 2
2

−

−

,

−

1

1 −

1 1

′1 1 2
1+ 1 1+ 2 2+ 1 2 1 2

− 2 2− 2 2
1+ 3 2+ 4 3+ 3 4 2 3
′1 1 2
2 2 3
−
+
1+ 1 1+ 2 2+ 1 2 1 2 1+ 3 2+ 4 3+ 3 4 2 3
50

,

4 =2

Ω

3−

− 3 3+

3

−

′2 2 3
1+ 3 2+ 4 3+ 3 4 2 3

′2 2 3
1+ 3 2+ 4 3+ 3 4 2 3

.

Then, from the boundary condition (3-5),
1 = 2

3
=1

1 −

Ω

∆

1−

For the other terms 2 , 3 , 4 , since
Then

−

=

−

Ω

1 −

1

2+ 3+ 4 =2

+

+
+

+

1

1 1 2

−

1−

−

1

Ω

2−

2

2

2−

−

1−

1

2

Ω

=1

=

−

−

∇

1−

+

≤ 0.

−

1 +

1

2

1

1 1 2

1+ 1 1+ 2 2+ 1 2 1 2
2

2−

2

2+

2

′1 1 2
′1 1 2
−
1+ 1 1+ 2 2+ 1 2 1 2 1+ 1 1+ 2 2+ 1 2 1 2
2 2 3

1+ 3 2+ 4 3+ 3 4 2 3

+2
−

≤−1=2

1+ 1 1+ 2 2+ 1 2 1 2

+2

3

Ω

3−

3

− 3

3 −

′2 2 3
1+ 3 2+ 4 3+ 3 4 2 3

51

−

3

2 2 3

1+ 3 2 + 4 3+ 3 4 2 3

+

′2 2 3
1+ 3 2 + 4 3 + 3 4 2 3

≤2
+2

2

1 −

2

2 −

1 4

2

+2
+2

Ω

1−

Ω

2−

≤

1
2

+

1

2

′1 1

2

3−

3

′2 3

Ω

1

2

+

1+ 1 + 2

2 −

2

+ 1
+ 1 2

′1 1 ′2 + ′1 ′
1+ 1 + 2 + 1 2 2

+2

′ 2

2 −

1−

1 2

−

′2 3 ′ 2 + ′ 2 ′
1+ 3 + 4 + 3 4 2

+

and

Ω

2−

1

2

+ 2
1+ 3 + 4 + 3 4

−

where

Ω

1 −

3 −

Ω

3

2

′ 2

− 3

′2 + ′1 ′ − 1 2 2 + 1
2
1 + 2 + 1 2

2

2

′2 + ′2 ′ − 1 4 2 + 2
2
3 + 4 + 3 4
+

3−

3

2

=

,

′ are minimum and maximum value of the system, respectively,

is the maximal eigenvalue of the following matrix
11

12

21
31

and
11 = 2

1 −

22 = 2

2 −

1

−

2

+

33 = 2 − 3 +

and

12 =

21 =

1

1

1 2

1+ 1 + 2

13

22

23

32
2

+ 1
+ 1

33

,

′ 2

2

2
′1 1 ′2 + ′1 ′
+ 2
2 4
−
1+ 1 + 2 + 1 2 2
1+ 3 + 4 + 3 4

′ 2 3 ′2 + ′2 ′
,
1+ 3 + 4 + 3 4 ′ 2

13 =

′2 + ′1 ′ − 1 + 1 1
2
1 + 2 + 1 2

52

2

31 = 0,

,

′ 2

,

23 =

′2 + ′2 ′ − 2 + 2 2
2
3 + 4 + 3 4

′2 3

32 =

Then, one can see that

+1

and
+

=

3
=1

+

≤

≤

where

1,

+1

Ω

exp

+1

−

3

exp

3

= max ∈Ω , 1 , 2 , 3 ∈ 2

.

+1 −

2,

+1

2

2

=1

+1
+1 −

+

3

1,

2,

3

+

,
2

′

=1 =1

2

3

.

In the following, let us estimate the variation of the function over the period. We have
+

≤

Obviously, if the condition
+

Therefore,
,

→ 0 for all

≤

=

∗

∗

=

=1 ln

+

→ 0 as

= 1,2,3 as

exp

=1

.

< 0 holds, then we have

→ ∞, which implies that

→ ∞, where

·

is the norm of space

∗ < 1.

,

2

−
.

From Lemma C (see Appendix C), solutions of system (3-1)-(3-5) are bounded in the
space

1+

. Therefore,
sup ∈

,

−

,

Consider the following sequence
1

, ,

1,

2,

3 ,

2

, ,

1,

2,

→ 0, = 1,2,3, → ∞.
3 ,

, ,

3

1,

2,

3

,

0

From Lemma C (see Appendix C), it is compact in the space
= lim →∞ ⁡

be a limit point of this sequence,
Since the fact that

and as

→∞

,

,
,

=

0

,

0
53

−

,

,
,

0

=

(3-17)

×

, then

0

→ 0.

,

×

0 ,

∈

. Let

,

=

.

.

→ ∞,

Then, one can obtain that, for
,

−

≤
+

The sequence
sequence
and

has

1

, ,

−

1,

2,

,

0

,

two

= lim →∞ ⁡

Hence,

,

3 ,

,

∈

,

,

,

−

0

0

limit

points,

=

and

−

1,

+

0

that

0 . From (3-17) and

, ,

,

,

0

−

→0

has a unique limit point. Otherwise, suppose the

,

≤
2

−

2,

,

0

3 ,

3

=

+

, ,

is,

1,

2,

,

0

,

3

= lim →∞ ⁡

, then we have

,

→ 0,

solution

−

the

0

→ ∞.

is the unique periodic

solution of system (3-1)-(3-5), and from (3-17), it is asymptotically stable. Therefore,
suppose that the conditions (C1)-(C5) and the conditions in Lemma C (see Appendix
C) hold, system (3-1)-(3-5) has a unique, globally asymptotically stable, and strictly
positive piece-wise continuous

-periodic solution if

< 0 holds.

Remark 3.2.4.1 Assume that conditions (C1)-(C3) holds, and the system (3-1)-(3-3),
(3-5) is permanent. Then the system (3-1)-(3-3), (3-5) has a unique, globally
asymptotic stable, and strictly positive

-periodic solution if
< 0.

3.3. Experimental simulations
Although some interesting results of system (3-1)-(3-5) have been achieved by
means of modeling analysis, it is difficult to further study the dynamical properties of
the system due to its complexity. Fortunately, experimental simulations can help us
provide more in-depth insights on the dynamics of the system. In this section,
therefore, some numerical simulations are carried out to study the effects of the key
factors on the dynamics of the system and show the effects of impulsive control and
environmental heterogeneity on the survival of population and the dynamic
mechanisms of the spatial and temporal distribution of population.
54

3.3.1 Impact of impulsive control on the dynamics of system (3-1)-(3-5)
In this subsection, some numerical simulations are carried out to study the effect
of impulsive control on the dynamics of system (3-1)-(3-5). In these numerical
simulations, the following parameter values are used:
,

= 1.5,

2 cos

+ 5,

0.5 sin

+ 4, c′2 t, x = 0.4 cos

3

1.5 sin

1 =

2 =

+ 10, 1 ,

1.2 sin

2

,

+ 4, 2 ,

0.5 sin

sin

3 = 1,

= sin

= 2 cos
= cos

,

= 2, Ω = −1,1
+ 2 sin

+

+ 0.5 sin

+ 6, 1 ,

+ 8,

+ 0.1 sin

,

1

2

,

+ 4, ′1 ,
+ 4.5,

= 2 ,

= 3 ,

= 2 cos

+

1

,

= 1.5 cos

= 1.5 cos
3

= 0.5 cos
,

=

+

=

+

+

+ 0.1, and other parameters are chosen as control parameters.

Obviously, the conditions (C1)-(C5) in the preliminaries subsection can be verified.
To study how impulsive control affect the dynamical behaviors of system (3-1)-(3-5),
we firstly consider that the system (3-1)-(3-5) does not have impulsive control, that is,
,

1,

2,

3

= 1,2,3,

∈

are identically equal to zero. Thus, the Remarks

3.2.2.1 and 3.2.4.1 hold by direct computation, which are sufficient to establish
permanence for system (3-1)-(3-3) and (3-5), that is, all the species can coexist at a
stable state under some conditions (see Fig. 3.1.).

55

Fig. 3.1 Dynamic behaviors of species 1 , 2 and 3 of system (3-1)-(3-5) without impulsive
effects, and the initial conditions 1 0, = 2, 2 0, = 2, 3 0, = 6.85 for all ∈ . (a):
permanence of species 1 ; (b): permanence of species 2 ; (c): permanence of species 3 ; (d):
time series of 1 , 2 and 3 on the population dynamics with = 0.

In the following, the impact of impulsive control on the dynamics of the system
1

(3-1)-(3-5) will be shown. Choosing

0.95, = 1,2, ⋯ , 3 , 1 , 2 , 3 = 0.95 for all
find that the species

1

,

2

,

,

2

,

and

3

,
,

1,

2,

3

= 1,2, ⋯ ,

= 0.9,

1

,

,

and

3

,

,

1,

2,

3

=

= 8, it is not difficult to

can coexist at a stable state under

some conditions (see Fig. 3.2). Actually, due to the conditions
the species

2

= 1,2,3 > 0, all

are permanent, which are consistent with

our experimental simulations. Comparing Figs. 3.1 and 3.2, it is obvious to find that
the impulsive control can affect the temporal spatial dynamics of the system. That is,
the impulsive control has a profound effect on population dynamic evolution
mechanism. Furthermore, it is obliged to be stressed that the impulsive control not
only can aggravate the emergence of pulse oscillation, but also can change the
periodicity of population density. Thus, it is worth pointing out that the results from
Figs. 3.1 and 3.2 can support that the reaction-diffusion impulsive hybrid system can
56

depict the interaction mechanism between populations.

Fig. 3.2 The effects of impulsive effects on the dynamic behaviors of species
system

(3-1)-(3-5),

1,2, ⋯ ,

3

species

2 ; (c): permanence

2,

2 0,

,

1,

= 2,

2,

3 0,

3

1

here

,

= 0.95 for all

= 4.2 for all

of species

1,

2,

= 1,2, ⋯ ,

permanence of species
= 0.

3;

2

,

1,

3 ; (d): time series
1 ;

1,

(d): time series of

2,

3

2

of

1,

2

and

3

3

of

1 0,

=

1 ; (b): permanence of

and

3

on population

(b): permanence of species

2

and

= 0.95, =

= 8 and initial conditions

∈ : (a): permanence of species

= 0 permanence of species

dynamics with

= 0.9,

3

1,

2 ;

(c):

on population dynamics with

On the other hand, when we control some a parameter values satisfying the

condition of
two species

1

4 < 0, then the species

,

and

2

,

3

,

undergoes extinction, but the other

generate periodic oscillation (permanence), as are

shown in Fig. 3.3. It is easy to know from Fig. 3.3 (d) that the time series solutions for
species

1

,

and

2

,

have a stable periodic oscillation over time. In contrast,
57

species

,

3

will rapidly decrease to extinction over time. Moreover, comparing

Figs. 3.2 and 3.3, it is obvious to find that the dynamical behaviors of the system
= 40 . Therefore, it is

(3-1)-(3-5) have been changed by controlling the value of

necessary to point out that some critical parameters have a profound impact on the
persistence and extinction of populations.
In order to further study how the impulsive parameter
,

impulsive perturbation constants

1,

2,

in a period and

= 1,2,3,

3

∈

affect the

population density dynamic evolution trend, it is necessary to present a more in-depth
analysis of the relationship between the parameter value and the populati on density.
Fig. 3.4 depicts the variation of species

1

,

,

2

,

and

3

,

with

increasing , where the red lines represent the initial value of the corresponding
population. It is obvious to survey from Fig. 3.4 that the maximum density of species
1

,

and

,

2

can remain unchanged, but species

decline and finally will undergo extinction when
that impulsive parameter

3

has a great influence on the maximum of species

Fig. 3.5 depicts how the impulsive perturbation constants
affect the density distribution of species
3

species

,

3

,

. When

= 0, species

3

,

3

,

1,

3

2,

,

,

1,

and
2,

3

2

,

,

,
.

∈

> 0 denotes the number of

,

3

3

,

1,

3

,

2,

will undergo extinction as
∈

3

→∞

. However, when 0 ≤

<

will develop a periodic oscillation. Thus, it is obvious that the
3

impulsive perturbation
of species

3 , where

3

1

3

released each time, and the red line represents the initial value of

without impulsive perturbation
0.5, species

will rapidly

increases. Thus, it is easy to see

but it has almost no effect on the maximum density of species

species

,

,

1,

2,

∈

3

has a profound effect on extinction

. These results suggested that the impulsive perturbation constants

survival of species

∈

3

and impulsive parameter
,

can play a restrictive role for

, which also verifies that the system has potential to be

applied for actual biological control issues.

58

3.3.2 Impact of environmental heterogeneity on the dynamics of system
(3-1)-(3-5)
The distribution of population relies strongly on the heterogeneity and
homogeneity of their living environments, thus, the dynamical behaviors of
population distribution in a homogeneous environment are studied. It is obvious to
find that the system (3-1)-(3-5) exists spatially homogeneous periodic solutions and
species

,

1

,

2

,

and

,

3

are distributed uniformly over space at the

same time, as shown in Fig. 3.6. However, when the environment is heterogeneous,
the system (3-1)-(3-5) can generate a solution which is heterogeneous in space
direction and periodic in time direction, as it is shown in Fig. 3.7. In such
circumstances, species

1

,

,

2

,

and

3

,

is heterogeneously distributed

in space with high density on the edge and cyclical variations over time. From the
comparative analysis of Figs. 3.6 and 3.7, it is not difficult to find that the living
environment plays an import role in the spatial distribution of populations.

Fig. 3.3 The effects of impulsive control on the dynamic behaviors of species
system (3-1)-(3-5), here
4.2 for all

∈

= 40 and initial conditions

: (a): permanence of species

1 0,

= 2,

2 0,

1 ; (b): permanence of species

59

1,

2

= 2,

and
3 0,

3

of
=

2 ; (c): extinction

of species

3

as

→ ∞; (d): time series of

1,

2

Fig. 3.4 The effect of impulsive control parameter
her the initial conditions

1 0,

= 2,

2 0,

= 2,

and

(0 ≤

3

with

′2 ,

fixed

= 0.4 cos

conditions

1 0,

3

= 1,

+ 0.1 sin

= 2,

2 0,

,

1,

2,

+ 3,

= 2,

3
3

3 0,

,

≤ 30) on the density of all species ,

3 0,

=1+

= 4.2 for all

= 1,2,3,

Fig. 3.5 The effect of impulse perturbation constants
3

0≤

= 0.5 sin

= 1 for all

= 0.

3 on population dynamics with

∈

≤ 0.5
sin

∈ .

∈ .

the density of species
= 1,2, ⋯ , ;

for

all

+2

and the initial

Fig. 3.6 Numerical simulation of spatially homogeneous stable periodic solutions of the system
(3-1)-(3-5) without spatial representation for
sin

+ 6, 1 ,

= 2 cos

+ +5, 2 ,

= 8, 1 ,

= 2 cos

60

= 1.5 cos

+ 8, 2 ,

+ 10, 1 ,

= 1.5 cos

+

=

4, 2 ,

= cos

0.5 sin

+ 4, ′1 ,

= 0.5 cos

+ 4, ′2 ,

+ 0.1 and the initial conditions

∈ .

1 0,

= 2,

= 0.4 cos
2 0,

= 3,

+ 4,

3

3 0,

,

=

= 1 for all

Fig. 3.7 Numerical simulation of spatially non-homogeneous periodic solutions of the system
(3-1)-(3-5)
2 sin

8, 2 ,

+ 6, 1 ,

0.5 cos
0.5 sin
all

= 8,

with

∈ .

= 1.5 cos

+ 0.5 sin

sin

= 2 cos

1

+ 1.2 sin

,

= 1.5 cos

+ cos

+ 4, ′2 ,

+ 1.5 sin

+ 5, 2 ,

+ 4, 2 ,

= 0.4 cos

+ 0.1 and initial conditions

= cos

= 2 cos

+ 0.5 sin

+ 0.1 sin

1 0,

+ 10, 1 ,

= 2,

+ sin

+ 4.5,

2 0,

3

= sin

+

+ 4, ′1 ,

= 2,

,

3 0,

=

+
=

= 4.2 for

4.3. Conclusions
An impulsive reaction-diffusion predator-prey system with Crowley-Martin
functional response is proposed, and we study the dynamics of the system analytically
and numerically in this thesis. Using the upper and lower solution method and
comparison theory of differential equations, the boundedness, persistence, and
extinction are analyzed theoretically, which provided some sufficiency conditions. To
prove the existence, uniqueness, and globally asymptotic stability of positive periodic
solutions, compactness theory and a method based on constructing a proper auxiliary
function are applied. Numerical analysis verifies the theoretical results and further
indicates that the change of population density is periodic oscillation in time direction
whether their spatial distribution is heterogeneous or not, but the heterogeneous
environment indeed has influence on the spatial distribution of populations.
61

Furthermore, it is investigated how the dynamics of system (3 -1)-(3-5) strongly
depends on pulse parameter
1,2,3,

∈

and the impulsive perturbation constants

, as well as other parameters. Choosing

and

= 1,2,3,

∈

=

as control parameters, the dynamics of the system were analyzed numerically,
particularly species extinction and permanence. According to the Figs. 3.1 and 3.2, it
is obvious to find that the impulsive control can significantly affect the temporal and
spatial dynamics of the system. From Figs. 3.2 and 3.3, by controlling the impulsive
control parameter , the dynamical behaviors of the system (3-1)-(3-5) can be greatly
changed. That is, when

>

∗

, the prey and intermediate predator can coexist, while

top-predator undergone extinction rapidly, where

∗

is some a critical value of . It

should be stressed that the top-predator will trend to extinct without impulsive effects
and a periodic oscillation can be generated by controlling the released number
under some conditions. From an ecological viewpoint, uniform persistence implies
that the prey, intermediate predator and top-predator populations can coexist at any
time and any location of the inhabited domain. Therefore, we can utilize some key
arguments to control population permanence and extinction by means of some
reaction-diffusion impulsive hybrid systems, which are expected to be useful in the
studies on the dynamic complexity of ecosystems.

62

Chapter 4 DYNAMICS INDUCED BY ENVIRONMENTAL
STOCHASTICITY IN A PHYTOPLANKTON-ZOOPLANKTON
SYSTEM WITH TOXIC PHYTOPLANKTON 2
Abstract
Environmental stochasticity and TPP are the key factors that affect the real
aquatic ecosystems. To investigate the effects of environmental stochasticity and TPP
on the dynamics of plankton populations, a stochastic phytoplankton-zooplankton
system with two TPP is studied theoretically and numerically in this thesis.
Theoretically, we first prove that the system possesses a unique and global positive
solution with any given positive initial values, and then derive some sufficient
conditions guaranteeing the extinction and persistence in the mean of the system.
Significantly, it is shown that the system has a stationary distribution when toxin
liberation rate reaches some a critical value. Additionally, numerical analysis shows
that the white noise can affect the survival of plankton populations directly.
Furthermore, it has been observed that the increasing one toxin liberation rate can
increase the survival chance of phytoplankton and reduce the biomass of zooplankton,
but the combined effects of two liberation rates on the changes in plankton
populations are stronger than that of controlling any one of the two TPP.
Keywords: Toxin-producing phytoplankton, Phytoplankton-zooplankton system,
White noise, Extinction, Stationary distribution

2

This work has been published as Liu, H., Dai, C.J., Yu, H.G., Guo, Q., Li, J.B., Hao,
A.M., Jun, K., Zhao, M., 2021, Dynamics induced by environmental stochasticity in a
phytoplankton-zooplankton system with toxic phytoplankton. Mathematical
Biosciences and Engineering 18(4) 4101-4126,. DOI: 10.3934/mbe.2021206.
63

4.1. Introduction
Plankton, the organisms that the freely floating and weakly swimming in aquatic
environments, occupy the first trophic level and the second trophic level of any
aquatic food chains. Phytoplankton are the photosynthetic microorganisms and
commonly unicellular and microscopic in size, and zooplankton are the heterotrophic
plankton that live on phytoplankton. In addition to recognizing the importance of
plankton for the wealth of the aquatic ecosystems and ultimately for the planet itself
(Huppert et al., 2002), the variation of plankton biomass is an important factor
influencing the real aquatic environments, and understanding of plankton dynamics
can be helpful to estimate the productivity of aquatic ecosystems (Behrenfeld and
Falkowski, 1997; Hoppe et al., 2002) and regulate the balance of plankton ecosystems.
However, planktonic blooms can occur under some conductive environments, which
may cause seriously environmental issues and threat to human health. But the
processes underlying the formation of planktonic blooms are not yet well understood.
In this respect, thus, the great effort has been made towards the understanding of the
complex dynamics of plankton, and then mathematical models can be acted as a
useful tool to investigate the dynamics of plankton ecosystems, which can provide a
deeper understanding of the dynamic mechanisms of changes in plankton populations.
Actually, many mathematical models have been constructed to study the
dynamical behaviors of plankton since the pioneering work of Riley et al. (1949), and
many physical and biological processes underlying the mechanisms of plankton
dynamics in the aquatic environments have been investigated (Huppert et al. 2002;
Dai et al., 2016; Caperon, 1969; Guo et al., 2019; Lin et al., 2005; Liao et al., 2020;
Zhao et al., 2020). For example, in order to study how the nutrient affects the
dynamics of phytoplankton blooms, Huppert et al. (2002) presented a simple
nutrient-phytoplankton model and identified an important threshold effect that a
bloom will only be triggered when nutrients exceed a certain defined level using
mathematical model analysis. Caperon (1969) concluded that the time-lag effect
exists in the growth process of phytoplankton, and further suggested that models play
64

an important role in understanding the growth dynamics of phytoplankton
characterized

by

time

delays.

Lin

et

al.

(2005)

used

a

nutrient-phytoplankton-zooplankton model to examine the patterns and consequences
of adaptive changes in plankton body size and suggested that evolutionary
interactions between phytoplankton and zooplankton may have contributed to
observed changes in phytoplankton sizes and associated biogeochemical cycle over
geological time scales.
In recent years, the dynamical behaviors of phytoplankton-zooplankton systems
with various biological factors, such as stability, bifurcation and spatiotemporal
pattern, have been explored extensively (Caperon, 1969; Zhao et al., 2020; Zhao et al.,
2016; Han and Dai, 2019). Nevertheless, some phytoplankton species are harmful
phytoplankton that can produce potent toxic or allelopathic substances during
phytoplankton blooms (Hallam and Luna, 1984), which can affect species interaction
by suppressing the growth and establishment of other phytoplankton species (Hallam
et al., 1983). Moreover, some laboratory experiments (Huntley et al., 1986;
Nejstgaard and Solberg, 1996), as well as field observation (Estep et al., 1990), have
suggested that the toxicity may be as a strong mediator in the zooplankton feeding
rate. As a result, some researchers have taken this important factor of toxic production
released by TPP into account when studying the phytoplankton-zooplankton systems
(Scotti et al. 2015; Banerjee and Venturino, 2011; Khare et al., 2010). For example,
Scotti et al. (2015) indicated that a toxic phytoplankton may destabilize the spatially
homogeneous coexistence and trigger the formation of spatial pattern, and further
concluded that local blooms more likely occur when the strength of the toxicity is of a
certain level. Additionally, some results from field observations and model analysis
concluded that the toxic substances can affect the interaction between phytoplankton
and zooplankton and reduce the growth of zooplankton, indicating TPP may act as a
biological control way for the termination of planktonic blooms (Chattopadhyay et al.,
2002; Chattopadhyay et al., 2002; Sarkar and Chattopadhyay, 2003; Chattopadhyay et
al., 2004). Sarkar et al. (2005) proposed the following mathematical model consisting
65

of two harmful phytoplankton and zooplankton species:
1
2

where

1

1− 1

= 1 1

1− 2

= 2 2

=

, 2

−

1

1

and

+

2

2

−

1

2

1

−

1

−

2
1

+ 1

−

2

−

,

1
2

2

−

+ 2

intrinsic growth rates of two TPP, respectively;

harmful phytoplankton;

and

1 and

2

,

1

and

2

1

and

2

denote the

are their corresponding

represent be the inhibitory effects of two

are the maximum zooplankton ingestion rates for

both two TPP species, respectively;
conversion rates, respectively;

(4-1)

are the population densities of two harmful

phytoplankton and zooplankton species at time , respectively;

environment carrying capacities;

,

and

are the maximum zooplankton

is the natural death rate of zooplankton;

are the rates of toxin liberation by two TPP species, respectively;

and

and
denote

the half-saturation constants for two TPP species. The authors studied the asymptotic
stability of the system (4-1) and claimed that the presence of two harmful
phytoplankton populations has a positive impact for the termination of planktonic
blooms (Sarkar, 2005).
In the real world, however, the unpredictability and ubiquity of environmental
fluctuations in the natural aquatic ecosystems, for example, the necessary nutrient
availability, water temperature, light and turbulence, can greatly cause the growths of
plankton populations to experience random fluctuations. Systems with such kinds of
environmental fluctuations can be described by stochastic differential equations,
which play a significant role in the population dynamics as they can provide some
additional degree of realism compared to their corresponding deterministic
counterparts (Renshaw, 1991). Thus, stochastic population systems, as an important
application in ecological and biological systems, have attracted increasing attention
(Sun et al., 2020; Deng and Liu, 2020; Jiang et al., 2020; Yu et al., 2019; Gao and
Wang, 2019; Liu et al., 2017; Liu and Liu, 2019). Especially, stochastic plankton
systems with white noise have been the common area of interest among researchers
66

(Liao et al., 2020; Sarkar and Chattopadhyay, 2003a, 2003b; Yu et al., 2019; Zhao et
al., 2017; Chen et al., 2020; Chen et al., 2020; Liao et al., 2019; Xia et al., 2020; Zhao
et al., 2020; Yu et al., 2020; Wang and Liu, 2020) in recent years and many interesting
results have been shown. For example, Sarkar and Chattopadhayay (2003b) proposed
a

toxic

phytoplankton-non-toxic

phytoplankton-zooplankton

with

stochastic

perturbation around the positive equilibrium, and they concluded that TPP and
stochastic fluctuations can significantly affect the coexistence of species. Yu et al.
(2018) investigated a nutrient-phytoplankton system with TPP under environmental
fluctuations, and they obtained some conditions for extinction, persistence and the
existence of ergodic stationary distribution. All these works greatly stimulate
researchers to explore the way how environmental stochasticity and toxin production
affect the coexistence and survival prospect of plankton populations in the presence of
harmful phytoplankton. Obviously, it is meaningful to further incorporate the
environmental fluctuations into the underlying model (4-1). Moreover, there are few
literatures to study the dynamics of the stochastic phytoplankton-zooplankton system
with two harmful phytoplankton populations, and the dynamics of the stochastic
phytoplankton-zooplankton system with two harmful phytoplankton is still not very
clear currently. Hence, we mainly present the influence of the effects of
environmental white noise and toxic liberation rates produced by two TPP on the
dynamics of phytoplankton-zooplankton system in this paper. Motivated by the works
above, we assume that the intrinsic growth rates of two harmful phytoplankton and
the death rate of zooplankton are influenced by the environmental fluctuations effect,
and thus introduce the white noise into underlying system (4-1), resulting in the
following form:

67

1
2

1

1− 2

= 2 2

=
where

1− 1

= 1 1

2

+

1

−

+ 1

−

+ 2

1
2

2

+ 3

1

1
2

1

1

−

1

2
2

2

+ 1
3

,

,

,

−

1

−

−

2
2

+ 2

(4-2)
−

are mutually independent standard Brownian motions with

(Gikhman and Skorokhod, 1979), and
are their intensities,

are standard white noise and

= 1,2,3.

0 =0
2

( )

By now, we have successful introduced a stochastic phytoplankton-zooplankton
system with two toxic phytoplankton focusing on the effects of environmental
stochasticity and TPP, and our research questions include: (i) How does the
environmental stochasticity affects the dynamics of plankton populations? (ii) What
influences the peak of the outbreaks of planktonic blooms in a fluctuating
environment? The rest of this paper is organized as follows: Section 4.2 presents the
basic assumptions firstly, and then we investigate the existence and uniqueness of
global positive solutions, and apply the Ito's formula to obtain the sufficient
conditions for the extinction and persistence in the mean of system (4-2), and the
existence of a unique ergodic stationary distribution by establishing a appropriate
stochastic Lyapunov function. A series of numerical simulations are carried out to
further study the dynamics of system (4-2) in Section 4.3. In Section 4.4, we
summarize the results and present our conclusions.
4.2. Dynamic analysis
The survival of plankton populations is a interesting topic in the biology and
ecology, and phytoplankton population plays an important role in the balance of
aquatic ecosystems, but the rapid growth of harmful phytoplankton can lead to the
occurrence of HABs and pose threat to our living environments. Thus, it is necessary
to discuss the survival chance of plankton populations under some conditions in the
68

model. In this respect, we firstly investigate the existence and uniqueness of global
positive solutions, and then discuss the evolution process of plankton growth,
especially for the extinction and persistence in the mean of system (4-2), and the
positive recurrence and ergodic property of system (4-2) in this section. Before it, we
introduce some preliminaries for the following discussion.
4.2.1. Preliminaries
Denote ℝ+ = 0, +∞ and ℝ+ =

=

=1

2

assume that

1, ⋯ ,

∈ℝ :

> 0, = 1,2, ⋯ ,

and

. Throughout this paper, unless otherwise indicated, we always

,ℱ , ℱ

≥0 ,

is a completed probability space with a filtration ℱ

satisfying the usual normal conditions (i.e. it is right-continuous and increasing while
ℱ0 contains all

-null sets). For convenience, if
1

ℝ+, we define

= ∫0

with initial value

=

Generally, we consider the

> 0.

-dimentional stochastic differential equation:

,

=

0

,

is a integrable function on

0 ∈ ℝ

+

,

, while

,

is

∈

ℝ × ℝ+, ℝ

(4-3)

-dimentional standard Brownian
,ℱ , ℱ

motion defined on the completed probability space
2,1

0,

the family of all non-negative functions

ℝ × ℝ+ such that they are continuously twice differentiable in
Define a differential operator
=
Let

( , )∈

+
2,1

where
=

+

1
2

+

1
2

(ℝ × ℝ, ℝ), then
=

,

. Denote by

( , ) defined on

and once in .

associated with Eq. (4-3) by (Mao, 1997b) as follows:

,

=1

≥0 ,

+

,

=

,

1

2

,

, =1

2

,

,

,⋯,
69

,

,
=

2

.

,

×

.

By Ito‟s formula, if

( ) ∈ ℝ , then

=

,

+

,

,

.

Next, we introduce the criterion of positive recurrent and the ergodic properties.
Before it, we consider the stochastic equation:

where

=

+

(4-4)

=1

is a homogeneous Markov process in

ℝ . The diffusion matrix is

=

-dimentional Euclidean space
=

, and

. Thus,

=1

a lemma which describes the criterion of stationary distribution can be given.
Lemma 4.2.1.1 (Khasminskii, 2012) Suppose that there exists a bounded open set
⊂ ℝ with a smooth regular boundary
(i) the diffusion matrix

is strictly positive definite for all

(ii) there exists a non-negative
that

≤−

satisfying the following conditions:

2

( ) and a positive constant

-function

for ∀ ∈ ℝ ∕ .

process with a stationary distribution
with respect the measure , we have

→∞

1

such

( ) of the system (4-4) which is a stationary Markov

Then there exists a solution

lim

∈ ;

0

(·) and for any given integrable function
=

ℝ

(

(·)

) = 1.

4.2.2. Existence and uniqueness of global positive solutions
Before investigating the stochastic dynamics of system (4-2), we should first
guarantee whether the solution of the system is global and positive. Therefore, based
on the biological interpretation, in this subsection, we just take the non-negative
solutions into account for system (4-2) and discuss the existence of global positive
solutions in system (4-2) for any given initial values in the following.
Actually, from the method of the Lemma 2.1 in the work of Ji et al. (2009), it is
obvious to obtain that, for any given initial values
70

1 0 , 2 0 ,

0

∈ ℝ3+, all the

coefficients of system (4-2) are locally Lipschitz continuous and the system admits a
unique local solution

, 2

1

,

∈ 0,

on

, where

represents the

explosion time. In the following, thus, we need to illustrate the solution is global, that
= ∞, . . Let

is, we only need to prove
1 0 , 2 0 ,

0

∈

1

0

,

time by the following form:
= inf

∈ 0,

: max

0

1

0 ≥ 1

≥

and for each integer

, 2

,

≤

1

large enough such that
0 , we define the stopping

or min

, 2

1

,

≥

∞ ≤

a.s. If

and set inf ∅ = ∞ (∅ denotes the empty set). Obviously, one can obtain that
→ ∞ and

increasing as

<

. Thus, let

∞ = ∞ a.s., then

we can show that

∞ = lim →+∞

= ∞ and

1

, then

, 2

,

all ≥ 0. In other words, to complete the proof, we only need to proof
If the statement is false, then there exist two constants
∞ ≤

that

∞ ≤

that

Define a

> . Hence, for all
≥ .

2 -function

1, 2,

=

Obviously, the function
log

> 0 holds for all

where

1, 2,

=

+

3

≥

1,

V: ℝ3+ → ℝ+ by

1 − 1 − log 1
1, 2,

+

there exists an integer

2 − 1 − log 2

−1

3

: ℝ3+ → ℝ is defined by

+

,

1

71

1 −1

1

+

∞ = ∞ a.s.

∈ 0,1 such
1 ≥

0

+

− 1 − log

1, 2,

yields

2

is

∈ ℝ3+ a.s. for

2 −1

such

.

−1−

is non-negative since the inequality

> 0. Applying the Ito‟s formula to

1, 2,

> 0 and

,

2

1, 2,

1

1−

1

1

−

1 2 −

2 −1

2

1−

2

2

−

2 1−

+

−1

1 +

2 −

−

1

≤

−

=

1 −1

+

+

2

−

where
=

Notice that

−

1 +

1

−

1

2
2

−

2

2
2 +

+

1

1

+

+

+

1 ∈ 0,+∞

max

− 1 − log

≤

+2

Υ = max

2
1

2

+

2

2
2

1

≤

+

2

2
2

+
+

1

1

2

2 +

2

+ 2 log 2 ≤ 2

+

−

log 2 + 2

+2

+

−

+

+

2
3

2

+

+

2
1 +

+

2

+

−

+

2
3

2

2
2

+

1

1

1 −

1 +

2 ∈ 0,+∞

≤2

2
1

−

max

+

2

2−

then one can obtain that

where

+

+

2
1

+

2−

+

log 2 , 2

+

2

2+

1−

2−

1, 2,

−

+

+

−

1−

+

2

2

2
3

2

2

,

+

2

1
2

1

2

1−

2
1

2
2

.

+ 2 log 2 for all

2

> 0,

≤ Υ 1+
−

.

The remainder of the discussion follows that in the Theorem 3.3 (Chen et al., 2020),
here, we omit it. Therefore, for any given initial values
system (4-2) exists a unique solution

1

, 2

,

solution will remain in ℝ3+ with probability one, that is,
for all

≥ 0 almost surely.

72

1 0 , 2 0 ,

0

∈ ℝ3+,

on ℝ+ and the positive
1

, 2

,

∈ ℝ3+

4.2.3. Extinction and persistence induced by white noise
The extinction and persistence in the mean are two important aspects in a
population system and it is vital to discover whether the white noise has important
effect on the survival of population (Tapaswi and Mukhopadhyay, 1999). Therefore,
based on the previous analysis, we will further discuss and analyze the extinction and
persistence in the mean of plankton populations in system (4-2). The following
analysis shows that white noise intensity can significantly affect the dynamical
behaviors of stochastic system (4-2). Here we give a detailed analysis:
Firstly, suppose that
initial value

1

yields

0 ,

ln 1
ln

and

=

2

1− 2

1

+

ln

=

= 1−

1

2
−
2 1

1 1

2 0

= 2−

1

2 1

0

=−

+

1

ln

1 0

1 ln

2

ln

and
1 ln
ln

where

+

+

1

∫0

1

2

2
−
2 2

1

2
2 3

2( )

+ 2( )

= ∫0

1

2

+

1
2

2

∫0

1

∫0

2

+

1
3

be the solution of system (4-2) with the

−

Integrating the above from 0 to
1 ln

,

∈ ℝ3+. Applying the Ito‟s formula to system (4-2)

0

1− 1

1

=

2

0 ,

2

, 2

1

∫0

−

1 2

−

2 1

−

−

−

+ 1( )

1( )

and dividing

1

−

−

1

−

2

1

1

+

∫0

2

∫0

1

1

73

2
2 1

−

−

∫0

, = 1,2,

1

1

2
2 2

2( )

+ 2( )

1

+

2
2 3

−

−

1

−

1

1

2

2

,

+

3

3

,

∫0

(4-5)

∫0

(4-6)

1

− ∫0

= ∫0

,

1

on both sides, we have

2

3

+

3

1( )

−

+ 1( )

3

.

(4-7)

( = 1,2,3) satisfy

Moreover, the quadratic variation of
,

2

= ∫0

∗ 2

≤

, = 1,2,3.

By the strong law of large numbers for martingales (Khasminskii, 2012) yields
lim →∞

=0

. . , = 1,2,3.

lim sup →∞

ln

≤

Thus, according to (4-5), we have

1

Obviously, we can obtain that lim →∞
phytoplankton population

1

= 1−

=0

(4-8)

1

2
2 1

. . if

tends toward extinction.

1

. .

< 0, that is, the toxic

Next, we analyze the persistence in the mean of the toxic phytoplankton
population

where

1

and

. By making some estimation of (4-5), we have
1 ln
ln

1

1 0

≥

1

− 1 ∫0
1

− 1 2 −

1

2

lim →+∞

ln

1 0

1

= lim →+∞

2 > 0, there exists a constant

≤

2

2

+ 2,

+ 2,

≤

1

ln

1 0

≥ − 2 , and

Substituting above inequalities into (4-9) and for all ≥
1

where
2

2

−

1 =

(4-9)

= 0.

such that
2

1

will be determined later. In addition, since the fact that

Thus, from the properties of the limit, for arbitrary
2 >0

+

−

1 2
2

ln 1
−

≥

,

1

1

− 1 ∫0
1

= 2−

1

1

2
,
2 2

=

1

2

2

2 , we have

+

+

1

1

≥ − 2.

,

2
,
2 3

and

=

1

1

+

From the Lemma 4 (Liu and Wang, 2011), it is obvious to find that, if

1 > 0, we have

1

lim →∞ inf ∫0

1

74

≥

1

1

1 > 0.

This implies that the toxic phytoplankton
For the toxic phytoplankton
toxic phytoplankton

2 > 0, where

under the condition

, the same to analysis of

2

1

According to Eq. (4-7), we have
1 ln
ln

0

1

≤

1

≤− +

−

− 2 1

2 =

In view of the zooplankton, we have
lim sup →∞

, we have the

1

< 0, while it is persistence in the mean

is extinct if

2

is persistence in the mean.

1

.

, lim sup →∞

1

+

1

2

≤

2

+

3

=

, . .

2

(4-10)

2

.

(4-11)

Combining with (4-8) and (4-10), and taking upper limit on both sides of (4-11) yields
lim sup →∞

Obviously, if the condition
zooplankton is extinct.

ln

1

≤

2

+

1

−

2

< 0, we have limt→∞

= 0. This means that the

Now, we show the persistence in the mean of zooplankton. Computing
(4-5) × 1
1

1

1

1

ln 1

1 0

1

1
1

1

2

+(4-6) ×
1

+ 2
2

− 1
1

+

+(4-7), we can observe that

2

ln 2

2 0

2

2

2

t

2

∫0

2

1

+ ln
+

2

0

2

ds +

≥

1

−
1

2

1

1

1

2
1

2

− + + −
+

3

= 0, limt→∞

3

+

2

2

(4-12)

By the strong law of large numbers for martingales, we can derive that
limt →∞

1

1

1

= 0, limt→∞

2

2

2

From the Lemma 2.3 (Zhao, 2016), we can obtain that
1

limt→∞ sup ln 1

1 0

1

≥ 0, limt→∞ sup ln 2
75

2 0

= 0, . .
1

≥ 0, limt→∞ sup ln

0

≥ 0.

Thus, taking the limit superior in (4-12) and from the lemma 4 (Liu and Bai, 2016),
and if

=

1

−

1

2 2

limt→∞ inf

2

≥

2

1

1

1

−

1

which implies

2

+

+

1 1

−

1

2

2

2 2

−1

+

2

×

− − − > 0, one can see that

2

−

2

1 2

2

− − −

> 0,

limt→∞ inf ( ) > 0.

That is, the zooplankton is persistence in the mean.
To sum up, for any initial values

1

following conclusion:
I. the harmful phytoplankton
(i) extinct if

< 0;

(ii) persistent in the mean if

(i) extinct if

2

< 0;

is

<0;

(i) extinct if

(ii) persistent in the mean if
=

1

1

−

0 ,

0

∈ ℝ3+, we can obtain the

1 > 0.

is

2 > 0.

(ii) persistent in the mean if
III. the zooplankton

2

is

1

II. the harmful phytoplankton

0 ,

2 2

2

> 0,

+ 2

> 0, and
−

2

1 1

1

− − − > 0.

4.2.4. The ergodic stationary distribution of plankton
It is well known that ergodicity is one of the most important and significant
characteristics for population systems described by stochastic differential equations,
which means that the system has a unique stationary distribution, providing a
biological perspective of cycling phenomena of a population system for long time. In
this subsection, therefore, we discuss the ergodic property for stochastic system (4-2)
by constructing a suitable Lyapunov function. The following modeling analysis shows
76

that system (4-2) admits the stationary distribution which is ergodic when the white
noise intensity and toxin released rate are not particularly large.
Actually, to prove ergodic stationary distribution, we need to verify the
conditions of Lemma 4.2.1.1. According to the Lemma 4.2.1.1 in preliminaries
subsection,

it

is

obvious

2 2 2
1 , 2, 3

( , ) = diag

to

obtain

that

the

diffusion

matrix

of system (4-2) is positive definite, which implies that

the condition (i) in Lemma 4.2.1.1 holds.
Next, we prove the condition (ii) in Lemma 4.2.1.1. By constructing a
2

-function
1, 2,

where

: ℝ3+ → ℝ as follows:
1

=

+1

(

1+

) +1 −

2 +

is positive constant satisfying 0 <

2

construct a non-negative
1, 2,

=

=

1, 2,

1
(
+1

−

ln 1 +

3 2

2
2
2
1∨ 2∨ 3

1, 2,

−

=(

-function defined

−

1 +

1, 2,

=

1

+ (
2
≤
−

2

2

1 2 −

1
2

2

1 +

1(
2

+2

−

2 +
1

+

)

1, 2,

+

1

1

1

−

2

+

−1
2+ )

1 +

+1

1 −
2

2
1

2 + ) −

+ (
2

1 +

77

1

−

2

2

− ,

> 0 will be

and

1, 2,

: ℝ3+ → ℝ by

) +1 −

2+

1, 2,

1 +

2

−

ln 2 + ln

is continuous, then there exists a

Using the generalized Ito‟s formula, one can see that
1

2

2

in ℝ3+ which is the minimum value of

1, 2,

unique point

<

1, 2,

determined later. Since the function

1

1

1

1

ln 1 +

1, 2,

2

1

2
1−

2

1

1

1

1

+

+2

2 +

) +1

2
2

2(

2

2
1 ∨

ln 2 + ln
1, 2,

3

1 2 +

2
2

+

+

2

2

. Thus,

+

.

2

2 −

2
3

2

1 +

2
2 ∨

2
3 .

2 +

2

2
2

2

)

Noting that the inequality
1

1, 2 ,

≤−
+

−
+

2

2 2

3
2

2

2

2
2

1

1

+2

+1

2

2

+2

2(

2

=−

2

+

≤−

where

1

1

2

=1

2

2

+ 2
−

1
2

2

+

and
2

1, 2 ,

1

Thus,

1, 2,

1
1

=− +

=−

2

1

1

+2

+1

2 +

)

2

2

1

2

2

1

1 +

+ 2

1

+

2

−

+

2
2

2

2
2

+

2
1 ∨

−

+1

4

2
2 ∨

2
3

−

−

1 +

2+

+2

−

+1

−

−

+1

2

2

+2

2

2

2 +

2

−

2

+2

2
2 ∨

ln 2 − ln

1
1

+

78

1

1

+

2 +
2

+

2

+

2

2

2

1

1

)

3
2

1

+2

2
1 ∨

2
2 ∨

2
3

2 +

)

+1

+ ,

4

2
3

+

−

1

−

2

) +

−

4
+1

2

2

2
1 ∨

−

2

2

2

2

1+

+1

2

1(

2

2

−

2

2

+2

ln 1 −

1

2

, we can obtain that

=1

1

+

2

−

1

+1

+2

+

2

2

2

2

+1

2

1

1

+

3

+1

2

1(

3
2

=−

1

1

1

=−

−

2

2 2

4

+2

= sup ( 1 , 2 , )∈ℝ3+ −
+

−

1 +

1

1

2

−1

≤

+1

2(

1 +

< ∞.

2

2

1

1

≤

2

1 +

1

2

+

2

1

+

1

1

2

,
2

+

.

1, 2,

≤−
+

+

1

2

2

+

2

2

−

2

1 1

1+

2 2

1

1

1

1

+2

2+

−

2

2

2 2

2

2

1

+2

−

1

+
+1

2

2

2

+

+ .

Considering the following compact subset :

where

=

1, 2,

−

+

∈ ℝ3+:

1 <

1

, <

2 <

1

, <

<

1

is a sufficiently small constant satisfying the following conditions:

−

2 2
2

+

−

2−4
1

+

4−4
1

−

+

−

+

6 −4

1 = sup ( 1 , 2 , )∈ℝ3+

1 1

+

1

+

−

2

1

2

1

1

2

+

2

2

2

2

+

2

2

1

2

2

1

+2

≤ −1,

+

(4-14)

3 ≤ −1,

− −2

≤ −1,

2

+

2 +

+

1 +

2

2

− −2

(4-13)

(4-15)
(4-16)

+ +

+

5 ≤ −1,

(4-17)

− −1 ≤ −1,

2

+

+ 1 ≤ −1,

2

2 2

2 = sup ( 1 , 2 , )∈ℝ3+

+

2

+

1 1

+

−

where

<

2

−

2

2

2
2 +

2
,

2

(4-18)

+

+1

2
1 +

+
79

1

1

+

1 1

+

,

1

2

+

2

2 +

2

2

2

2

+

2
2

−

4

2

2

1

1

1

+2

2 2

3 = sup ( 1 , 2 , )∈ℝ3+

+

2

4 = sup ( 1 , 2 , )∈ℝ3+

+
−

2

1 +

2

+

2+

2 2
1

1

1 +

2

+

2

4 2 2 2

2

+2

2

2

−

2
1 +

,

2

+

+1

2

1

1

+

2

1 1

2
1 +

1

+

+

+

2

,

2 +

2

2

2−

2 2
2

1 +

2
1 +

1 1

6 = sup ( 1 , 2 , )∈ℝ3+

2 2

1

2 +

2

1 +

2
1 +

1 1
1

2 ++

+
−

Then

2

2

2

1

1

1

1

1

ℝ3+

with

+2

2

−

=

2

2

2

2

2

+

+2

2

2

4

+1

∈ ℝ3+: 1 ≥

1

6 =

1, 2,

∈ ℝ3+:

1

.

∈ ℝ3+: 0 <

≤

∈

2
2

1, 2,

1, 2,

1, 2,

,

2 =

∈ ℝ3+: 0 <

=

1

1

for any
1, then 1

≤

2 ≤

+

2

2

1, 2,

≤

80

,

,

4 =

−

−

1+

2

∈ ℝ3+

5∪

−

+2

,

5 =

1, 2,

.

2

2

2

1

1

4 ∪

1 ≤

negativity of

+

2
2 +

1

3∪

∈ ℝ3+: 0 <

Now, under the condition

+

2

2

2∪

1, 2,

1, 2,

+

2

2

1 ∪

1 =

3 =

Case I If

+

2
2

2

5 = sup ( 1 , 2 , )∈ℝ3+

2

+

6

1, 2,

−

∈ ℝ3+:

2 ≥

≥

1

,

,

> 0, we need to prove the

in the following cases:

, and one can obtain that

1, 2,

≤−
+

≤−
+
+

By (4-13), we have
Case II If
1, 2,

1, 2,

≤−
+

≤−
+
−

2

1, 2,

By (4-15), we have
Case IV If

1, 2 ,

2

+

1

≤−
∈

2

1, 2,

+

2 2

2

∈

∈

2

2

+

+2

1, 2,

1, 2,

−

2

+2

1

2

2 2

+

−

2

−

2

1

2 2 1 1
1 +

2
2 +

2

2

+1

−

+

2

2

+2

+

1 ≤ −1.

1, 2,

−

2

2

2

2

2 +
1

1

+2

+

1, 2,

−

+2

∈

2

+

+1

2

+

2

2

+

+2

+

−

2

+1

1.

1

+

−

2

2

1

4

2

2

1

1

2 −

2

2

2 2 2 2

1

4

−
2

∈

2 2 2 2

1

≤−

≤ −1 for any

2 +

−

+

1

1

+2

1 1
1

+2

1

2 2 1 1

1

1

2

2

2 +

2

1

4

2

2

1 1

1 +

2

2 +

−

≤ −1 for any

2 2
2

1

1 1

2
2 +
2

+

2

+

1

2

2

2.

2
+1
2

1

2

+

+
+

+2

1

1

− −2

≤ −1,

3 , the similar analysis to case I, we can obtain that

≤−

1, 2,

1 1

+

1

≤ −1 for any

+

1, 2,

+

∈

3 ≤ −1.
3.

4 , the similar analysis to case II, we have

1, 2,

∈

1

+

≤−

+

which follows from (4-16), we have
Case V If

1

2 , we have

2

2

1

2

1, 2,

2
1 +

2

+

+

+

1

2

2

2

2

2 2

1+

1 1

1+

−

2

+

By (4-14), we have
Case III If

2 2

+

5 , then 1

≤

4 −

1, 2,

1 ≤

81

4

2

2

2

2

− −2

≤ −1,

≤ −1 for any

1+

2
1

, 2 ≤

1, 2,

2 ≤

∈

1+

4.

2
2 , we

have
1, 2,

≤−
+

≤−
+
+

+

2 2

+

2 2

2

+

1

1+

−

2

2

2
1

≤−

1, 2,

1, 2,

≤−
+

≤−
+
−

4

∈

1

1

1 +

+

1+

+

1

1

2 2

+

2

+

+

2

−

2

2 2
2

+1 +

+

2

1, 2,

+2

−

1 1
1

−

+

2 +
2

2

2

2

2

2

2

2+
1

1

1

+

+

1 +

1 1

2

2

2

1

1 +

2

1

2

+

1

−

1

+2

2 +

−

2

1 1
1

4

6 −

+1

4

+2

+2

≤ −1 for any

2

2

2

2 +

−

2

−
1

1

−

2

+
2

2

2

2

1

1

1

+2
1

2 2 1 1

1, 2,

2

∈

+

−
1

2

+

+2

−

2
+1
2

2

+

− −1 ≤ −1.

≤ −1 for any

+

1, 2,

2

2

1

1

2 2 2 2

1, 2,

2
2 +

≤−

1

2
2

6 , we have

2
1 +

From (4-18), we have

2

1 1

2 2 1 1

According to (4-17), we have
Case VI If

1 +

2

+

+2

2

+
+

−

2

+

5 ≤ −1.

∈

5.

2

2
+1
2

2

2

+
2

+1

+
+

2 2 2 2

+2

6.

Hence, the condition (ii) of Lemma 4.2.1.1 is verified. Thus, for any given initial
value

1

0 , 2 0 ,

0

∈ ℝ3+ , we can obtain that, if

admits a unique ergodic stationary distribution.

> 0, the system (4-2)

Note: When the environmental fluctuations are not particularly large, there exists a
unique ergodic stationary distribution, implying that the biomass of phytoplankton
and zooplankton can be persistent and coexistence in the long term. Additionally, it
also shows that the adaptability of plankton to environmental fluctuations is limited.

82

4.3. Experimental simulations
In the previous subsections, we have analyzed the effects of environmental
fluctuations and TPP on the dynamics of plankton ecosystem and verified the intensity
of white noise and the liberation rate produced by TPP can lead to the extinction,
persistence in the mean and even the stationary distribution of plankton populations
by modeling analysis. In order to further study how the white noise and TPP affect the
dynamics mechanisms of the formation and evolution process of planktonic blooms,
we perform some experimental simulations for system (4-2) based on the Milstein's
Higher Order Method (Higham, 2001) in this section. In the following experimental
simulations,
1 0 , 2 0 ,

unless
0

otherwise

specified,

we

take

the

initial

value

as

= 0.5,0.5,0.5 , and the parameter values are always used in

Table 4.1 and other parameters are chosen as control parameters.
4.3.1 Impact of white noise on the dynamics of system (4-2)

Due to in a phytoplankton-zooplankton system that takes toxic phytoplankton
into consideration, the intrinsic growth rates of two toxic phytoplankton and the death
rate of zooplankton are parameters that are most susceptible to environmental
influences and are relatively important. Therefore, we only consider the intrinsic
growth rates and death rate affected by white noise. In order to study how the white
noise and two TPP affect the dynamics of system (4-2), we firstly consider the system
does not experience the white noise, that is, system (4-2) becomes its corresponding
deterministic system. According to the work of Sarkar et al. (2005), we can obtain that
the
∗

system

(4-1)

possesses

0.8178,1.8066,2.1071

a

unique

positive

interior

equilibrium

which is locally asymptotically stable, depicting the

coexistence of all three species at a stable state and indicating the changes in the
biomass of plankton populations are static.
Table 4.1 Parameter values
Parameters

Biological meaning

Unit
83

Value

Sources

0.55

Sarkar et al. (2005)

Growth rate of other TPP

day −1

0.5

Sarkar et al. (2005)

Inhibitory effect

day −1

ml/nos.

0.005

Sarkar et al. (2005)

Inhibitory effect

ml/nos.

0.004

Sarkar et al. (2005)

Zooplankton

ingestion

ml/nos.

0.15

Sarkar et al. (2005)

ingestion

ml/nos.

0.15

Sarkar et al. (2005)

the conversion of one

ml/nos.

0.09

Sarkar et al. (2005)

TPP into zooplankton

/day

the conversion of other

ml/nos.

0.075

Sarkar et al. (2005)

TPP into zooplankton

/day

Death rate of

0.09

Sarkar et al. (2005)

zooplankton

day −1

Half-saturation constant

ml/nos.

0.1

Sarkar et al. (2005)

Half-saturation constant

ml/nos.

0.12

Sarkar et al. (2005)

Environmental capacity

nos./ml

2

Estimated

nos./ml

5

Estimated

Growth rate of one TPP

rate
Zooplankton
rate

of one TPP
Environmental capacity
of other TPP
Note: the estimated values in the table indicate that the parameter has been
self-estimated in this thesis or has been selected as a control parameter in
experimental simulations.

84

Fig. 4.1 The effect of
1 = 0.1, 3 = 0.1,
2

on the stochastic dynamical behaviors of the system (4-2) with

= 0.06,

= 0.07, 0 ≤

2 ≤ 1.2. (a)

The dynamical behaviors of species
,

change from persistence in the mean to extinction in different areas of

for 0 ≤
for

2

2 ≤ 1.2. (b) The same path of species

2 = 0.1, 2 = 0.8, 2 = 1

and

system (4-1).

on

2

and

for system (4-2) with respect to Fig. 4.1(a)

1000,2000

and its corresponding deterministic

Next, we explore the impact of the white noise on the stochastic dynamics of
system (4-2). We first fix

1 = 0.1 and

3 = 0.1, and let

2 0≤

2 ≤ 1.2

vary to

see how the white noise influences the survival of plankton populations. According to
the condition of

2

= 2 − 2 , we can obtain that all three species of system (4-2) will
2

undergo extinction when the white noise reaches some a critical value. Obviously, we
can find from Fig. 4.1 (a) that the species
in the mean in the area of
area of

, but species

2( )

of system (4-2) is always persistence

and persistence in the mean or extinct alternating in the

2 ( ) dies out rapidly in the space

when

2

is beyond

2 = 1. Fig. 4.1 (b) depicts that the stochastic dynamical behaviors of species

with respect to Fig. 4.1 (a) for

2 = 0.1, 2 = 0.8, 2 = 1.1

and

on

2( )

1000,2000 .

Moreover, we can observe from Fig. 4.1 (b) that, with the increase in the magnitude of
the environmental fluctuations, the random variation of plankton density becomes
more significant, which implies that white noise can accelerate the stochastic
oscillation of plankton density. For example, let

2 = 0.1, it is not difficult to find

that the two TPP and zooplankton of system (4-2) can coexist at a relatively stable
state and their densities exhibit oscillation around the deterministic steady state values
85

∗
∗
1 = 0.8178, 2 = 1.8066, and

(c)). Actually, following the

∗

= 2.1071, respectively (see Fig. 4.2 (a), (b) and

1 > 0,

2 > 0,

and

> 0, then system (4-2) is

persistence in the mean and has a unique ergodic stationary distribution, which is
consistent with our experimental simulations. From the stationary distribution of all
three species, it can be seen clearly that they are distributed normally around the
values 0.8, 1.8 and 2.1, respectively, which illustrates that the standard deviation
and

3

can keep processes

1( ) ,

2 ( ) and

1, 2

( ) moving around the solution of

deterministic system (4-1). In other words, system (4-2) can preserve some stability in
the random sense when the intensities of white noise are relatively weak. Now let

2

vary within some a level, it can be concluded that the weaker the environmental
fluctuations are, the closer the solutions of system (4-2) are to steady state

∗

(The

Figures here are not given due to the similarity to Fig. 4.2). However, when we
increase the density of environmental forcing
can easily to get that

2 > 1,

2 = 1.1, for example, we

= −0.105 < 0, which implies that the species

2 ( ) tends to

go rapid extinct, even if its corresponding deterministic system (4-1) still presents
obvious stability, indicating a different phenomenon from its deterministic system (see
Fig. 4.3). This also shows that white noise intensity can help to control the density of
toxic phytoplankton. Comparing Figs. 4.1, 4.2 and 4.3, it is obvious to find that the
intensity of white noise cannot only aggravate the stochastic oscillation of plankton
density, but also significantly change the dynamics of the plankton system. That is, a
high-level white noise intensity can accelerate the extinction of the plankton
populations, which implies that the white noise can help control the biomass of
plankton populations and may provide a guide for us to the termination of planktonic
blooms. This is consistent with the results obtained by the work of Sarkar and
Chattopadhayay (2003), who demonstrated the controlling of planktonic blooms by
artificial eutrophication or the intensity of white noise from their experimental and
field observations. Thus, it is worth pointing out that the results from the Figs. 4.1, 4.2
and 4.3 can support that the plankton systems incorporating white noise can better
simulate planktonic blooms than its corresponding deterministic counterparts.
86

Similarly, if we impose the intensities of white noise on species

1 ( ) and

respectively, we can easily obtain the similar results, thus here we omit it.

( ),

Fig. 4.2 (a), (b), (c) the solution trajectories of system (4-2) and its corresponding deterministic
system (4-1). (d), (e), (f) the probability density function diagrams of
the system (4-2) with

1 =

2 =

3 = 0.1,

probability density functions for system (4-2).

= 0.06,

1

1 =

0.07, 2 = 1.1, and its corresponding deterministic counterparts on
1( )

and

3 = 0.1,

system (4-2). (b) The persistence of deterministic system (4-1).

= 0.06,

=

∈ 0,1000 . (a) The

( ) and extinction of species

87

( ) for

= 0.07, and the red smoothed carves are

Fig. 4.3 Stochastic dynamical behaviors of system (4-2) with

persistence in the mean of species

, 2 ( ) and

2( )

of stochastic

4.3.2 Impact of TPP on the dynamics of system (4-2)
In order to study how the effect of one toxin liberation rate on population density
dynamic evolution trend under the environmental fluctuations, we choose

as a

control parameter and all other parameters are the same as Fig. 4.2. Clearly, we can
1 ( ) and

observe from Fig. 4.4 that the species

2 ( ) are persistence in the mean

and their biomass will increase as the increasing value of
undergoes extinction when
biomass of species
that
if

1 ( ),

, while the species

( )

beyond a certain value, here the color-bars denote the
2 ( ) and

( ), respectively. Actually, it is easy to obtain

> 0 under the condition of 0 <

< 0.3846 and

= 0 if and only

≈ 0.3846, which indicates that system (4-2) is persistence in the mean and has

the stationary distribution under the condition of

< 0.3846, in contrast, species

of system (4-2) will die out. Therefore, it can be asserted that TPP can
significantly affect the coexistence of all the three species. For precisely, we take three
different values of

= 0.1,0.2,0.35 , then system (4-2) has a unique stationary

distribution. Fig. 4.5 depicts the relative frequency density of

1 ( ),

2 ( ) and

( )

with these different values, respectively, where the smoothed curves are the
probability density functions of system (4-2). More importantly, we can obtain the
result from the Fig. 4.5 that with the increasing value of , the distributions of two

Fig. 4.4 The effect of toxin rate
behaviors of the system (4-2) with
mean of species
0≤

1( )

and

produced by population
1 =

2 ( );

2 =

3 = 0.1,

0≤

1 ( ) on the stochastic dynamic

≤ 1. (a)-(b) The persistence in the

(c) The persistence in the mean of population

≤ 0.3846 and extinction for 0.3846 ≤

≤ 1.

88

( ) for

Fig. 4.5 The effect of one toxin rate

on the dynamics of system (4-2). (a), (b), (c) The

histograms of probability density functions for
Here,

= 0.07, and

1 =

2 =

3 = 0.1.

1

, 2 ( ) and

( ) with different values of .

TPP appear closer to the normal distribution, but the distribution of zooplankton
becomes more skewness, which implies that the increase of the liberation rate

can

increase the survival chance of two harmful phytoplankton but decrease the biomass
of zooplankton. Additionally, it can be seen from Fig. 4.5 that the peak value of the
probability density functions of system (4-2) will be higher as

increases. All these

results indicate that for the value of the toxin liberation rate
conditions of

satisfying the

> 0, its enhancement will contribute to the persistence in the mean

of system (4-2) though the termination of planktonic blooms. In addition, from the
theoretical analysis, we want to know what happens if
89

< 0? Selecting

= 0.39,

we can easily check
conditions of

> 0 and

≈ −0.0054 < 0 and

≈ −0.0194 < 0 . Although the

> 0 are not satisfied, the system (4-2) has a stationary

distribution, indicating all the three species are persistence in the mean (see Fig. 4.6).
However, when we choose
0 and

= 0.42, similarly, we can obtain that

≈ −0.0354 <

≈ −0.0494 < 0. Obviously, the two toxic phytoplankton can coexist while

the zooplankton tends to go extinct (see Fig. 4.7). For the case of

= 0.06 and

changing the value of , we can easily get the similar results, which are omitted here.

In additionally, the combined effects of two toxin liberation rates on the
dynamics of system (4-2) are studied as well. Fig. 4.8 depicts how the combined role
of

and

affect the dynamics of system (4-2), where the red smoothed curves are

probability density functions of system (4-2). By a straightforward computation, the
condition of

> 0 can be verified, which means system (4-2) has a unique

stationary distribution (see Fig. 4.8). Comparing Figs. 4.5 and 4.8, one can see that the

mean values of two harmful phytoplankton populations are larger than the case
of

= 0.2,

= 0.07, while that of zooplankton is smaller than that case. Thus, we

can obtain that by controlling any one of the harmful phytoplankton, the mean values
of both the harmful phytoplankton are smaller than the value observed when
considering in the case of both two harmful phytoplankton populations, while that of
zooplankton is larger than the case of both harmful phytoplankton are present.
Therefore, the introduction of two harmful phytoplankton can be contribute to
persistence of the system (4-2) and play an important role in the termination of
planktonic blooms.

Fig. 4.6 The effect of one toxin rate

on the dynamics of system (4-2). (a), (b), (c) The

histograms of probability density functions for

1

90

, 2 ( ) and

( ) with different values of .

= 0.39,

Here,

= 0.07, and

1 =

2 =

3 = 0.1.

Fig. 4.7 Stochastic dynamical behaviors of system (4-2) and its corresponding deterministic
counterparts on

∈ 0,1000 . Here,

= 0.42, 1 = 2 = 3 = 0.1,

Fig. 4.8 The combined effects of two toxin rates

and

on the dynamics of system (4-2). (a),

(b), (c) The histograms of probability density functions for
= 0.2,

= 0.2, 1 = 2 = 3 = 0.1, respectively.

= 0.07.

1

, 2 ( ) and

( ) with

4.4. Conclusions and discussion
It is now well recognized that stochastic population dynamics play a significant
role in population dynamics, since environmental fluctuations can affect the growth
process of species, such as the growth rate and death rate, which can be described by
white noise (May, 1973). And the Gaussian white noise can be theoretically preferred
to model environmental fluctuations because of its irregularity and thus a good
approximation to the phenomena of rapid fluctuations (Jonsson and Wennergren,
2019). The study of stochastic population dynamics goes back to the pioneering work
by Haminskii (1980), who introduced two white noise to stabilize a linear system.
After that, lots of attention has been paid on stochastic population dynamics studies
91

(Mao et al., 2002; Wang et al., 2020; Lee et al., 2020). Mao et al. (2002) pointed out
that stochastic noise can suppress potential explosion in population dynamics. Wang
et al. (2020) showed that time-periodic forcing can lead to the transitions from a
spatially homogeneous stationary state to a periodic oscillation in time. Additionally,
lots of stochastic plankton growth systems have been derived by numerous
researchers (Zhao et al., 2020; Wang and Liu, 2020; Zhao et al., 2016), and stochastic
plankton systems involving toxin-producing phytoplankton have become a hot topic
in ecological studies due to harmful phytoplankton can significantly affect the
dynamics of plankton systems (Scotti et al., 2015; Banerjee and Venturino, 2011;
Khare et al., 2010; Chattopadhyay et al., 2002; Chattopadhyay et al., 2002; Sarkar and
Chattopadhyay, 2003a, 2003b; Chattopadhyay, 2004; Sarkar et al., 2005; Yu et al.,
2019).
In this thesis, therefore, we first propose a stochastic phytoplankton-zooplankton
system with two harmful phytoplankton populations, where the intrinsic growth rates
of two harmful phytoplankton populations and the natural death rate of zooplankton
are influenced by the environmental noise, and then we study the effects of TPP and
white noise on the dynamics of system (4-2) theoretically and numerically. In order to
ensure that the system is biologically meaningful, the existence and uniqueness of
global positive solutions of system is discussed, and the results demonstrate that for
any initial value

1

0 , 2 0 ,

0

∈ ℝ3+, the solution will remain in ℝ3+ with

probability one. Based on this situation, we derive some sufficient conditions for the
extinction and persistence in the mean of the system. Obviously, those conditions are
great significance to study the extinction and persistence in the mean for the
phytoplankton-zooplankton system (Yu et al., 2019; Sarkar and Chattopadhayay,
2003). Significantly, when the system is persistence in the mean, we also investigate
the existence and uniqueness of positive recurrent of solution for the system, which
implies that the system has a unique stationary distribution under some conditions.
Numerical analysis illustrates our theoretical results and further indicates that both
two TPP and environmental fluctuations have a significant effect on the controlling of
92

planktonic blooms.
On the one hand, from our dynamical analysis, which follows that, when the low
level intensity of white noise satisfies the conditions

1 > 0, 2 > 0

and

> 0,

system (4-2) is persistence in the mean and exists a stationary distribution which is
ergodic, indicating the coexistence of all those three species in the random sense for a
long time (see Fig. 4.2). However, when we increase the intensity of environmental
2 <0

forcing satisfying the condition of

holds, the harmful phytoplankton

2( )

undergoes extinction although other two species are persistence in the mean, as it is

shown in Fig. 4.3. Comparing Figs. 4.2 and 4.3, it can be asserted that white noise can
aggravate the emergence of stochastic oscillation and significantly change the
dynamics of phytoplankton-zooplankton system. Especially, the strong intensity of
white noise can accelerate the extinction of the plankton populations. Consequently,
these results may be more realistic than that of in (Sarkar et al., 2005), which implies
that the controlling of random environmental fluctuations may be a good way in the
termination of planktonic blooms. Therefore, it is great ecological significance to
consider environmental noise when studying phytoplankton-zooplankton interaction
in the presence of harmful phytoplankton.
On the other hand, it is investigated how the dynamics of system (4-2) strongly
depends on TPP. By controlling one toxin liberation rate, the dynamic behaviors of
system (4-2) can be changed. That is, when the toxin liberation rate is beyond some a
critical value, two harmful phytoplankton can coexist, while zooplankton tends to
extinction (see Fig. 4.4). Moreover, when controlling any one of the two TPP, it is
obvious to survey from Fig. 4.5 that the increasing value of one toxin liberation rate
can reduce the biomass of zooplankton, while increase the survival chance of two
phytoplankton populations. In addition, in the presence of both two TPP, it can be
seen from Figs. 4.5 and 4.8 that the combined effects of two liberation rates on the
changes in plankton populations are stronger than that of controlling any one of the
two TPP. Thus, the introduction of two harmful phytoplankton populations is
conducive to the persistence of the system (4-2) through the termination of planktonic
blooms.

Therefore,

TPP

has

a

profound
93

impact

on

the

dynamics

of

phytoplankton-zooplankton systems and may be used as a biological way to control
planktonic blooms.
There are some interesting topics waiting for us to further explore. For example,
the zooplankton mortality will occur after some time lapse due to the bloom of toxic
phytoplankton (Chattopadhyay et al., 2002), it seems to more reasonable to study a
stochastic toxic-producing phytoplankton-zooplankton system with time delay.
Another problem of interest is to consider impulsive perturbations into the system. We
leave those for our future research goals.

94

Chapter 5 DYNAMICS OF A STOCHASTIC PHYTOPLANKTON
–TOXIC PHYTOPLANKTON-ZOOPLANKTON SYSTEM UNDER
REGIME SWITCHING
Abstract
In this thesis, a stochastic phytoplankton-toxic phytoplankton-zooplankton
system with Beddington-DeAngelis functional response, where both the white noise
and regime switching are taken into account, is studied analytically and numerically.
The aim of this research is to study the combined effects of the white noise, regime
switching and toxin-producing phytoplankton (TPP) on the dynamics of the system.
Firstly, the existence and uniqueness of global positive solution of the system is
investigated. Then some sufficient conditions for the extinction, persistence in the
mean and the existence of a unique ergodic stationary distribution of the system are
derived. Significantly, some numerical simulations are carried to verify our analytical
results, and show that a high-level intensity of white noise is harmful to the survival
of plankton populations, but regime switching can balance the different survival states
of plankton populations and thus decrease the risk of extinction. Additionally, it is
found that an increase in the toxin liberation rate produced by TPP will increase the
survival chance of phytoplankton, while it will reduce the biomass of zooplankton. All
these results may provide some insightful understanding on the dynamics of
phytoplankton-zooplankton systems in randomly disturbed aquatic environments.
Keyword: Stochastic phytoplankton-toxic phytoplankton-zooplankton system, White
noise, Regime switching, Extinction, Stationary distribution
5.1. Introduction
Plankton are the basis of the freshwater and seawater food chains, and their
importance for the wealth of aquatic ecosystems and ultimately for the planet itself is
95

nowadays widely recognized (Huppert et al., 2002). Phytoplankton, particularly, can
create energy for the aquatic life through photosynthesis and produce large amounts
of oxygen by absorbing carbon dioxide from their surroundings. However, the rapid
growth of phytoplankton can cause large-scale blooms in one area and the
occurrences of harmful phytoplankton blooms have been reported globally with an
increasing frequency in the past decades (Hallegraeff, 1993), which are detrimental to
the public health, fisheries, tourisms as well as the balance of ecosystems (Anderson
et al., 2000). For example, some freshwater lakes in China, such as Lake Taihu, Lake
Poyanghu, Lake Chaohu, etc., have suffered varying degrees of toxic cyanobacterial
blooms in recent years. In 2011, Lake Erie experienced the largest harmful algal
blooms (HABs) in its recorded history, with a peak intensity over three times greater
than any previously observed bloom (Michalak et al., 2013). Based on the huge
effects of planktonic blooms and the mechanisms behind them are not yet clearly
understood, it is necessary and important to understand the dynamics mechanisms of
changes in plankton populations.
Many researchers have attempted different approaches to explain the dynamics
mechanisms of planktonic blooms in the past decades. The results from experiments
suggested that the toxic or noxious chemicals produced by blue-green algae may
reduce the grazing pressure of zooplankton population and even cause their mortality
for a long time (Fulton and Paerl, 1987; Lampert, 1981), which could be one of the
key parameters for planktonic blooms (Krik and Gilbert, 1992). Some experimental
evidence demonstrated that the grazing pressure by micro-zooplankton can represent
an important factor for the controlling and regulation of HABs (Calbet et al., 2003;
Johansson and Coats, 2002). In addition, there is an experiment revealing that under
some suitable conditions, the formation of Microcystis blooms is closely related to the
presence or absence of zooplankton population and to its selective grazing of the
naturally occurring zooplankton (Wang et al., 2010). Another approach the researchers
are trying to explain the bloom phenomenon is the role of toxicity. The result that
toxicity may be as a strong mediator in the zooplankton feeding rate is found in a field
96

observation (Estep et al., 1990), as well as a laboratory experiment (Huntley et al.,
1986). Moreover, their experimental findings and field study revealed that TPP can
suppress the grazing pressure of zooplankton and may act as a biological control way
for the termination of planktonic blooms (Chattopadhyay et al., 2004; Chattopadhyay
et al., 2002). All these results imply that toxin production plays a significant role in
the interaction between phytoplankton and zooplankton populations, which may
greatly stimulate researchers to explore the way these toxin production affect the
coexistence and survival prospect of plankton populations in the presence of
non-toxic and toxic phytoplankton.
Due to the complexity and openness of real aquatic ecosystems, establishing
mathematical models is now a classical way to study the planktonic blooms (Truscott
and Brindley, 1994), which can provide quantitative insights into the dynamic
mechanisms of changes in plankton populations. In recent years, many deterministic
mathematical models for plankton dynamics, such as delayed nutrient-phytoplankton
models (Dai et al., 2016; Guo et al., 2019), a diffusive nutrient-toxic phytoplankton
model (Chakraborty et al., 2015), viral infection nutrient-phytoplankton models (Li
and Gao, 2016; Chattopadhyay et al., 2003), a phytoplankton-toxin producing
phytoplankton-zooplankton model (Chattopadhyay et al., 2004), and so on, have been
developed and studied extensively, and many interesting results have been shown.
However, plankton populations in the real aquatic environments often fluctuate
unpredictably because of the unpredictability of environmental stochasticity, and these
deterministic models do not capture random environmental fluctuations which is an
important feature of aquatic ecosystems. In fact, some experiments shown that
environmental noise has a significantly effect on population systems in ecology
(Richardson and Heilmann, 1995; Carpenter et al., 2011). For example, the work of
Reichwaldt et al. (2013) demonstrated that the wind can be the most likely driver to
control the biomass of cyanobacteria. In addition, the growth rate of toxic Microcystis
and environmental biomass rely heavily on the temperature and nutrient concentration
(Davis et al., 2009; Fujimoto et al., 2007). Thus, plankton systems with such
97

environmental fluctuations are significantly more reasonable, and the issue of how
environmental fluctuations affect plankton systems have attracted increasing attention
and great effort has been made towards the study of the dynamics of stochastic
plankton systems recently (Yu et al., 2017; Zhao et al., 2020; Sarkar and
Chattopadhayay, 2003; Pal et al., 2009; Chen et al., 2020; Chen et al., 2020; Wang and
Liu,

2020).

But

the

study

of

stochastic

phytoplankton-toxic

phytoplankton-zooplankton system is still in its infancy, especially the dynamics of
the phytoplankton-toxin producing-phytoplankton-zooplankton system with white
noise and regime switching is currently unclear. Thus, we mainly present the effects
of white noise, regime switching and toxic substances produced by TPP on the
stochastic phytoplankton-toxic phytoplankton-zooplankton system in this paper.
The rest of this paper is organized as follows. The model is presented in Section
5.2. Section 5.3 introduces some preliminaries firstly, and then we give the main
results, including the existence and uniqueness of the global positive solution,
extinction and persistence in the mean as well as the stationary distribution and
ergodicity of the system. Some numerical simulations are carried out to study the
dynamics of the system in Section 5.4. We summarize the results and present our
conclusions in Section 5.5.
5.2. Model formation
In this section, we will establish a stochastic two preys-predator model in which
the zooplankton feeds on two types of phytoplankton species, including a non-toxic
phytoplankton (NTP) and a toxic one. The ecological construction of the stochastic
phytoplankton-toxic phytoplankton-zooplankton system is based on the following
assumptions:
1.

It is assumed that

1

, 2

and zooplankton, respectively;
2.

,

are the population densities of NTP, TPP
is the natural death rate of zooplankton.

It is considered that the growths of NTP and TPP in the absence of the grazer
zooplankton are generally considered as logistic type with the intrinsic growth
98

rates

1

and

capacities
3.

1

2 respectively,

and

It is assumed that

2.

1

and

2

and their corresponding environmental carrying

measure the competitive effects of TPP on NTP

and NTP on TPP, respectively. In fact, these competitions have been introduced
into ecological systems to explore the properties of plankton dynamics, such as
stability (Pal et al., 2009), oscillation and chaos (Huisman and Weissing, 1999),
etc.
4.

Behavior of the entire community is assumed to arise from the coupling of these
interacting species. Both groups of phytoplankton exhibit Beddington-DeAngelis
functional
1+

and

response

to

+

2 2

1 1

the

grazer

and

zooplankton

as

1+

2 2

1 1

+

given

by

, where

are the attack rates of zooplankton on NTP and TPP, respectively;

1 and

are the product of attack rate and handling time on NTP and TPP, respectively.

2

In addition, the term

1+

2

1 1 +

2 2

, which describes the resultant

reduction for the growth of zooplankton due to the ingestion of TPP, where
the
1

the

inhibition

rate

of

1+

2 2

can be regarded as the growth form of zooplankton in

present

1 1 +

of

NTP,

zooplankton

where

is

the

growth,

while

conversion

the

is

efficiency.

term

The

Beddington-DeAngelis functional response (Beddington, 1975; DeAngelies et al.,
1975) here is appropriate in the case of plankton population due to the fact that
the predator individuals either search, consume or interfere with each other (Pal et
al., 2009).
5.

It is assumed that environmental noise exists in the realistic aquatic ecosystems
because of the unpredictability of the environmental stochasticity, such as
nutrients supply, water temperature, and some other small environmental
fluctuations, which may affect population growths of the system. Actually, the
work of May (1973) pointed out that these small environmental fluctuations can
affect the ecological parameters of a model positively or negatively, which can be
described by white noise. Thus, following the idea of (Yu et al., 2017; Zhao et al.,
99

2020; Chen et al., 2020; Chen et al., 2020), the convenient formulations, which
describe the intrinsic growth rates of phytoplankton populations and the death
rate of zooplankton population that are influenced by white noise, are taken as
→

terms
noises,
6.

+

= 1,2 and − → − +

enote the white noises and
= 1,2,3.

2

3

3

, respectively. Here the

> 0 are their intensities of white

We further consider the regime switching into the model, where the biomass of
plankton often suffer from switch abruptly to a contrasting alternative stable state
in the real world due to some kinds of moderate environmental fluctuations, such
as environmental pollution (Scheffer and Carpenter, 2003), rain falls (Du et al.,
2004; Slatkin, 1978) and biotic exploitation (Scheffer, 2001). The plankton
population models in this case can be characterized by the telegraph noise or
colored noise (Mao et al., 2003), which may cause the population systems
switching from one environmental regime to any other regimes (Mao et al., 2003;
Luo and Mao, 2007). In addition, the switching is generally memoryless and the
waiting time between two shifts follows exponential distribution. The convenient
formulation here is to take

, ≥ 0 as regime switching, which is a

continuous-time Markov chain with state space
7.

It is assumed that the Markov chain

= 1,2, ⋯ ,

and the Brownian motions

1,2,3 are defined on a completed probability space
filtration ℱ
of

,1 ≤

,ℱ , ℱ

satisfying the usual normal conditions, and

≥0 ,

< ∞.

, =

with a

is independent

, = 1,2,3.

Based on above assumptions, a stochastic phytoplankton-toxic producing
phytoplankton-zooplankton system under regime switching is presented as follows:

100

1

2

=
=
=

where

1

1−

1

1+ 1
2

1+ 1

+ 3
,

2

+ 2

1

+ 2

1

1

1+ 1

,

,

1

1
2

1−

2

−

2

2

−

3

1

+

1

+

2

−
2

+ 2

−

2

1

,

∈ .

are all positive constants for each

−

1

−

1

2

= 1,2 ,

2

2

,

,

1

(5-1)
2

and

= 1,2,3

5.3. Dynamic analysis
Due to the sudden and rapid changes of the external environment in a real
aquatic ecosystem, such as nutrients supply, water temperature and some other small
environmental fluctuations (white noise), and even rain falls (Du et al., 2004; Slatkin,
1978), environmental pollution (Scheffer and Carpenter, 2003), biotic exploitation
(Meijer et al., 1994) and other kinds of moderate environmental fluctuations (colored
noise or telegraph noise) that can be described by regime switching, which may
significantly affect the growths of plankton populations. In this section, therefore, we
will discuss how these changes (white noise and colored noise or telegraph noise)
affect the dynamics of the system. Before it, we firstly investigate and guarantee the
existence and uniqueness of global positive solutions of the system, and then discuss
the extinction and persistence in the mean, and the positive recurrence and ergodic
property of solution of the system under the effect of environmental fluctuations.
5.3.1. Preliminaries
ℝ+ = 0, +∞

=

=1

2

and

ℝ+ =

1, ⋯ ,

∈ℝ :

> 0, = 1,2, ⋯ ,

. For convenience, we introduce the following notations. If

bounded and integrable function on ℝ+ , we define
= lim⁡inf →∞

, here

1

= ∫0

101

,

> 0.

= lim⁡sup →∞

,

and
is a
, and

, ≥ 0 be a right-continuous Markov chain on the probability space

Let
,ℱ , ℱ

≥0 ,

1,2, ⋯ ,

0 =

with

,1 ≤

0,

taking values in a finite-state space

=

=

< ∞ with the transition rate

following form:

+Δ

=

=(
+

1+

) × of
,

+

given by the

,

is the infinitesimal of higher order, Δ > 0 and

where

transition rate from

to

≠

if

=−

while

=

≠ ,
= ,

≥ 0 is the

. Throughout this paper,

≠

, ≥ 0 is irreductible, which means that

we always assume that Markov chain

the system can switch from any regime to any other regime, indicating that there exist
1, 2, ⋯ ,

finite number

∈

such that

Under this assumption, the Markov chain
distribution

= π1 ,

linear equation
vector
min ∈

=

.

,1

1, 2

⋯

,

> 0 , for any

, ∈ .

, ≥ 0 has a unique stationary

∈ ℝ1× , which can be determined by solving the

2, ⋯ ,

= 0 subject to

1 ,⋯,

,

we

= 1 and

=1

∗

define

> 0, ∀

= max ∈

∈

. For any

and

∗ =

Now, we introduce some fundamental results on the stationary distribution of
,

stochastic differential equations under regime switching. Let
diffusion process described by the following equation:

where

=
0 =

· and

· are the

,

0 ∈ℝ

,

+
0 =

0 ∈

,

,

,

→ ℝ × satisfy

,

,

=

,

stands for the transpose of a matrix or vector. For each

continuously differentiable function
define a operator

:

(5-2)

-dimentional Brownian motion and right continuous

Markov chain in the above discussion, respectively.
·,· : ℝ ×

be the

,
102

∈ ℂ2 ℝ ×

·,· : ℝ ×

→ ℝ , and

, where the superscript
∈

and for any twice

that are non-negative, we

,

where
Γ

,·

=

=1

=

=1

,

,

,

+

1
2

=

2

,

, =1

,

≠ , ∈

,

+Γ

−

,

,·

,

,

∈ .

From Theorems 3.13 (Zhu and Yin, 2007), the following lemma which gives a
criterion for the ergodic stationary distribution of system (5-2) can be presented:
Lemma 5.3.1.1. If the following conditions are satisfied:
≠ ,

(i) for

(ii) for each

constant

> 0, , ∈ ;
2

∈ ,

≤

,

∈ 0, 1 for all

∈ℝ ;

≤

−1

2

(iii) there exists a bounded open subset Ξ of ℝ
: Ξ → ℝ such that

that for some ς > 0,

with a regular (i.e. smooth)

is twice continuously differentiable and
∈

× .

recurrent. That is, there exists a unique stationary distribution

·,· =

Then the diffusion process

·,

·,

∈ ℝ , with some

∈ À , there exists a nonnegative function

boundary satisfying that, for each
·,

, for all

≤ −ς, for any

,

for system (5-2) is ergodic and positive

and for any Borel measurable function

1

lim →∞ ∫0
Lemma 5.3.1.2. Let
linear system:

=

has a solution if and only if
1,

2, ⋯ ,

,

∫
∈À ℝ

we have

∈ ℝ1× .

,

·,· : ℝ ×
,

,

=

be irreducible and

=

→ℝ

such that

∫

,

·, : ∈

< ∞,

∈À ℝ

,

,

= 1.

∈ ℝ . Then the following

= 0 , where the stationary distribution
103

,

=

5.3.2. Existence and uniqueness of global positive solutions
Before investigating the dynamics of system (5-1), we should first guarantee the
existence of global positive solutions according to the biological interpretation.
Therefore, in this subsection, we discuss the existence of global positive solutions of
system (5-1) in the following.
Actually, from the method of the Theorem 3.15 (Mao and Yuan, 2006), obviously,
we can verify that all the coefficients of system (5-1) are locally Lipschitz continuous
and system (5-1) admits a unique local solution
∈ 0,

for any given initial value

1

1 0 , 2 0 ,

0 ,

0 ≥ 1

enough

0

, 2

,

,

on

∈ ℝ3+ × , where

represents the explosion time. In order to illustrate the solution is global, we only need
to

= ∞, . .

prove
1

0 , 2 0 ,

0 ,

0

following stopping time:
= inf

∈

∈ 0,

≤

Let
1

,

0

1

0

≥

. For each integer

: max

, 2

1

or min

, 2

1

,

,

large

,

0 ,

,

≥

and the set inf ∅ = ∞ (∅ denotes the empty set). Obviously,
→ ∞. Let

show

∞ = lim →+∞

that

∞ =∞

1 0 , 2 0 ,

0 ,

0

, then we can obtain that

a.s.

∈ ℝ3+ ×

the proof, we only need to proof

≥

the

1, 2,

2

,

∗

Obviously, the function
Ito‟s formula to

1 ≥

1, 2,

1 − 1 − log 1

,

1, 2,

,

is increasing as

a.s. Thus, If we can
=∞

then

and

≥ 0. In other words, to complete

∈ 0,1 such that

-function V: ℝ3+ → ℝ+ by
=

we define the

∞ = ∞ a.s. Otherwise, the statement is false, then

1 , there exists an integer

Define a

following,

a.s. for all

> 0 and

there exist two constants
for all

in

∞ ≤

satisfying

,

+

0

such that

2 − 1 − log 2

∞ ≤

∞ ≤

+

∗

> . Hence,

≥ .

− 1 − log

.

is non-negative. By applying the generalized

, we have
104

1, 2,

where

,

=
+

1, 2,

2

2−1

: ℝ3+ → ℝ is defined by

1, 2,

,

=
−

∗

1+

+
−

+

−

where
=

∗

1

1+

+

1

1

+

∗

∗

∗+

∗ ∗
1

1 ∗

2 ∗

+

2 ∗
∗
2

2
2+

∗

∗

+

∗

∗

2 ∗

∗ ∗

−

∗

+

1 ∈ 0,+∞

max

+

max

2 ∈ 0,+∞

1

∗
∗
2

+

2

1 −

1 −

−

1 ∗−

∗ ∗
1

−

1

−

+

2
2
2

1
2

2

∗ ∗
1 +

1 ∗
2 ∗

+ 2∗ +

105

2

≤

+

∗
2

1

1 ∗
∗
1

2 ∗+

,

3

∗ 2
1

2 ∗+

∗ ∗

−1

2

2

2

∗

1

2

+

1 ∗−

∗

∗ 3

+

2

1

−

∗ ∗
1 +

2

2

1+

1

1−1

+

1

2

1−

1

−1

∗

2

2

1

2

1−

1

2 −1

≤
+

1 −1

∗

+

1
2

2
1 +

2
∗ 3

−

+

∗ ∗+
1

∗
2 +

∗
2

2 ∗

∗ ∗

∗ ∗
1 +

∗
2 +

1−

∗ ∗
1

2−

∗
∗ 3

+ 2∗ +

+

∗
2

2

∗

+

∗

−

∗
∗ 3
1 ∗
∗
1

2
1

2
2

.

2 ∗
∗
2

∗ ∗
1
∗ ∗

,

2

≤2

Notice that

− 1 − log

> 0, then one can obtain that

where

≤

+2

∗ ∗

Υ = max

∗

+

−

2

∗ ∗ log 2 +
∗

∗ ∗+

+2

2

+ 2 log 2 ≤

∗

−

1, 2 ,

∗

∗ ∗

∗

∗ ∗ log 2 ,

∗

+
2

−

,

≤ Υ 1+

∗ ∗

∗ ∗+

∗

+ 2 log 2 for all

∗

∗

−

∗ ∗

,

.

The reminder of the proof follows that in (Yu et al., 2018), here, we omit it. Therefore,
for any given initial value
solution
solution
1

, 2

will

, 2

,

,

1

,

,

remain

0 , 2 0 ,

1

0 ,

∈ ℝ3+ × , there exists a unique

0

of system (5-1) on

in

∈ ℝ3+ ×

ℝ3+ ×

with

∈ ℝ+ and the positive

probability

one,

namely,

≥ 0 almost surely.

for all

5.3.3. Extinction and persistence of plankton
Based on the previous analysis, and the perspective of the study of population
dynamics, it is necessary and important to consider whether a population can sustain
development or become extinct in the long time under the effect of the environmental
fluctuations in the model. Thus, we will discuss the persistence in the mean and
extinction of system (5-1) in this subsection, and the following analysis shows that the
environmental fluctuations (white noise and regime switching) can significantly affect
the persistence in the mean and extinction of plankton populations, which implies that
the environmental fluctuations can control the growth of harmful phytoplankton
population using modeling analysis. For convenience of discussion in the following,
we define
=

=1

−

1

=

1
2

=1

2
1

,

=
+

106

1
2

=1
2
3

2

.

−

1
2

2
2

,

Now, we discuss the statement. Suppose that
solution of system (5-1) with the initial value
firstly consider the toxic phytoplankton species
system (5-1), one can easily obtain that
ln 1

=

1−

1

−
+

1+

1

1

1

1

Integrating the above from 0 to
1 ln
ln

1

1 0

=

where

−

1

−

1

1+ 1

1

−

.

1 0 , 2 0 ,

2
2 1

1

+ 2

= ∫0

1

is the

∈ ℝ3+ × . We

on both sides yield
−

1

+

2

0

,

1 2
2 1

−

2

1

1

0 ,

,

2

2

1

−

, 2

. Applying the It ‟s formula to

1

and dividing

1

1

+

1

1

2

(5-3)

1( )

1

By the strong law of large numbers for martingales (Mao, 1997c) yields
lim →∞

1

=0

According to the ergodic theorem of Markov chain
to find that, if

< 0, we have

lim sup →∞
which implies lim →∞

1 ln
ln

1

1

1 0

≤ lim →∞
=

=1

1
1

. .

−

−

1

(5-4)
and (5-3), (5-4), it is obvious

1

2
2 1

2
2 1

=

< 0, . .

= 0, . . , that is, the toxic phytoplankton

1

is

extinct, which implies that the white noise and regime switching can affect the
dynamics of the system and provide a biological way to control the growth of harmful
107

phytoplankton.
2 ( ) and zooplankton

For other species phytoplankton
< 0 holds, we have

condition

lim sup →∞

1 ln

≤ lim →∞

2

ln

2 0

=

and
lim sup →∞

1 ln
t ln

0

This imply that lim →∞

=−
2

that

1 0 , 2 0 ,

0 ,

under the conditions of

1
2

+

=1

1

1
2
2

+

2

2
2

2

1
2

2
3

=
2
3

< 0, . .

< 0, . .

= 0, . . and lim →∞

= 0, . . , respectively.

and zooplankton

2

<0 .
0

−

−

2

=1

≤ − lim →∞

Namely, phytoplankton
provided

2

Thus,

for

any

( ), similarly, if the

tend toward extinction
given

the

2

< 0 and

< 0.

3 =

∗

1 ∗
∗
1

−

< 0 hold.

Actually, suppose that

with the initial value

, 2

1

1 0 , 2 0 ,

1 =

−

,

,

0 ,

0

et al., 2011; Liu and Bai, 2016), we can obtain
1

1 ∗
∗
1

≤

We first consider the zooplankton

,

2

ln

0

≤−

and non-toxic

+

1

2
2 3

≤

∗
1

2 ∗
∗
2

> 0,

∗

− 2 ∗1 ∗

2 =

1

>

is the solution of system (5-1)
∈ ℝ3+ × . From the Lemma 4 (Liu
2 ∗
∗
2

(5-5)

. Applying the It ‟s formula to system (5-1)

and then integrating the above from 0 to
1 ln

1

can coexist at a stable state, while the zooplankton

undergoes extinction if the conditions
0, and

value

∈ ℝ3+ × , all the populations will undergo extinction

Interestingly, we find that the toxic phytoplankton
phytoplankton

initial

and dividing , we have

108

+

1+ 1

1

+ 2

2

1

−

1+ 1

+ 2

1

+

2

2

3

(5-6)

Taking upper limit on both sides of (5-6) and using the strong law of large number of
local martingale yields
ln
3 < 0 holds,

if the condition
zooplankton

∗

≤

1

is extinct.

1

≥ ln 1 0 +

limt →+∞

and from the definition of

≤

2 ∗
∗
2

+

ln

2

2 1∗

∗

− 1 ∗2 ∗

1

1 ∗

= limt→+∞

1

ln 1

1 0

≤
1 0

t

2

2

∗

,
3

−

1

≥ − 2,
∗

≥

− 1 ∗2 ∗

1

≥

This implies that toxic phytoplankton
2

1

,

, then we can obtain that for arbitrary

1

1

2
2 1

≥ − 2.
3

∗
1

−

2

1 ∗

1 > 0 and by (5-5), one can obtain that

For the species

+

=0

Substituting the above inequalities into (5-7) and for all
1

∗

−

2

such that

,

ln

∗
1

1 0

and

2 >0

there exists a constant

Obviously, if

= 0 . ., namely,

. By (5-5) and integrating (5-3) on the

1

−

In addition, since the fact that

and

< 0, . .

and making some estimations, one can obtain that

ln 1

2

−

which implies that lim →∞

Now, we consider the species
interval 0,

1 ∗
∗
1

1 ∗
∗
1

1

∗

− 1 ∗1 ∗
1

1

− 2,

≤

≥

++

3

2 , we have
1

.

> 0.

is persistence in the mean.

, the same analysis to the species
109

1

, we have

(5-7)

2 > 0,

≥

2
2 >0

if the condition

2 ∗
∗
2

−

∗
2

2 ∗
∗
2

holds. That is, the species

> 0,

is persistence in the mean.

2

5.3.4. Stationary distribution and ergodic property of plankton
In aquatic ecology, the coexistence of plankton populations is strongly related to
the sustainable development of aquatic ecosystem and it is important to study the long
time statistical behaviors of plankton populations under the effect of environmental
stochasticity in the model, that is, the positive recurrence and the existence of
uniqueness of stationary distribution of the system. Thus, by constructing a suitable
Lyapunov function and using Khasminskii's method (Khasminskii, 1980), we will
discuss and analyze the existence of the stationary distribution and ergodic property of
the system in this subsection. The following analysis shows that the regime switching
and toxin released rate by toxic phytoplankton can significantly affect the existence of
the stationary distribution and ergodic property of plankton populations.
In order to prove our statement, we only need to prove the three conditions in
Lemma 5.3.1.1 one by one. Obviously, the condition (i) of Lemma 5.3.1.1 is satisfied
> 0 for

by the assumption

≠ , , ∈

in subsection 5.3.1. On the other hand,
,

it is easy to verify that the diffusion matrix

= diag

2
1

, 22

, 32

of

system (5-1) is positive definite, which implies that the condition (ii) of Lemma
5.3.1.1 holds.
In the following, we prove the condition (iii) of Lemma 5.3.1.1. By constructing
a

2 -function
1, 2,

where
=

1

1,

,

and

: ℝ3+ ×

→ ℝ as follows:

=

1 1 + 2+ 2

=

1

− ln 1 + 1 + ln 1 + ln 2 + 1 + ln

2, ⋯ ,

2

1, 2,

,

+

2

1, 2,

,

are positive constants and
,

=

2
1 +

2
2 +⋯,

110

+

2

3

+ 1 + ln 2 +
,

> 0 satisfying –
and

∈

+
≤ −1 ,

will be

determined later and the reason for

1, 2 ,

non-negative. Obviously, the function
Ito‟s formula to
1

1, 2,

,

1

and

2 , we have

= 1 1

1 −

+ 2
+

1+

≤−
−

and
2

1, 2,

,

1

2

1 1

1

1+

−

1

1

≤
+

1

2

1

1 +

2
2 −
2

2

1

+

1

2

− 2
1 +

+

1+

1

2

2

1 −

1 +

1

+

2

− 1

2

2 +
2

is non-negative. Applying the

1

2

,

2

2

2

2

+

111

1+

2

1 +

1+

1

2
2 + 2

2

1 +

+

2

1

2 +

2

1 +

1+

1 −

1

2

− 2

−

2

1 2−

1 2 −

2

1 +

1

,

2
1 − 1 1

2
1 + 1 1

=−1
− 2

Thus,

2 −

1 1

+

being here is to make

1+

1+
+

+
+

2
3

1

2

1
2
1

2

1

2

1 +

2

2

2

2

2

− 2

2 − 2

1 +

2 +

+

1

1 +

1 +

1

1

2
2

2

2

2

2

2

+

+

2
3

+

+

2
1
2
2

2

.

2

1, 2,

1

,

+

1

+
−

1, 2,

1

1

≤−
−

1

−

2

+
+

+

∗

1 =

Choosing

∗

,

2

1

1+

1

2

+

2

+

2

Moreover,

−

42

1

− 1

2

4

2

+

2

22
1

2

1

1,

2, ⋯ ,

2

− 2

,

1

+

2

1 −

+

− 2

2
1

+

2
2

+

2

2
2

2

+

2

, one can obtain

+

2

2

Note that

=

1

+

1

+1

−

2

+

2
3

.

1

+

1

1

2+

1

2

∗

3

where

+

+1 +

2
1

=1

+

1 1

∗ + ∗ +1

,

+
2

+

−

− 2
1+

+1

1

2 −

2

1, 2,

1

2

2

2 =

and

+

1

2

+

2

,

2
1 +

1

1

2

+

1 1

≤−

1, 2,

= 1,

1

+

2
2

−
=

2

1

+1

2

+

+1

+

= 1,1, ⋯ ,1
112

1

1

+

+

2

2

2

2

2
3

2

.

.

=1

Λ−

1

+

Λ

= 0,

∈ ℝ . Using the Lemma 5.3.1.2, we

=

can obtain that the following equation
1,

2, ⋯ ,

∈ ℝ , which implies that
−

Then

+

1

=1

−

1

2

+

1

1

and
2

=−

+ 2

2

2

Thus, we have
− + 1

1

if the condition

+

2

2

1

=−

,

4

2

+

+

2

42
+1

+1 +

2

1

2

2

1

2 −

+

1

+ 2

1

−

2

=1

+1

+

,

1

+
+

+1

2

22

− + 1 1 + 2 → −∞,
− + 1 1 + 2 ≤ − ≤ −1,
≤
− + 1 + 2 2 → −∞,
− + 1 + 2 2 ≤ − ≤ −1,
=

=

has a solution

,

2

2
3

+
1

−

2

+ 1

+1

2

+

1

2

2

1

+
2

2

=1

2
2

−

Λ
= −Π,

≤− + 1
2

1

−
1

2
1

−

−

1

=1

1 , 2,

where
=

=−

−

> 0 holds, where
113

2

+

1

+

1

2

2

2

,

.

1 → +∞,
+
1 →0 ,

2 → +∞,
+
2 →0 .

=

1

−
−
−

2
1

+

2

2

,

1

×

+

2

× ,
=

2
1

Therefore, we can take

where

−

,

1

42

2
2

−

2

4

2

2

+

1

+

+1

2

1

2
3

+

+

,

2

−
1

+

+1

2

1

+

1

> 0 sufficiently small such that for any

×

,

1

1, 2,

,

2

2

1, 2,

,

∈

≤ −1,

. Hence, the condition (iii) of Lemma 5.3.1.1 is

verified. It follows from Lemma 5.3.1.1 that system (5-1) admits a unique ergodic
stationary distribution for any given initial value

1

0 , 2 0 ,

0 ,

∈ ℝ3+ × .

0

5.4. Experimental simulations
In the previous subsections, we have investigated the effects of environmental
fluctuations (i.e. the white noise and regime switching) and TPP on the dynamics of
plankton ecosystem and verified the intensity of white noise, regime switching and
the liberation rate produced by TPP can lead to the extinction, persistence in the mean
and stationary distribution of plankton populations using modeling analysis. In order
to further study how the white noise, regime switching and TPP on affect the
dynamics mechanisms of the formation and evolution process of planktonic blooms,
we perform some experimental simulations for system (5-1) based on Milstein's
Higher Order Method (Higham, 2001) in this section. In the following experimental
simulations, unless otherwise specified, we always assume that the right-continuous
Markov chain

takes values on state space

= 1,2

parameters are listed in table 5.1, the initial condition is
and other parameters are chosen as control variables.

114

and the values of

1 0 , 2 0 ,

0

= 1,2,1

5.4.1. Impact of regime switching on the dynamics of system (5-1)
In order to study how the white noise, regime switching and TPP affect the
dynamics of system (5-1), we firstly consider that there is no regime switching in
system (5-1).
1.5,1.5 ,

Fixed

σ2 1 , σ 2 2

1 ,

2

= 0.2,0.3

= 1.3,1.4

,

σ1 1 , σ1 2

and choose

σ3 1 , σ3 2

=

= 0.7,0.8 . By direct

computation, the sufficient conditions of the extinction for both Subsystem 1 and 2

are easily to verify. Thus, all the species of the Subsystems 1 and 2 tend to extinct (see
Fig. 5.1). Furthermore, suppose that the generator
=

is

−

1

1
8

12

1
8

−

1

12

, by the irreducible property, we can easily to obtain that the

stationary distribution of
< 0,

< 0, and

of the Markov chain

is

= 0.4,0.6 , which satisfies the conditions of

< 0, then all the species of system (5-1) undergo extinction

(see Fig. 5.1). This result suggests that the regime switching cannot change the

extinction behavior of system (5-1) in this case, that is, system (5-1) is extinct if both
two Subsystems die out simultaneously.
Table 5.1 Parameter values
Parameters

Biological meaning

1

The intrinsic growth rate day−1
of NTP

1
2
2
1
2

Units

Environmental
carrying nos.ml−1
capacity of NTP
The intrinsic growth rate day−1
of TPP

Environmental
carrying nos.ml−1
capacity of TPP
The competitive effect of ml
TPP on NTP
nos.−1 day −1
The competitive effect of ml
NTP on TPP
nos.−1 day−1
The
attack
rate
zooplankton on NTP

of ml
nos.−1 day−1
115

Values
1 1

0.85

= 0.8, 1 2 =

1 = 1.8, 1 2 =

1

3.8
2 1 = 0.65, 2 2 =
0.8
2 1

3.5

1 1

2 1

= 2.5 ,

2 2

= 0.28,

2 2

=

= 0.01, 1 2
= 0.01

1 = 0.85,

= 0.08

2
= 0.95

1 = 0.08,

2
= 0.01
1 1 = 0.2, 1 2
= 0.1

The
attack
rate
of ml
zooplankton on TPP
nos.−1 day−1
The mutual interference nos. ml −1
between individuals of
NTP

1

The mutual interference nos. ml −1
between individuals of
TPP

2

2 1

= 0.5,

1 = 0.6,

2 2

= 0.8

2
= 0.75
1 = 0.28, 2 =
0.2

The conversion efficiency ml
of NTP into zooplankton
nos.−1 day−1
The
death
rate
of day−1
zooplankton

On the other hand, to illustrate the effect of regime switching on the dynamics of
system (5-1), we choose
3 1 , 3 2

1

1 , 1 2

= 0.1,0.05 ,

2 1 , 2 2

= 0.1,0.05 ,

= 0.1,1.8 and all other parameters remain unchanged. By a simple

computation, we can easily verify the conditions of
implying that both species
mean, while species

and

1

1 > 0,

2 > 0 and

3 < 0,

of system (5-1) are persistence in the

2

tends to extinction, as shown in Fig. 5.2. From Fig. 5.2, it

is clear that Subsystems 1 and 2 have different persistence-extinction behaviors and
system (5-1) can switch from one state to another state due to the regime shift, which
implies that regime switching can balance the density of the population under
different regimes. Significantly, it should be pointed out that the zooplankton of
system (5-1) is extinct due to the extinction of zooplankton in Subsystem 1. This
indicates that the regime switching may not change persistence-extinction behaviors
in this case. So the question may arise: what role does the regime switching play in
the dynamics of system (5-1)? Changing the generator
by controlling the value of
distribution of

is

=

0≤

1,

2

to

=

−

100
100 −

−

100

100
100 −
100

≤ 100 , it is easy to obtain that stationary

=

observe that with the increasing value of

100 −
100

,

100

. From Fig. 5.3 (a), one can

, the dynamical behaviors of species

change from persistence in the mean to extinction in different areas of
116

,

and

and Fig. 5.3 (b) depicts the dynamical behaviors of species
5.3 (a) for

= 1,

= 15 and

= 25, respectively. Taking

with respect to Fig.
= 1 for example, we

can see from the Fig. 5.4 that system (5-1) becomes persistence in the mean, whereas

other Subsystems remain unchanged and almost all of the sample trajectories of
system (5-1) are in that of Subsystem 2 due to

1 >

2 . This means that plankton

species can choose a better living environmental state to survive due to Markov chain.
For the case of

= 25, we can obtain that the zooplankton of system (5-1) becomes

extinction again (Figures here are not given due to the similarity to Fig. 5.4). Thus,

under the effect of the regime switching, we can obtain the result from Figs. 5.3 and
5.4 that even if one population undergoes extinction in one state, it will become
persistence in the mean in another state because of its staying longer in a better living
environmental state. Therefore, it can be asserted that the regime switching can
change the persistence-extinction behaviors of system (5-1) and the distribution of
Markov chain

is beneficial to the survival of plankton.

Fig. 5.1 (a), (b) and (c) denote the solution trajectories of
with

1 ,

σ 3 1 , σ3 2

2

= 0.2,0.3

and

= 0.7,0.8 , respectively.

σ1 1 , σ1 2

117

1

, 2

and

for system (5-1)

= 1.5,1.5 , σ2 1 , σ2 2

= 1.3,1.4 ,

Fig. 5.2 (a), (b) and (c) denote the solution trajectories of
with

1 ,

σ 3 1 , σ3 2

2

= 0.2,0.3 , σ1 1 , σ1 2

= 0.1,1.8 , respectively.

1

, 2

and

= 0.1,0.05 , σ2 1 , σ2 2

for system (5-1)
= 0.1, .05 ,

and

Fig. 5.3 The effect of regime switching on the stochastic behaviors of zooplankton species
for system (5-1). (a) denotes the stochastic behaviors between extinction and persistence in the
mean zooplankton species
, ,

for system (5-1) with different values of

and other parameters as in Fig. 5.2; (b) denotes the solution trajectories of zooplankton

species

with respect to Fig. 4.3 (a) for

= 1,

= 15 and

Fig. 5.4 (a), (b) and (c) denote the solution trajectories of
with

in different areas of

1 ,

σ 3 1 , σ3 2

2

= 0.2,0.3

and σ1 1 , σ1 2

= 0.1,1.8 , respectively.

118

1

, 2

= 25, respectively.

and

= 0.1,0.05 , σ2 1 , σ2 2

for system (5-1)
= 0.1, .05 ,

Fig. 5.5 (a), (b) and (c) denote the solution trajectories of system (5-1) and its corresponding
deterministic counterparts, and (d), (e) and (f) denote the density function diagrams of system (5-1)
with

1 ,

σ 3 1 , σ3 2

2

= 0.2,0.3 , σ1 1 , σ1 2

= 0.1,0.05 in regimes

= 0.1,0.05 ,

= 1 and

= 2, respectively.

Fig. 5.6 (a) denotes the movement of Markov chain in the state
the probability density function (PDF) of

( ).

119

σ 2 1 , σ2 2

= 0.1, .05 ,

= 1,2 over time. (b) denotes

Fig. 5.7 The effect of the toxin released rate

on the probability density function of system (5-1).

(a), (b) and (c) denote the histograms of probability density function for

of system (5-1) with

1 ,

2

= 0.5,0.55 , respectively.

1

, 2

and

5.4.2. Impact of white noise on the dynamics of system (5-1)
Does the white noise affect the survival of plankton populations when the system
has one alternative stable state? In the following, we see the impact of the white noise
on the persistence-extinction properties of system (5-1). Choosing
0.1,0.05 ,

2 1 , 2 2

= 0.1,0.05 ,

3 1 , 3 2

1 1 , 1

2

=

= 0.1,0.05 , it can been

seen from Fig. 5.5 that the NTP, TPP and zooplankton populations can coexist at a

relatively stable state when the intensities of white noise are comparatively small and
all other parameters remain unchanged as in Fig. 5.1. Actually, according to

> 0,

system (5-1) has a unique ergodic stationary distribution, which are consistent with

our experimental simulation. Moreover, it is clear to see from Fig. 5.4 (a)-(c) that the
white noise keeps the stochastic processes

1

, 2

and

moving up and

down randomly and the solution (the red lines) of system (5-1) fluctuates in a small
neighborhood of that (the blue lines) of its corresponding deterministic system. Thus,
we can obtain that white noise can affect the distribution of phytoplankton and
zooplankton populations. That is, white noise can significantly affect the dynamic
evolution mechanism of plankton populations. Significantly, we can observe from Fig .
5.4 (d)-(f) that the probability density functions of NTP, TPP and zooplankton have
two wave curves that are corresponding to the two states

= 1,2 of the Markov

switching, respectively. Comparing Figs 5.1, 5.2 and 5.5, it is obvious that a high
120

density of white noise can accelerate the extinction of the plankton populations and be
advantageous to the rapid disappearance of planktonic blooms, which may help us
control the density of plankton populations in real aquatic ecosystems. Therefore, it
can be asserted that the plankton systems incorporating white noise can better
simulate planktonic blooms than its corresponding deterministic counterparts. Fig. 5.6
describes that system (5-1) switches from one state
the law of Markov chain

over time.

= 1 to another state

= 2 by

5.4.3. Impact of TPP on the dynamics of system (5-1)
The influence of the toxin liberation rate produced by TPP under the effects of
the white noise and regime switching is also studied. Choosing
0.5,0.55 , a simple computation shows that

1 ,

2

=

> 0, which means that system (5-1)

has a unique stationary distribution and the probability density functions of NTP, TPP

and zooplankton populations have two wave curves due to the regime shift (see Fig.
5.7). Comparing Figs. 5-5 and 5-7, we can observe that the peak values of the
probability density functions for
than that in the earlier case

1

1 ,

, 2
2

and

of system (5-1) are higher

= 0.2,0.3 . In addition, we can also

observe that with the increasing value of , the mean values of
system (5-1) are getting larger, while that of

1

and

2

of

is becoming smaller. Therefore, it

is clear that the introduction of TPP can be benefic to the persistence in the mean of
three species through the termination of planktonic blooms and may be used as a
controlling agent to control planktoic blooms.
5.5. Conclusions
The occurrences of harmful phytoplankton blooms have been reported globally
with an increasing frequency in the past decades (Hallegraeff, 1993), and TPP are
among the contributors in these blooms (Hallegraeff, 1993; Philips et al., 2004;
Hallam and Luna, 1984). Moreover, plankton populations in real aquatic ecosystems
often fluctuate unpredictably because of environmental stochasticity, which plays an
121

important role in the ecosystems (Carpenter, 2011). In order to better understand the
effects of environmental fluctuations and TPP on the dynamics of plankton systems,
in

this

thesis,

we

propose

a

stochastic

phytoplankton-toxic

producing

phytoplankton-zooplankton system with Beddington-DeAngelis functional response,
which incorporates with white noise and regime switching, and study how these
factors affect the dynamics of system (5-1) analytically and numerically. We firstly
investigate the existence and uniqueness of global positive solutions, and then derive
some sufficient conditions for the extinction and persistence in the mean of system
(5-1). To prove the existence of the stationary distribution, the theory of Khasminskii
(Khasminskii, 1980) for periodic Markov process and a method based on constructing
a Lyapunov function are employed. Numerical analysis illustrates our theoretical
results and further indicates that the white noise, regime switching and TPP play an
important role in controlling planktonic blooms as follows:
Ø Regime switching plays an important role in the balance of the different survival
states of plankton populations. On one hand, Subsystem 1, Subsystem 2 and
system (5-1) have the same persistence-extinction behaviors. Actually, the regime
switching cannot change the persistence-extinction behaviors of these systems
(see Figs. 5.1 and 5.5), which means that system (5-1) still becomes persistence in
the mean (or extinction) if Subsystems 1 and 2 becomes persistence in the mean
(or extinction). On the other hand, the persistence-extinction behaviors of system
(5-1) rely heavily on that of Subsystem 1 and Subsystem 2 due to the role of
regime shifts. In the case of Subsystem 1 is persistence in the mean and
Subsystem 2 dies out, then system (5-1) will tend to extinction (see Fig. 5.2).
However, system (5-1) becomes persistence in the mean although one Subsystem
is extinct by controlling the value of

(see Fig. 5.4). Thus, the presence of

regime switching in the stochastic system can change the survival of plankton
populations and reduce the risk of extinction. Therefore, it can be asserted that
whether the regime switching is conducive to the survival of plankton populations
or not strongly depends on its staying longer in a 'good' or 'bad' environmental
122

state.
Ø White noise is adverse to the survival of plankton populations. As the Fig. 5.5
points out, if the white noise densities are relatively small satisfying the
conditions of Theorem 5.3.4.1, then system (5-1) has a unique ergodic stationary
distribution, which means the NTP, TPP and zooplankton can coexist at a stable
state for a long time. However, by enhancing the intensity of white noise on
zooplankton only or on all three species simultaneously, the zooplankton of
Subsystem 2 or all the three species of every system will go to extinction (see
Figs. 5.1 and 5.2). From Figs. 5.1, 5.2 and 5.5, by controlling the intensity of
white noise, the dynamic behaviors of system (5-1) can be significantly changed.
That is, high intensity of white noise is disadvantageous to the development of
plankton and increases the risk of extinction. This is ecologically meaningful as
the species deteriorates drastically because of high environmental fluctuations.
Thus, it is obliged to be stressed that the controlling of the white noise may be
acted as a possible biological way to control planktonic blooms.
Ø TPP can increase the survival chance of phytoplankton but reduce the biomass of
zooplankton. With the increasing value of the toxin liberation rate
condition of

ensuring the

> 0, system (5-1) has a unique stationary distribution (see Figs.

5.7), which describes the long time asymptotic behaviors of the system (5-1) from

a statistical viewpoint. Additionally, comparing Figs. 5.5 and 5.7, we can
conclude that the toxin liberation rate is conducive to the persistence in the mean
of phytoplankton but is adverse to the survival of zooplankton population.
Therefore, TPP plays an important role in controlling planktonic blooms.

123

Chapter 6 DYNAMICS OF A STOCHASTIC NON-AUTONOMOUS
PHYTOPLANKTON-ZOOPLANKTON SYSTEM INVOLVING
TOXIN-PRODUCING PHYTOPLANKTON AND IMPULSIVE
CONTROL
Abstract
This thesis describes an analytical and numerical investigation of a stochastic
non-autonomous phytoplankton-zooplankton system involving TPP and impulsive
perturbations. White noise, impulsive perturbations, and TPP were incorporated into
the model to stimulate natural aquatic ecological phenomena. The aim of this thesis
was to analyze how these factors affect the dynamics of the system. Mathematical
derivations were utilized to investigate some key threshold conditions that ensure the
existence and uniqueness of a global positive solution, population extinction, and
persistence in the mean. In particular, we determined if there is a positive periodic
solution for the system when the toxin liberation rate reaches a critical value. The
numerical results indicated that both white noise and the impulsive control parameter
can directly influence population extinction and persistence in the mean. Enhancing
the toxin liberation rate of TPP increases the possibility of phytoplankton survival but
reduce zooplankton biomass. These results improve our understanding of the
dynamics of complex of aquatic ecosystems in a fluctuating environment.
Keywords:

Stochastic

phytoplankton-zooplankton

system;

Toxin-producing

phytoplankton; White noise; Impulsive perturbations; Extinction; Periodic solution
6.1. Introduction
It is widely recognized that plankton populations play an important role in the
wealth of aquatic ecosystems and ultimately the planet itself (Huppert et al., 2002).
Phytoplankton not only generate organic compounds by absorbing carbon dioxide
124

dissolved from their surroundings, but also carry out photosynthesis, which may have
a significant effect on large-scale global processes such as the global carbon cycle,
climate change and ocean-atmosphere dynamics (Subhendu et al., 2015). However,
phytoplankton blooms, a natural phenomenon involving the rapid increase and almost
equally rapid decrease of a certain dominant phytoplankton species in aquatic
ecosystems, occur frequently and can persist under certain conditions, disrupting the
ecological balance of aquatic ecosystems and becoming detrimental to public health.
Harmful algal blooms (HABs), for example, have been widely reported and have
become a serious environmental problem worldwide (Anderson, 1997). To understand
the mechanisms of planktonic blooms and to regulate the ecological balance of
plankton ecosystems, it is crucial to perform a deeper analysis of the function of
aquatic ecosystems, and especially the dynamics of plankton populations.
Mathematical models can be a powerful tool for revealing plankton dynamics. Indeed,
since the pioneering work of Riley et al. (1949), mathematical models have been
developed to reveal the dynamic behaviors of plankton populations. In particular,
researches on phytoplankton-zooplankton systems have made great progress, in which
various biological factors have been taken into account, yielding important results in
recent years (Abhijit et al., 2021; Zhao et al., 2016; Han and Dai, 2019; Agnihotri and
Kaur, 2019; Zhao et al., 2018).
Toxin-producing phytoplankton (TPP) are a well-known group of phytoplankton
that have the ability to release toxic chemicals into aquatic environments during
HABs, which may inhibit predation pressure from zooplankton and other predator
populations in planktonic systems (Huppert et al., 2002; Falkowski, 1984; Colin and
Dam, 2003; Fulton and Paerl, 1987). For example, the results from experimental
observations (Colin and Dam, 2003) indicated that the toxic dinoflagellate
Alexandrium fundyense can negatively affect the growth rate of the copepod Acartia
hudsonica, and the toxic effects may have profound implications on the ability of
grazers to control HABs. Moreover, some experimental evidence has revealed that
TPP are one of the contributors to the formation of HABs (Hallegraeff, 1993; Philips
125

et al., 2004), which play a significant role in determining the dynamics of plankton
populations. For example, Roy et al. (2007) suggested that the presence of TPP might
be a potential reason for the generation of complex interactions between
phytoplankton and zooplankton populations, which could lead to the long-term
coexistence of plankton populations in a fluctuating biomass. By analyzing a
nutrient-plankton model, Jang and Allen (2015) concluded that TPP is conducive to
the stability of planktonic interaction and plays a key role in bloom termination.
Based on field observation and a model analysis, Chattopadhyay et al. (2002)
proposed the following nonlinear coupled ordinary differential equations:
=

where

=

and

respectively;

1−

(6-1)

−

are the population densities of TPP and zooplankton,

and

are the intrinsic growth rate and environmental carrying

capacity of TPP, respectively;
the rate of predation and
liberation rate;

−

−

and

denotes the natural death rate of zooplankton;

is the conversion rate of zooplankton;

is

is the TPP toxin

represent the predation response function and the

distribution of toxic substances, respectively. A previous study (Chattopadhyay et al.,
2002), investigated the existence and local stability of positive equilibria, and the
existence of the Hopf-bifurcation of the system, by considering different
combinations of functional response

and

. It was concluded that TPP may

act as a biological control for the termination of planktonic blooms. In recent years,
many studies have incorporated TPP into phytoplankton-zooplankton systems by
considering various factors, including time delay, plankton diffusion, infected
phytoplankton, and phytoplankton refuge, yielding interesting results (Wang et al.,
2014; Li et al., 2017; Jia et al., 2019; Agnihotri and Kaur, 2019).
It should be noted that most previous studies have focused on continuous
systems; however, in the real world, populations undergo inherent discontinuity
phenomena due to natural and anthropogenic factors, such as predation, planting, and
harvesting, which lead to rapid population decrease or increase over a fixed period of
126

time. Systems with such kinds of discontinuous changes relate to impulsive
differential equations, which have attracted research attention over the past decades,
since they provide a natural description of the observed evolutionary behavior of
certain real-world problems (Zavalishchin and Sesekin, 1997; Samoilenko and
Perestyuk, 1995). For example, many biological phenomena involving thresholds,
bursting rhythm models in medicine and biology, optimal control models in
economics, pharmacokinetics, and frequency modulation models can exhibit
impulsive effects, and considerable research has been conducted into incorporating
these impulsive effects into models (Tang and Chen, 2002; Shulgin et al., 1998;
Yenicerioglu, 2019; Jatav and Dhar, 2014; Liu et al., 2019). Furthermore, some
experimental observations shown that the toxic substances released by TPP is not a
constant but change over time (Graneli and Johansson, 2003; Johansson and Graneli,
1999), and seasonal periodicity can either terminate or initiate periodic outbreaks of
plankton populations (Mcgillicuddy et al., 2003; Phlips et al., 2004). Therefore, it is
meaningful to introduce the impulsive effects and periodicity into the underlying
system (6-1). As a result, system (6-1) can be extended into a non-autonomous
impulsive system as follows:
=

=

+

+

1−

Note that

= 1+
= 1+

−

and

–
,
.

−

+

,

≠

,

∈ ℕ,

=

,

∈ ℕ.

(6-2)

in system (6-1) adopt a linear and Holling II functional

response in system (6-2), respectively, which is presented in the case 4 of a previous
study (Chattopadhyay et al., 2002). Moreover, it is assumed that and system (6-2) is
subjected to short-term external influences at a fixed time
of real numbers with 0 = 0 < 1 < ⋯ <

, comprising a sequence

< ⋯, and lim →+∞

biological perspective, we impose the following restriction on
1+

> 0,

1+

> 0,

∈ ℕ.

= +∞. From a

and

:

This means that the impulsive perturbations become a descriptive process of species
127

introductions if

,

> 0, while harvesting if

,

< 0 (Liu and Wang, 2012).

On the other hand, planktonic systems in nature are inevitably influenced by
environmental fluctuations due to the stochasticity and unpredictability of their
surroundings. Regarding environmental fluctuations, for example, the availability of
necessary nutrients, temperature, artificial eutrophication, and many other physical
factors exist in real-world aquatic ecosystems, which may affect the growth rates of
plankton populations. For example, some experimental evidence has shown that the
growth rate of toxic Microcystis spp. and environmental biomass rely heavily on
temperature and nutrient concentration (Davis et al., 2009; Fujimoto et al., 1997). The
stochastic model analysis of Melbourne and Hastings (2008) demonstrated that the
extinction risk in natural populations depends strongly on a combination of factors
that contribute to stochasticity. Therefore, increasing attention has been paid to
studying the influence of environmental fluctuations on aquatic ecosystems (Abhijit et
al., 2021; Sarkar and Chattopadhayay, 2003; Ji et al., 2016; Zhao et al., 2017; Tapaswi
and Mukhopadhyay, 1999). Thus, it is reasonable and valuable to further incorporate
environmental fluctuations into system (6-2), which may provide a deeper insight of
the dynamics of phytoplankton-zooplankton systems in fluctuating environments.
Based on previous studies (Ji et al., 2016; Zhao et al., 2017), we assume that the
intrinsic growth rate of HAB phytoplankton and the death rate of zooplankton are
influenced by environmental factors, and we introduce white noise into the
deterministic system (6-2), expressed as follows:
=

−

+ 2
= 1+
+
= 1+

+

where

=

+ 1

and

1−
+

,

1
2

−

−

+

≠

,

∈ ℕ,

=

,

∈ ℕ.

(6-3)

( = 1,2) are mutually independent one-dimensional

standard Brownian motion and standard white noise, respectively, and
128

2

>

0

= 1,2 is the white noise intensity.

In recent years, many researchers have investigated the combined effects of

stochastic fluctuations and impulsive perturbations on population dynamics (Liu and
Wang, 2012, 2013; Zhang et al., 2017; Wu et al., 2015; Zuo and Jiang, 2016; Zhang et
al., 2015). For example, Liu and Wang (2012) studied a stochastic logistic system
with impulsive perturbations, and obtained some conditions for the extinction,
non-persistence in the mean, weak persistence, persistence in the mean and stochastic
persistence of the system. Zuo and Jiang (2016) discussed the periodic solutions for a
stochastic non-autonomous Holling-Tanner predator-prey system with impulses, and
obtained some conditions for positive periodic solution. All these studies have
stimulated further research into the dynamics of stochastic impulsive systems.
However, studies investigating the effect of environmental fluctuations on the survival
of plankton populations in an impulsive control environment with seasonal
disturbance are lacking. Therefore, in this study, we investigated the dynamics of a
stochastic non-autonomous phytoplankton-zooplankton system involving impulsive
perturbations and TPP. The aim of the study was to determine the influence of
impulsive perturbations, white noise, and TPP on the dynamics of this system (6-3).
Our research questions included: (i) How do environmental noise and the impulsive
control parameter affect the dynamics of plankton? (ii) What influences the peak of
the cyclic outbreaks of planktonic blooms in an impulsive perturbation and fluctuating
aquatic environment?
The rest of this paper is organized as follows: Section 6.2 presents the basic
assumptions and useful lemma firstly, and then we investigate the existence and
uniqueness of global positive solutions by establishing the equivalent system without
impulses, apply Ito's formula to obtain the sufficient conditions for the extinction and
persistence in the mean of the system, and the existence of positive

-periodic

solution are obtained by constructing a suitable stochastic Lyapunov function and
using the theory of Khasminskii (2012). A series of numerical simulations are carried
out to verify the theoretical analysis in section 6.3. In section 6.4, we summarize the
129

results and present our conclusions.
6.2. Dynamic analysis
Environmental stochasticity and impulsive perturbations are the key factors
affecting the real aquatic ecosystems, and the growth rates of phytoplankton and
zooplankton may be affected by the environmental fluctuations (Tapaswi and
Mukhopadhyay, 1999). Thus, it is interesting to study how these factors influence the
survival of plankton populations. In this section, therefore, we investigate mainly the
existence and uniqueness of global positive solutions, the extinction and persistence in
the mean, and the existence of positive periodic solutions of the system (6-3).
6.2.1. Preliminaries
Let ℕ, ℝ be the set of positive integers and real numbers, respectively, and

denote ℝ+ = 0, +∞ . Throughout this paper, unless otherwise indicated, we always
,ℱ , ℱ

assume that

≥0 ,

is a completed probability space with a filtration

ℱ satisfying the usual normal conditions (i.e. it is right continuous and increasing,

while ℱ0 contains all

-null sets). Moreover, it is assumed that a product equals

unity if the number of factors is zero and the following hypotheses are given:
(H1) the functions

,

,

,

,

,

,

, 1

, and

2

are

,

,

,

,

,

,

, 1

, and

2

are

all bounded positive-valued functions on ℝ+ = 0, +∞ ;
(H2) the functions

periodic with a common period

> 0;

(H3) there exists a positive integer
,

+

=

,

∈ ℕ and 0,

∩

,

such that

∈Ν =

1, 2, ⋯ ,

.

+

=

For convenience, we introduce the following notations. If

-periodic onℝ+, we define

integrable function, we define
1

=

= max ∈ℝ+ , and
1

= ∫0
2

− 1

2

130

+

1

+ ,

+

=

is a continuous

= min ∈ℝ+ , and if it is an

, > 0. Moreover,
=1 log 1 +

and
=

2

1

=1 log 1 +

2

+ 2

−

2

.

Now, we introduce the Ito‟s formula for general stochastic differential equations
and the definitions for the extinction and persistence in the mean of the system. We
define the

-dimensional stochastic equation as follows (Zuo and Jiang, 2016):
=

with the initial value
a

0

=

,

+

,

0 . Here,

=

,

1

2

,⋯ ,

is a

associated with Eq. (6-4)

,

=

1

is a

,

, 2 ,

×

+

=1

,

+

1

=

+

=1

,

+

1

2

≥0 ,

2

∈ ℂ2,1 ℝ × ℝ, ℝ , then

where

,

=

,

,

=

is

. We define the differential

=1

,

,

, =1

=1

,

,

Lemma 6.2.1.1 (Tang et al., 2015; Mao, 2008) Let
,

,

matrix function and

, =1

∈ ℂ2,1 ℝ × ℝ, ℝ , resulting in

Thus, the Ito‟s formula can be presented.

function

,⋯,

-dimensional standard Brownian motion

=
,

Let function

×

,ℱ , ℱ

defined on the probability space
operator

,

,

-dimensional vector function;

(6-4)

,

1

+
,

,
,

2

,⋯,

2

2

.

.

satisfies Eq. (6-4) and the

,

.
,

.

Definition 6.2.1.2 (Khasminskii, 2012) System (6-4) is said to be extinct if
lim sup →∞

= 0 a.s.

System (6-4) is said to be persistent in the mean if lim inf →∞
131

τ > 0 a.s.

This presents a lemma that describes criteria for the existence of the periodic
solution of stochastic differential equations without impulses (Li and Xu, 2013; Jiang
et al., 2008).
Lemma 6.2.1.3 Suppose that the coefficients of (6-4) are all continuous
function in

-periodic

and system (6-4) has a global solution, and further suppose that there

conditions:

>

× ℝ, ℝ) satisfying the following

′ such that ℒ

(i) there exists a constant
(ii) inf

1,2 (ℝ

,

exists a periodic function

( , ) → ∞ as

,

→ ∞.

≤ 0,

≥

′;

Then the system (6-4) has a periodic solution.
We then consider the following periodic Logistic system:
=

+ 1

with an initial value

+
1

0 =

1

=1 1 +

1 0

−

1

(6-5)

1

> 0. By the Lemma 2.2 (Mao and Yuan, 2006)

and the Lemma 2.1 (Liu et al., 2011), we can easily obtain that, if
can obtain that
lim →∞ sup

=

1

1

.

1 > 0, then one

(6-6)

6.2.2. Existence and uniqueness of global positive solutions
Before investigating the stochastic dynamics of system (6-3), we should first
guarantee the existence of global positive solutions. Therefore, we will discuss the
existence of global positive solutions in system (6-3) by constructing the equivalent
system without impulses in the following.
Actually, suppose that the conditions (H1)-(H3) hold, and consider the following
stochastic differential equation without impulses:

132

1
1

=

−

=

1
1

−

with the initial value

and

2

−
1

+ 1

+

1

1

1

1

+

1

1 0 , 1 0

=1 log 1 +

+

1

1

+

2

1

−

=1 log 1 +

=

0 ,

1

=

=1 1 +

2

=

=1 1 +

1

1

1

+

1

0≤ <

1+

,

0≤ <

1+

.

2

−

(6-7)

1

0 , where

−

The remainder of the proof follows that in the Theorem 2.2 of (Zuo and Jiang, 2016),
here, we omit it. Therefore, for any given initial value
(6-3) exists a unique solution

,

on

remain in ℝ2+ with probability one, namely
surely.

0 ,

0

∈ ℝ2+, the system

∈ ℝ+ and positive solution will

,

∈ ℝ2+ for all

> 0 almost

Remark 6.2.1.2 From the above equivalent in Theorem 6.2.2.1, we can conclude that
we only need to consider the asymptotic properties of system (6-7). Then, system (6-3)
has similar properties.
6.2.3. Extinction and persistence induced by impulsive control and white noise
From the perspective of population dynamics, it is necessary and important to
predict and control the development of populations. Therefore, we are now in position
to discuss the properties of extinction and persistence in the mean of system (6-3) in
this subsection, and the following modeling analysis indicates that impulsive control
and environmental fluctuations can significantly affect the dynamics behaviors of
plankton populations, such as the extinction, persistence in the mean, and the
existence of periodic solution of the system, indicating harvesting or artificial remove
133

harmful algae is a better biological way to control algal growth in a fluctuating
environment.
Actually, suppose that the conditions (H1)-(H3) hold. Applying Ito‟s formula to
the first equation of system (6-7), we have
log 1 t ≤

+

1

=1 log 1 +

2
1

−

2

Integrating both sides of the above inequality on the interval 0,
1

log 1 t − log 1 0

≤

+

1

1

−

∫0

+

1

yields

2
1

=1 log 1 +

2

1

+ ∫0

1

.

1

By the strong law of large numbers (Li et al., 2009), one can obtain that
1

lim →∞ ∫0

1

= 0, . .

1

According to the condition (H2), we have
2

1

→∞

undergoes extinction.

= ∫0

2

1 < 0, we can easily to get

By comparison principle, and if

which implies lim 1

2

1

− 1

lim →∞ ∫0

lim →∞ sup

log

1

.

1

≤

1 < 0,

=1 log 1 +

− ∫0

− 1

.

2

= 0 . . , that is, the harmful phytoplankton species

Similarly, it can be obtained that
1

Since

log

1 t

lim 1
→∞

2006), we have

− log

1 0

≤

1
1

+ ∫0

1

= 0 . ., and lim →∞ ∫0

134

1

2

2

1

1

1

+ ∫0

2

+ 2

2

2

2

.

= 0, . . (Mao and Yuan,

provided that

lim sup
→∞

log

2 < 0. In other words,

1

≤

lim 1
→∞

1 < 0

2 < 0,

and

=0 . .

0 ,

To sum up, for any given initial value
if

2 <0

∈ ℝ2+ , we can conclude that

0

( ) and zooplankton

both harmful phytoplankton

undergo extinction almost surely.

( )

Now, we analyze the case of toxic phytoplankton which is persistence in the
mean, while the zooplankton undergoes rapid extinction by artificial activities in a
fluctuating environment.
It is obvious to find that the conditions (H1)-(H3) is satisfied. Note that
1

≤

+

1

+ 1

1

=1 log 1 +

1

1

−

.

1

1

1

By the comparison principle of stochastic differential equation and (6-5), we have
1

lim sup

1

τ ≤

0

2

+ ∫0

→∞

1

. .

(6-8)

Taking Ito‟s formula to the second equation of system (6-7), we one obtain that
1

log 1 − log

≤

1

1

Together with (6-8) and limt→∞ ∫0
lim sup
→∞

provided that the condition
which means the species
value

0 ,

0

∈ ℝ2+ , if

2

log

2 +

1

1 >0

1
2

1

≤
1

1

2 +

1

1

+ ∫0

2

2

= 0, . . we have
1

1

<0

< 0 holds. That is, lim
→∞

1

.

= 0 . .,

dies out. Therefore, for any given initial
and

will tend to extinction almost surely.

2 +

1

1

< 0, then the species

( )

In the previous discussion, we have obtained that the cases of population
extinction. Then a certain question may arise and what people usually are interested in
135

is how does all the species of the system coexist in real life under some suitable
conditions? We now give a another discussion.
Considering the conditions (H1)-(H3) is satisfied. Applying Ito‟s formula to the
first equation of system (6-7) and (6-5), we have
1

log 1 t − log

1

0

=

− ∫0

log Φ t − log Φ 0

=

Thus, we have

0≥

1

1

1

1

−

1

1

1

= ∫0

1

≤

2

1

− ∫0

1

1

Φ

1

That is,
∫0 Φ

2

− ∫0

1

log 1 t − log Φ

1

1

− ∫0

1

+ ∫0

1

Φ

−

2

1

1

+ ∫0

1

· ∫0

1

1

,

1
1

1

.
.

1

.

(6-9)

Combining (6-9) and applying Ito‟s formula to the second equation of (6-7), we have
1 log

log

1 t

0

≥

2−

1

+ ∫0
−

1

2

+ ∫0
2

1

2

· ∫0

1

≥

Φ

1

2 −

+

1

+

2 2

− ∫0
1

.

∫0 Φ

By the definition of superior limit of (6-6), that is, ∀

for >

Thus,

0 . Thus, we have

1 log

log

1 t

0

≥

2−

+

∫0 Φ

>

1

−

−
136

1

1

− 1

> 0, there exists

that

1

Φ

1

−

0 > 0 such

1

2

1

· ∫0

1

+

2 2

.

log

1

≥

1

2−

+

+

1

−δ

2

−

∫0

+

1

2

2

.

According to the Lemma 4 (Li et al. (2009)) and for the arbitrariness of , if the
condition

2−

holds, we have

lim inf →∞

0 ,

for any given initial value
mean provided that

0

1 > 0 and ( 2 −

2−

≥

1

+

1

2

>0

∈ ℝ2+, the system (6-3) is persistent in the
)

+

1 > 0.

6.2.4. Periodic oscillations of plankton density
For the phytoplankton-zooplankton system, the existence of positive periodic
solution of the system implies that the populations are persistent. In this subsection,
we prove that the system admits a positive

-periodic solution by constructing a

suitable Lyapunov function and using the theory of Khasminskii (Khasminskii, 2012).
It is sufficient to show that a periodic solution exists for the equivalent system
(6-7) without impulses. According to the previous discussion, we have obtained that,
for any given value

0 ,

Construct a ℂ2 -function
, 1,

where

1

0

=

≜

∈ ℝ2+, the system (6-3) exists a unique solution.
−

1

2
1 : ℝ+ → ℝ

, 1,

ln 1 −

, 1,

2

1

+

> 0 will be determined later and
′

1

and
′2

=

2

− 1

2

=2

Obviously, we can verify that

2

− 1

−
−

2
2

2

ln

1 1+

2

−

− 2

defined by

1+

2

, 1,

2 1

2

2

+

2
2

= 1,2 are both

137

+

= 1,2 satisfy
2

−

1

1 ,

− 2

−

2

+

1

−

2
2

2

.

−

,

-periodic functions and

′2

> 0 such that

is a bounded function. Thus, there exists a constant

′2

≤

for all ≥ 0. To proof the condition (ii) of the Lemma 6.2.1.3, we only

need to prove that

as

inf , 1 , 1 ∈ 0,+∞ × ℝ2+\

→ ∞, where

=

1

,

1

×

,

, 1,

→∞

1

, which is clearly established since all the
, 1,

coefficients of the quadratic term in

are positive, and thus, we have

1

proved that the condition (ii) of the Lemma 6.2.1.3 is satisfied.
Next, we will verify that the condition (i) of Lemma 6.2.1.3 is satisfied.
Applying Ito‟s formula to
that

1

, 1,

1

≤

, 1,

1

−

+

+
−

+

and

≤

1

=1

=1

2
2

− 1+

2

, 1,

log 1 +

log 1 +
−

1

+
+

1

and

1

−

1

2

138

1 1

1 2
2 1

1 2
2 2

+

+

′

1

1

2

1 1 1
2

−

1

−

1 , respectively, one can obtain

−

,

1

=1

log 1 +

1

2

, 1,

1

2

=

′

2

·

2

+

1

+

≤

=1

Then, we can obtain

2

+

1 1+

2

· 1−

log 1 +

·

2 1

1

2
1

′2
2

1

1 1

1

1

+

+

−

+

+

1

3
1

1

2

2

−

+

2

log 1 +
2

2 2 2
1 1 +

1

+

′2
2

2

2

1 +

1

2

log 1 +

=1

=1

2 2 2
2 1

+

log 1 +

1 2
2 1

1 1

1

1

2

log 1 +

=1

2

′2

2

2

=1

1

log 1 +

1

1

=1

+

1

2

2

+
−

1 1

1

−

+

1

1

+

≤

1+

2

1

1

2 1

1 1

−

2

−

+

1 +

1

−

+

1 1 +

2

=1

+

log 1 +

+

1

=1

+

1 2
2 2

log 1 +

1 1

+

+

2

1

1

=1

3
1 +

139

log 1 +

+

2

2
2 1

2

1

.

2

2

2
1

2
1

2
1

, 1,

where

≤−

1

2

1 =

+
+

2 =

1

2

1
2

=1

1 =

and

1

2

2

+

=
2

1

max

2
1 , 1 ∈ℝ+

2, 2 12 −

2

1

2

1

+

2

log 1 +
1

2

3
1 −

2

2

2

2
2 1

2

1

Let

=

2
2 1 −

1 + 1 1 1+

+

1 1,

2

1

+

2

1

−

+

−

+

2

1

=1

log 1 +

=1 log 1 +

+

, 2=− 2

=1 log 1 +

1

1

,

1

2

2

,

.

3
1 −

2

2

2

2

2
2 1

. (6-10)

To confirm the condition (ii) of Lemma 6.2.1.3, we choose a sufficient small
,

> 0 such that

where

0<

≤

−

1 +

=

1

4 1

1, 2

2

+1 ≤

2

2
1
1 , 1 ∈ℝ+ 5
2

2

2

2

2, 2
1
2 3

min
5
1 2 +

max 2

−

2

2

2

2

1

2
2 1 −
2

Define the following bounded open set

2
2 1

140

2

2
.

2

,

2

1
2 2

2 3

1

1

,

(6-11)

,
2

(6-12)

1

3
1 +

3
5 1 1

=

and consider
1
ℰ =

1, 1

3
ℰ =

1, 1

1,

<

1

∈ ℝ2+ 0 <

ℰ =

1
ℰ ∪

In the following, let the conditions of
prove that
, 1,

Case 1 If

, 1,

1

, 1,
1

≤−
+

≤ −1 on ℝ+ ×

1

1
ℰ , then

∈ ℝ2+ ×

4

1

+ −
−

1

+ −

2

1

4
2

1

2

+

+

2
ℰ =

2
ℰ ∪

3
ℰ ∪

1

1 ≥

4
ℰ =

,

.
∈ ℝ2+ 0 <

1, 1

∈ ℝ2+

1, 1

4
ℰ.

ℰ , respectively.

1

2
2 1 −

1

1 <

1 > 0, 2 > 0, and

1 1 ≤

2

, <

,

1 ≤

∈ ℝ2+

Obviously, we can obtain that

1

1 <

2

2

−

2

1 ≤

2

2
1

Case 2 If

, 1,

, 1,

≤−

1

1

−

4

2

+ −
Similarly, we have

2
ℰ , then

∈ ℝ2+ ×
1

+ −
2

4
2

2

1

+

1

2

1

1

3
1 −

3
1

2
1

2
2

1

+

2

2
2 1

≤−

2

1

2
2 1 −

4

1

1 1 ≤

+

1

2

.

and we have

2

According to (6-10) and (6-11), we have
, 1,

1

,

> 0 hold, we need to

1+

1

1 ≥

1 ≤

2

1

≤ −1.
1 ≤

−

1

141

2

2

2

1+

3
1
1

1

3
1 −

2

2

2

2
2 1

.

and we have
1

2

2

2

3
1

2

2

2
2 1

.

, 1,

1

≤−

4

5
2

3

On the other hand, by Young inequality, we have
1 1 ≤

Thus,

Case 3 If

Case 4 If
, 1,

, 1,

, 1,

, 1,
1

1

≤−

, 1,

1

1

1

≤−

∈ ℝ2+ ×

≤ −1.

Thus, we obtain that

3
ℰ

≤−
4
ℰ

−

, 1,

−

+
1

5

2

≤ −1.
5
3

5 1

2

1

.

2

3
1 −

1

2

2

and by (6-12) and (6-13), then we have

≤−

∈ ℝ2+ ×
1+

1+

2

1

1+

1+

−

2

−

2

1

2
1

2

1
2

2 3

2

2
2 1.

3
1
1

≤ −1.

and from (6-12) and (6-13), then we have

2
1

2

2

2

2
2 1 ≤−

≤ −1, for all

condition (ii) of Lemma 6.2.1.3 is satisfied.

1 +

, 1,

1

−

2

2

2

2

2

ℰ,

namely the

0 ,

0

∈ ℝ2+ and

suppose the conditions (H1)-(H3) hold, and further if the conditions of
> 0 hold, then system (6-3) has a positive

2

∈ ℝ2+ ×

To sum up, we can obtain that, for given initial value

0, and

2

-periodic solution.

1 > 0, 2 >

6.3. Experimental simulations
In the previous section, we have studied the effects of environmental fluctuations,
impulsive perturbations and TPP in a periodic environment on the dynamics of system
(6-3) using modeling analysis, including the existence and uniqueness of global
positive solutions, the extinction and persistence in the mean and periodic solutions of
the system. To find the treatment strategy of planktonic blooms, and the effect of
artificial interventions on the growth of plankton populations, the white noise,
142

artificial interventions and TPP are chosen as control parameters and other parameters
= 0.75 + 0.01 sin

are listed as follows:
0.85 + 0.0001 sin

40

,

= 0.2 + 0.005 sin

= 0.001 + 0.95

4
40

,
,

40

,

= 0.65 + 0.0001 sin

40

= 0.12 + 0.06 sin

,
4

= 0.75 + 0.05 sin
,

40

= 0.05 + 0.008 sin

= 80, and the initial condition is

=

,
40

= 0.05 + 0.005 sin

,

40

0 , (0) = 0.5,0.5 .

In this section, based on Milstein's method mentioned (Higham, 2001) by
supplementing impulsive perturbations into it, some numerical simulations are carried
out to further investigate how the factors influence the stochastic dynamics of system
(6-3).
6.3.1. Impact of white noise on the dynamics of system (6-3)
In order to study how white noise, impulsive control, and TPP affect the
dynamics of system (6-3), we firstly consider that system (6-3) does not experience
the white noise (σ1 t = σ2 t = 0), that is, the stochastic system (6-3) becomes its

corresponding deterministic system (6-2). As mentioned above, the conditions
(H1)-(H3) can be easily verified. Thus, by simple calculations, we can obtain that
both phytoplankton and zooplankton populations in system (6-2) are persistent and
the system has a periodic solution provided that the value of impulsive control
parameter is small ( = 2), depicting the coexistence of plankton populations at a
stable state (see Fig. 6.1).

Fig. 6.1 The dynamics of system (6-2) with impulsive control parameter
143

= 2. (a)-(b): The

periodic solutions of phytoplankton population

and zooplankton population

of

deterministic system (6-2), respectively.

Fig. 6.2 The stochastic dynamics of system (6-3) with

1

(a)-(b): The periodic solutions of phytoplankton population

=

2

= 0.005 + 0.005 sin

40

.

and zooplankton population

of system (6-3), respectively.

Now we explore the impact of white noise on the persistence-extinction
properties of system (6-3). Fig. 6.2 shows that phytoplankton and zooplankton
populations can coexist in a relatively stable state when the white noise intensity is
comparatively weak

1

=

= 0.005 + 0.005 sin

2

40

. Actually, according to

dynamical analysis, it is not difficult to find that system (6-3) is persistence in the
mean and has a positive periodic solution with a period

= 80, which are consistent

with our numerical analysis. Comparing Figs. 6.1 and 6.2, it is evident that white
noise can affect the distribution of phytoplankton and zooplankton populations. That
is, white noise can significantly affect the dynamic evolution mechanism of plankton
populations. Furthermore, when

1

, the effect of white noise intensity on

phytoplankton population varies within some critical level, and the amplitude of
random oscillation for the phytoplankton population increases with an increasing
value of

1

, while that of zooplankton population decreases (figures here are not

given because of the similarity to Fig. 6.2). However, when we change the white noise
1

to

1

= 1.8 + 0.005 sin , satisfying the conditions of
40

1 < 0 and

2 < 0,

as a result, both phytoplankton and zooplankton population of system (6-3) undergo
144

rapid extinction, whereas the corresponding deterministic system still presents
obvious periodicity, which is a different phenomenon from that of its deterministic
system (see Fig. 6.3). Comparing Figs. 6.2 and 6.3, it can be seen that white noise has
a significant effect on the dynamics of system (6-3), and a large white noise intensity
can accelerate the extinction of plankton and is advantageous for the termination of
planktonic blooms. Moreover, it should be emphasized that white noise can not only
aggravate the emergence of random oscillation, but also change the periodicity of
plankton density. Therefore, it is worth pointing out that the results from Figs 6.2 and
6.3 can support the notion that planktonic systems incorporating white noise can
better simulate plankton blooms than the corresponding deterministic counterparts.
6.3.2. Impact of impulsive perturbations on the dynamics of system (6-3)
In order to study how the impulsive perturbations on the dynamical behaviors of
the system, we control some a parameter values satisfying the conditions of
and

2 +

1

1

1 >0

< 0, the zooplankton population of system (6-3) will tend towards

extinction rapidly, but the phytoplankton population generates periodic oscillation in
the sense of joint distribution after a period of time and the corresponding
deterministic system (6-2) shares similar properties, shown in Fig. 6.4. Moreover,
comparing Figs. 6.2 and 6.4, the dynamical behaviors of system (6-3) are significantly
altered by controlling the impulsive control parameter value of

= 40 . Thus, it

should be stressed that some critical parameters have a profound impact on the

persistence and extinction of plankton populations in fluctuating environments. These
results suggest that the impulsive control parameter value plays a restrictive role in
the survival of species ( ), which verifies that the system has potential to be applied
in real-world biological control situations.

145

Fig. 6.3 The stochastic dynamics behaviors of system (6-3) with

1

its corresponding deterministic counterparts with initial value

= 1.8 + 0.005 sin

40

0 , (0) = (0.5,0.5) ,

respectively. (a)-(b): Blue curves are the extinction of phytoplankton population
zooplankton population

and

and

of stochastic system (6-3), respectively, and red curves represent

the persistence of phytoplankton and zooplankton populations for its corresponding deterministic
counterparts, respectively.

Fig. 6.4 The stochastic dynamic behaviors of system (6-3) with
deterministic counterparts with initial value
the mean of phytoplankton population

= 40 and its corresponding

0 , (0) = (0.5,0.5). (a)-(b): the persistence in

and the extinction of zooplankton population
146

of stochastic system (6-3), respectively; (c)-(d): the persistence of phytoplankton population
and the extinction of zooplankton population

of its corresponding deterministic counterparts,

respectively.

6.3.3. Impact of TPP on the dynamics of system (6-3)
Finally, the influence of TPP in a periodic environment under impulsive
perturbations and environmental fluctuations on planktonic blooms was studied. Fig.
6.5 depicts how the TPP toxin liberation rate
dynamic behaviors of system (6-3), where 0 ≤

( ) in one period affects the stochastic

≤ 1, and the colorbars represent the

biomass of the corresponding species. Fig. 6.5 clearly shows that the population
densities of the system (6-3) change periodically and the density of species
increases with the value of

, while that of species

finally undergo extinction when

will rapidly decline and

reaches a critical value

particular, when we select three different values of

( )

≈ 0.2971 . In

and the other parameters

remain unchanged, as in Fig. 6.2, system (6-3) exists as periodic oscillation solutions,
and the phytoplankton and zooplankton populations can coexist in a stable state when
= 0.06,0.18. For these two cases, one can observe that periodic solutions of system

(6-3) can fluctuate in a small neighborhood of the periodic solutions of its
corresponding deterministic counterparts, whereas the overall trend still presents
obvious periodic oscillatory while considering the effects of impulsive perturbations
and environmental fluctuations. However, when we continue to increase the value
of

= 0.5 , the zooplankton population will rapidly decrease to extinction, as

shown in Fig. 6.6. Moreover, Fig. 6.6 shows that the maximum random amplitude of

the phytoplankton population will increase, while that of zooplankton will decrease,
and finally declining to zero when

increases. This means that an increase in the

toxic phytoplankton population leads to massive die-off of zooplankton, depicting an
unstable situation for some critical values of the system parameters. However, the
decrease of the toxin liberation rate can reduce the peak of the cyclic breakouts of
planktonic blooms, but increase the survival of zooplankton. Thus, TPP plays an
147

important role in controlling planktonic blooms.

Fig. 6.5 The effect of TPP on the stochastic dynamic behaviors of system (6-3) with
0.06 sin

40

=

+

. (a) The persistence in the mean of population ( ); (b) The persistence in the mean
( ).

and extinction of population

Fig. 6.6 The effect of TPP on the stochastic dynamic behaviors of system (6-3). Panels (a) and (b)
denote the sample path of phytoplankton

( ) and zooplankton

( ) of the stochastic system

(6-3) and the periodic solutions of its corresponding deterministic system (6-2) with different
respectively.
and

= 0.06 + 0.06 sin

= 0.5 + 0.06 sin

40

40

(green curves),

= 0.18 + 0.06 sin

40

(red curves)

(cyan curves), and their corresponding deterministic counterparts

(blue, black and magenta curves). Here,

= 2.

6.4. Conclusions
Understanding the dynamic mechanisms of phytoplankton-zooplankton systems
involving TPP can help guide the control of planktonic blooms, and impulsive
perturbations and environmental stochasticity can significantly affect real-world
148

,

ecosystems (Liu and Wang, 2012; Carpenter et al., 2011). Taking these factors into
account, we proposed a stochastic non-autonomous phytoplankton-zooplankton
system involving TPP and impulsive perturbations, and studied the dynamics of
system (6-3) analytically and numerically. By establishing an equivalent system
without impulses, we initially investigated the existence and uniqueness of global
positive solutions, and then obtained the sufficient conditions for the extinction and
persistence in the mean of the system by the comparison principle and It o's formula.
In order to prove the existence of positive periodic solutions, the theory of
Khasminskii and a method based on Lyapunov function construction were employed.
Numerical analysis verified our theoretical results and further demonstrated that the
dynamics of the system are intimately associated with white noise, impulsive
perturbations, and TPP.
(i) When the densities of white noise are relatively weak, fulfilling the conditions
of

( 2−

)

+

1 >0,

1 > 0, 2 > 0 ,

and

> 0 , system (6-3) is

persistence in the mean and has a positive periodic solution, namely, the species

phytoplankton and zooplankton can coexist in a stable state (see Fig. 6.2). According
to Figs. 6.1 and 6.2, white noise can affect the distribution of plankton populations.
Moreover, by enhancing the intensity of white noise, we observed that both
phytoplankton and zooplankton populations tend towards rapid extinction (see Fig.
6.3), which are consistent with the analysis of the conditions of

1 < 0 and

2 < 0.

Comparing Figs. 6.2 and 6.3, a high intensity of white noise is disadvantageous to the
development of plankton populations and increases the risk of population extinction,

which implies that adjusting the white noise intensity can help control plankton
population densities. Therefore, controlling white noise is a key factor in the
termination of planktonic blooms, which is consistent with the results obtained by the
experimental and field observations (Sarkar and Chattopadhayay, 2003a, 2003b).
(ii) As impulsive control parameter value of

increases over time, it can be

seen from Fig. 6.4 that the zooplankton population will undergo rapid extinction, but
the phytoplankton population can generate periodic oscillation in the sense of joint
distribution. The impulsive control parameter can change the stochastic dynamic
149

behaviors of the system significantly (see Figs. 6.2 and 6.4). Therefore, this suggests
that the impulsive parameter value

may play a restrictive role in zooplankton

survival and has potential application in planktonic bloom control.
(iii) With increasing toxin liberation rate, the conditions of
2 +

1

1

1 >0

and

< 0 become increasingly difficult to be met. Thus, the zooplankton

population will tend towards extinction, but the phytoplankton population can
generate periodic oscillation in the sense of joint distribution (see Fig. 6.5). This
means that an increase in the toxic phytoplankton population leads to massive die-off
of zooplankton, depicting an unstable situation for some critical values of the system
parameters. However, a low toxin liberation rate can reduce the peak of the cyclic
breakouts of planktonic blooms, but increase the survival changes of zooplankton (Fig.
6.6). Therefore, TPP may be used as a biological way to control planktonic blooms.
To sum up, we presents an investigation on the effects of environmental
fluctuations, impulsive perturbations and TPP on a phytoplankton-zooplankton system.
Studies on the dynamic mechanisms of phytoplankton-zooplankton interactions have
been a key topic in theoretical ecology, and can enhance our general understanding of
real-world aquatic ecosystems. Thus, it would be interesting to further incorporate
some real-world factors into our proposed model, such as the time-lag effect (Caperon,
1969), prey refuge (Mullin et al., 1975), and cell size (Hart and Bychek, 2011), and to
investigate how these factors affect the dynamics of the complex models. We
recommend this as the focus of future research.

150

Chapter 7 CONCLUSIONS AND RECOMMENDATIONS
7.1. Conclusions
Based on the knowledge and methods of plankton dynamics models, impulsive
control dynamics and stochastic dynamics, the dissertation adapts the techniques of
dynamics modeling, dynamics analysis and experimental experiments to study the
nonlinear dynamics of the growth and evolution of plankton, and discuss the effects of
some key factors such as population diffusion, impulsive control, environmental
fluctuations (including white noise and regime switching) and toxins produced by
TPP on the dynamics mechanisms of the formation and disappearance of the
planktonic blooms, and predict the dynamics evolutionary processes of plankton
growth. The main results are as follows:
(i) To discuss the prevention and control strategies on planktonic blooms, an
impulsive reaction-diffusion hybrid system was developed. On the one hand, the
dynamic analysis showed that impulsive control can significantly influence the
dynamics of the system, including the ultimate boundedness, extinction, permanence,
and the existence and uniqueness of positive periodic solution of the system. On the
other hand, some experimental simulations were preformed to reveal that impulsive
control can lead to the extinction and permanence of population directly. More
prexisely, the prey and intermediate predator populations can coexist at any time and
location of their inhabited domain, while the top predator population undergone
extinction when the impulsive control parameter exceeds some a critical value, which
can provide some key arguments to control population survival by means of some
reaction-diffusion impulsive hybrid systems in the real life. Additionally, a
heterogeneous environment can influence the spatial distribution of plankton density
and change the temporal-spatial oscillation of plankton distribution. All results are
expected to be helpful in the study of dynamic complex of ecosystems.
(ii) A stochastic phytoplankton-zooplankton system with toxic phytoplankton
was proposed and the effects of environmental stochasticity and TPP on the dynamics
151

mechanisms of the termination of planktonic blooms were discussed. The research
illustrated that white noise intensity can aggravate the stochastic oscillation of
plankton populations density and a high-level intensity of white noise can accelerate
the extinction of plankton and may be advantageous for the disappearance of harmful
phytoplankton, which imply that the white noise can help control the biomass of
plankton and provide a guide to the termination of planktonic blooms. Additionally,
some experimental simulations were carried out to further reveal that the increasing
toxin liberation rate can increase the survival chance of phytoplankton and reduce the
biomass of zooplankton, but the combined effects of those two liberation rates
produced by TPP on the changes in plankton are stronger than that of controlling any
one of the two TPP. All results suggest that both white noise and TPP

play an

important role in controlling planktonic blooms.
(iii)

We

developed

a

stochastic

phytoplankton-toxic

producing

phytoplankton-zooplankton under regime switching and investigated how the white
noise, regime switching and TPP affect the dynamics mechanisms of planktonic
blooms. The dynamical analysis indicated that both high level of white noise intensity
and toxin liberation rate released by TPP are disadvantageous to the development of
plankton and may increase the risk of plankton extinction. Also, a series of
experimental simulations were performed to verify the correctness of dynamical
analysis and further reveal the effects of the white noise, regime switching and TPP
on the dynamics mechanisms of the termination of planktonic blooms. On the one
hand, the numerical study revealed that the system can switch from one state to
another due to regime shift, and further indicated that the regime switching can
balance the different survival states of plankton and decrease the risk of plankton
extinction when the density of white noise are particularly weak. On the other hand,
an increase in the toxin liberation rate can increase the survival chance of the
phytoplankton but reduce the biomass of zooplankton, which implies that the presence
of toxic phytoplankton may have a positive effect on the termination of planktonic
blooms. These results may provide some insightful understanding on the dynamics of
152

phytoplankton-zooplankton systems in randomly disturbed aquatic environments.
(iv) A stochastic non-autonomous phytoplankton-zooplankton system involving
TPP and impulsive perturbations was studied, where the white noise, impulsive
perturbations and TPP are incorporated into the system to simulate the natural aquatic
ecological phenomena. The dynamical analysis revealed some key threshold
conditions that ensure the existence and uniqueness of a global positive solution,
plankton extinction and persistence in the mean. In particular, we determined if there
is a positive periodic solution for the system when the toxin liberation rate reaches a
critical value. The results from experimental simulations revealed that both white
noise and impulsive control parameter can directly influence the plankton extinction
and persistence in the mean. Significantly, enhancing the toxin liberation rate of TPP
increases the possibility of phytoplankton survival but reduces zooplankton biomass.
All these results can improve our understanding of the dynamics of complex of
aquatic ecosystems in a fluctuating environment.
All these results suggest that population diffusion, impulsive control,
environmental stochasticity and toxins produced by TPP can significantly affect the
dynamics of plankton growth, which play a key role in the dynamics mechanisms of
the formation and disappearance of planktonic blooms. Therefore, it is necessary and
important to consider the population diffusion, artificial interventions, environmental
uncertainty and toxic phytoplankton into plankton dynamics models when studying
and predicting the growth and evolution mechanisms of plankton, which is conducive
to the better understanding of the interactions and dynamics mechanisms of the
formation and disappearance of planktonic blooms, and provide theoretical basis for
the prediction and control strategies of algal blooms.
7.2. Limitations and Future research
In this dissertation, we have obtained some interesting results, but considering
the complexity of stochastic modeling and dynamical behaviors analysis of plankton
under the effect of environmental fluctuations, there are still some further works in the
153

future to discuss.
Ø Considering the discrete time-delay or distributed time-delay into the model
Since the time delay exists in the growth response of some phytoplankton
populations (Caperon, 1969) and is indispensable for zooplankton population to digest
phytoplankton population, thus, the introduction of time delay can significantly affect
the dynamical behaviors of the original plankton systems, which may be a major part
of the controlling in phytoplankton blooms. For example, the interaction process of
allelopathic effect. For the stochastic plankton model with discrete or distributed
delay, whether can we apply some related theories such as the ergodicity theory given
by Hasminskii, Markovian semi-group theory and Hasminskii's periodic solution
theory, to study the stationary distribution, and the uniqueness and existence of
periodic solution of the plankton systems with time delay? which will be an important
content in our future research.
Ø Establishing a stochastic plankton dynamics model with Levy noise
Although the model (5-1) has achieved some excellent results, it cannot describe
the sudden and devastating disasters that the plankton populations may suffer in the
nature, such as tsunamis, floods, hurricanes, etc. These disasters can significantly
destroy the habitat of plankton populations, which may lead to the rapid extinction of
plankton. To describe the impact of these huge, sudden and devastating disasters,
therefore, it seems to be more realistic to combine the environmental noise (Levy
noise) on the basis of the model in this dissertation and study how the environmental
fluctuations affects the dynamics mechanisms of planktonic blooms.
Ø The dynamics of a stochastic nutrient-phytoplankton-zooplankton model under
the influence of environmental fluctuations
In addition to recognizing the availability of nutrient (nitrogen, phosphorous, etc.)
and nutrient input are important factors affecting phytoplankton growth (Huppert et
al., 2002), the litter fall and decomposition processes of dead phytoplankton and other
litters are also dominant pathways to accomplish nutrients to the aquatic environments.
154

Thus, nutrient input act as a key role in the quick appearance of algal blooms or limits
their growth. Based on bottom-up and top-down control mechanisms, and considering
the nutrient recycle and nutrient input in the real aquatic ecosystems, a stochastic
nutrient-phytoplankton-zooplankton model with environmental fluctuations and TPP
can be proposed, trying to investigate the effects of nutrient input and TPP on the
dynamics mechanisms of planktonic blooms
modeling analysis and experimental simulations.

155

in a fluctuating environment using

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201

Appendix A

Lemma

A1

,

1

=

(Walter,
,

,⋯,

1997)
,

Suppose
,

and

satisfy the following conditions:
(i)

2

∈

where
−

(ii)

1, ⋯ ,

=

1, ⋯ ,

−

−

1

,⋯,

,

,

≥ 1,

is a bounded domain with smooth boundary;

× ,

, ,

, where

,

∈

×

≤

and class of

,

×

in

∈

1

vector-functions

∈

−

=

=

the

,

they are of class

,

that

, ,

, and vector-function

, ,

in

=

1

, ,

,⋯,

×

,

, ,

is continuously differentiable and quasi-monotonically increasing with respect to

(iii)
then

=

= 0,

,

≤

,

,

: ≥ 0, , = 1, ⋯ ,
∈

for

,

,

×

, ≠ ;

,

∈

×

× .

×

, continuously differentiable in

Lemma A2 (Smith, 1999) Suppose that
,

is continuous on 0,
2

continuous derivatives

and

on 0,

and

the following inequalities:
−

+

and

where
,

,

,

≥ 0,

is bounded on

is strictly positive on 0,

0,

0,

×

are positive numbers, a function

≥ 0,

,

≥ 0,

∈

if

0,

,

∈ 0,

×

, then

202

× , and

∈ 0,

× ,

,

≥ 0 on

× ,
,

,

0,

∈

is not identically zero.

×

,, with
satisfies

, and

Appendix B

In the following, we consider the following impulsive logistic differential
equations:
=

+

=

∈

where

−
+,

,

≠

and

are positive constants, strictly increasing sequence

( 1)
∈

condition (C4), and
+

≥

=

, are continuous positive-valued functions such that

∈

for all

= min =0,1,2,⋯,p

+1 −

max =0,1,2,⋯, max ∈ 0,

+,

∈

,

. From condition (C4), we deduce that

≥1 .

= max

, and

=

Lemma B Every solution
<

0,

, 0, 0 , 0 =

for all ≥ 0.

Proof For ∈ 0, 1 , we can obtain that
=

−

0 1−

=

Denote

Then we have the following useful result.

(B1) satisfies 0 <

satisfies

0

+

,

1−

, where

−

0 is

,

+1 −

=

given below.

0+ > 0 of the system

0 =

.

−

Then, it is easy to find that the solution is positive-valued and no larger than
max

0,

=

0 1−

−

≤ max

,

In particular, 0 <
0<

that 0 <
,

≤ ≤ 1 , then

on the interval. If

+1 ,

2

0
1

+

−

≤

if

≤

≤

( 2)

−

1, 2

, and similar to (B2), we can verify

, hence, 0 <

∈

= .

1−

+
1

=

1

1

1

≤ . In the same manner we can show that 0 <

= 1,2,3, ⋯. This completes the proof of the lemma.

203

≤

. Therefore,

<

if

∈

Appendix C

+

Let
:

( ) is the space of

→

, where

×

×

-times continuously differentiable functions

is a positive integer and 0 <

< 1, which have

derivatives satisfying the Holder condition with exponent
,

>

is a positive integer. For some
1∆ −

0
∆
2 −
0

0
0

1 =

=

, and

0
0
3∆ −

1,

> 0, let

-order
2,

3

∈

,

æ æ
ö
ö
c1 (t , x )u1 u 2
ç u1 ç a1 (t , x ) - b1 (t , x )u1 ÷÷ + eu1 ÷
ç
1 + r1 (t , x )u1 + r2 (t , x )u 2 + r1 (t , x )r2 (t , x )u1 u 2 ø
ç è
÷
ç
÷
æ
c'1 (t , x )u1 u 2
ç u 2 ç a 2 (t , x ) - b2 (t , x )u 2 +
÷
ç
ç
1 + r1 (t , x )u1 + r2 (t , x )u 2 + r1 (t , x )r2 (t , x )u1u 2 ÷
è
F (t , w) = ç
÷,
ö
c 2 (t , x )u 2 u 3
ç
÷
÷ + eu 2
ç
÷
1 + r3 (t , x )u 2 + r4 (t , x )u 3 + r3 (t , x )r4 (t , x )u 2 u 3 ÷ø
ç
÷
æ
ö
c 2 (t , x )u 2 u 3
ç
÷
÷÷ + eu 3
u 3 çç - a 3 (t , x ) +
ç
÷
1
+
r
t
,
x
u
+
r
t
,
x
u
+
r
t
,
x
r
t
,
x
u
u
(
)
(
)
(
)
(
)
3
2
4
3
3
4
2 3 ø
è
è
ø

and
1

=

2
3

,

,

,

,

,

,

,

1

,

,

1

,

,

1

,

Thus, we can rewrite (3-1)-(3-3) in the form
=

+

1

=

+

The operator
2,

1

+

,

, ≠

, ∈

,

2

,

,

2

,

,

2

.

=

1

:

satisfy sup t

-period in . Denote the spectrum of
1981) that the sectorial operator
fractional power

−

is

1

−

,

3

1

by σ

−

1

2
3

,

,
,

.

Ω ,

( 4)

Ω = 0 , where

that have two generalized
,

<∞ ,

is

≤ − . For any

> 0, the

1 , then it follows from (Henry,

satisfies Reσ
204

2,

∈

Ω is the Sobolev space of functions from
,

,

3

,

−

( 3)

has the domain

derivatives. Functions

,

3

1

−

=

Γ

1

−

where Γ is the gamma function. The operators
we define

operator, and
=

×

1

−
1

as

1

0
1

−1

, and

1

=

−1

1

−
1

−
1

,

are bounded and bijective, and

is the identity operator in X. For 0 ≤
=

with norm
×

.

1

· is the range of an

, where

, where,

≤ 1, we introduce the space

· is the norm in the space

=

From Theorem 9 of (Akhmet, et al., 2006), we have the following lemma.
Lemma C Assume that the functions
exists a positive-valued function

for some α ∈

1
2

+

sup

2

such that

2 −1− .

,

, 1 . Let

0 :

≤

≤

,

solution of (C3) and (C4), i.e.

Then the set

are continuously differentiable and there

,

0 ,
0

=
≤

10 ,
′

,

∈

20 ,

, > 0.

> 0 is relatively compact in

205

,
30

1+

∈

be a bounded

,

3

for 0 <

<