Modeling study of nonlinear dynamics in the growth of aquatic phytoplankton

Phytoplankton bloom has become a growing global concern in recent years due to the excessive growth of algae, causing significant negative impacts on aquatic ecosystem and threatening human health. Growing evidence suggests that algal blooms are a consequence of the interplay of various hydrodynamical, chemical, and biological processes in aquatic systems. The complexity and nonlinearity of aquatic ecosystems, and the complexity of climatic and hydrographic events, make interpreting and predicting the blooms a very challenging task. In recent years, many different strategies have been adopted to manage algal blooms. Among them, mathematical models are advantageous because they can capture the ubiquitous stoichiometric constraints for modeling species growth and interaction. Thus, mathematical models have been widely used to investigate the dynamics of phytoplankton growth. In this study, five mathematical models were developed based on population dynamics, ecological dynamics, dynamic modeling, and probability theory. The models were investigated theoretically and numerically in terms of the theory of partial differential equations, stochastic differential equations, impulsive differential equations, and numerical simulation. The objective of this dissertation research was to gain insight into plankton dynamics and explore potential management strategies for excessive algal growth in aquatic systems. The main results are presented as follows: Firstly, a nutrient-plankton model incorporating delay and diffusion was developed. The theoretical analysis revealed that delay can trigger the nutrient-plankton oscillation via a Hopf bifurcation. Especially, the stability switches for positive equilibrium occur with increasing delay, which indicates that delay can generate and suppress the unstable coexistence of species population. Numerical results reveal that the stability switches for positive equilibrium indeed exist in the model, and the homogeneous multiple periodic solutions, as well as chaos, can occur with different values of delay, which implies that the model exhibits delay-induced complex dynamics. Secondly, a stochastic Leslie-Gower phytoplankton-zooplankton model with prey refuge was developed. The dynamical analysis revealed the sufficient conditions for the persistence and extinction of plankton populations. The numerical simulations showed that environmental noise and prey refuge play a crucial role in the survival of plankton species. The results further demonstrate that prey refuge can enhance the oscillation range of phytoplankton population, but the variance of zooplankton tends to increase and then decrease as prey refuge increased. Thirdly, considering seasonal disturbances in aquatic ecosystem, a stochastic nutrient-phytoplankton model with seasonal fluctuation was developed. The results indicate that periodic solutions exist under certain conditions, suggesting that plankton populations are associated with periodic oscillations. Furthermore, numerical results showed that seasonal fluctuation can trigger periodic blooms of phytoplankton and promote the coexistence of plankton species. Specifically, the results indicate that phytoplankton is more sensitive to nutrient than to seasonal fluctuation. Fourthly, a stochastic nutrient-plankton model with regime-switching was developed by considering regime-switching plankton mortality. The results showed that the model admits a stationary distribution under certain conditions. Then the numerical analysis revealed that the persistence and extinction of plankton populations are sensitive to variations in nutrient input. The numerical results also indicate that regime-switching plankton mortality contributes to the survival of plankton populations in the aquatic system. Finally, a stochastic nutrient-plankton model with impulsive control was developed. The theoretical analysis established sufficient conditions for the existence of periodic solutions. In addition, the numerical analysis showed that nutrient impulse plays an important role in preventing and controlling algal blooms, and appropriate environmental fluctuation can promote the coexistence of phytoplankton and zooplankton populations. However, excess intensity noise can result in the collapse of the entire ecosystem.