Dynamics Of Two Reaction-Diffusion Plankton Models With Delay

The population exists in a constantly changing environment,where the interactions between different predators and prey populations are affected by many factors,such as population diffusion,interspecific competition,prey refuge,delay and environmental toxins and so on.Therefore,it is more practical to consider these factors in predator-prey models.Based on the general predator-prey model,two kinds of plankton models with diffusion and time delay are established in this paper,and the dynamical behaviors of the models are discussed.In Chapter 2,a phytoplankton-zooplankton model with phytoplankton refuge,state feedback controller,environmental toxins,delay and diffusion is developed.Firstly,the positivity and boundedness of the solutions of the model are proved,and the existence and stability of the positive equilibrium of the ODE model without control are analyzed.At the same time,the occurrence of Hopf bifurcation caused by environmental toxins is studied by using environmental toxins as a bifurcation parameter.Secondly,for the controlled diffusion model,the conditions of the Turing bifurcation caused by diffusion is obtained,and the effect of the controller on the Turing bifurcation is studied.Aiming at the controlled delay and diffusion model,the Hopf bifurcation caused by time delay and the influence of the controller on Hopf bifurcation are studied by taking the delay as a bifurcation parameter.Finally,numerical simulation is used to verify the correctness of the theoretical results.In Chapter 3,a phytoplankton-zooplankton-fish model with phytoplankton refuge,C-M functional response,delay and diffusion is established.Firstly,the positivity of the solutions of the model is proved,and the existence and stability of the equilibrium of the ODE model are analyzed.At the same time,the Hopf bifurcation caused by the refuge of phytoplankton is studied,and the influence of the refuge on the model stability is analyzed.Secondly,the existence of Hopf bifurcation caused by delay is studied with or without diffusion,respectively.The direction of Hopf bifurcation and the stability of periodic solutions of bifurcation are analyzed by using central manifold theorem and normal form theory.Finally,numerical simulation is used to verify the correctness of the theoretical results.The results show that the delay affects the stability of the model,and even causes bifurcations and chaos.Reasonable control measures can change the occurrence of bifurcation,which provides important reference value for the management of planktonic ecosystem.