Study On Bifurcations In Plankton Reaction-Diffusion Model With Time Delays

The frequent occurrence of algal blooms in oceans or lakes destroys aquatic food chains and ecosystems,and interferes with human production and life,which has become a concern of scholars in many fields.Therefore,it is of great significance to study the bifurcation and dynamical behaviors of plankton ecosystem for understanding the interaction of plankton and exploring the formation mechanisms of algal blooms.In this paper,we mainly study the dynamical behaviors of two kinds of plankton reaction-diffusion models with time delay from the perspective of bifurcation.First of all,the bifurcation of a class of plankton reaction-diffusion model with time delay and toxin release is studied,and the effects of toxin release rate and time delay on the system dynamics are mainly considered.By analyzing the characteristic equation,the local stability of the positive equilibrium is studied,and the conditions for the occurrence of Hopf bifurcation are obtained.Through the discussion of monotonicity,the critical value of time delay for the occurrence of Hopf bifurcation is ordered,and it is proved that time delay can induce the occurrence of stability switches,which means that time delay has an important impact on the formation and termination of blooms.In the process of stability switching,double Hopf bifurcation also appear.Taking the toxin release rate and time delay as the bifurcation parameters,the normal form of double Hopf bifurcation is derived,and the corresponding bifurcation set is obtained,thus the qualitative classification of dynamics on the plane of two parameters is obtained.The numerical simulation shows the complex dynamical behaviors near the double Hopf bifurcation points,such as periodic solutions,quasi-periodic solutions,and even chaos.Secondly,we study the bifurcation of a class of nutrient-phytoplankton model with time delay,and focus on the joint effect of nutrient cycle delay and diffusion on the system dynamics.Firstly,the existence of positive equilibrium and Turing instability caused by diffusion are analyzed when the time delay is zero.Taking the diffusion coefficient as the bifurcation parameter,the conditions for the stability of positive equilibrium and Turing instability of the system without time delay are given.When the time delay is not zero,the effect of time delay on the stability of the positive equilibrium point of the system is studied.The results show that the strength of the recycling rate plays an important role in whether the delay can cause Hopf bifurcation:when the recycling rate is small,the delay will make the positive equilibrium unstable and cause Hopf bifurcation;but when the recycling rate is large,the delay will not cause Hopf bifurcation.On the two-parameter plane with diffusion coefficient and time delay,the conditions for the occurrence of Turing-Hopf bifurcation are obtained by finding the intersection of Turing bifurcation curves and Hopf bifurcation curves.Taking diffusion coefficient and time delay as bifurcation parameters,the normal form of Turing-Hopf bifurcation is derived,and the corresponding bifurcation set is obtained.The numerical simulation verifies the results we have obtained,and shows that there are two stable spatial inhomogeneous periodic solutions or two stable non-constant steady-state solutions coexisting in the system.