Bifurcation Analysis Of Two Phytoplankton Models

The bifurcation problem is an important direction in the research of dynamical systems,and there are bifurcation phenomena in many disciplines.Due to the important role of Hopf bifurcation in explaining natural phenomena,biological phenomena,and so on,the research of Hopf bifurcation has been receiving much attention.Population dynamics models are one of the important research topics in the field of biomathematics,and are widely used in ecological theory,especially in the field of species protection.Among them,the interaction between plankton is a typical interspecific interaction.Continuously improve and improve the dynamic properties of traditional phytoplankton models in order to better protect the ecological environment,understand and explain the dynamic behavior and evolution laws of marine ecosystems,and guide the rational utilization and protection of marine resources.This paper considers two types of phytoplankton models,and the main research contents are as follows:1.A class of phytoplankton models with time delays and diffusion terms are established.The characteristic equations of the system are obtained by linearizing the system at the coexistence equilibrium point,and the conditions for the coexistence equilibrium point to be stable and Turing to be unstable are obtained by analyzing the characteristic roots.Further,relevant conclusions are drawn that the system generates Hopf bifurcation at the coexistence equilibrium point.On this basis,we study the properties of Hopf bifurcation and give the corresponding Hopf bifurcation canonical form.The results show that the phytoplankton model with time delay and diffusion terms exhibits complex spatiotemporal dynamics,such as generating spatially homogeneous periodic solutions.Finally,we conduct a numerical simulation to verify the correctness of the theoretical results.2.A class of phytoplankton models with nonlocal competition terms is established.Sufficient conditions for the existence of coexistence equilibrium points in the system are given,and conditions for the stability of coexistence equilibrium points are obtained.Next,we discuss the conditions for generating Hopf bifurcation and prove the corresponding theorem.The canonical form of Hopf bifurcation is calculated using the method of central manifold and canonical form.Finally,we conduct numerical simulations experiment and drew graphs in various intervals where the parameters belong,obtaining complex dynamic behaviors,such as spatial imhomogeneous periodic solutions,spatial homogeneous periodic solutions,etc.,to demonstrate the accuracy of the theoretical results.