Turing-Hopf Bifurcation Analysis Of Two Diffusive Predator-prey Models

In real world,many biological systems can be described by the reaction-diffusion equation.By analyzing the Turing-Hopf bifurcation of the reaction-diffusion equation,stable spatially inhomogeneous periodic solutions are obtained.The stable spatially inhomogeneous periodic solutions has periodic fluctuations in time with spatial inhomogeneity.So Turing-Hopf bifurcation is often used to explain biological phenomena,such as explained periodic outbreaks of pets with geographical inhomogeneity.Meanwhile,the study of the Turing-Hopf bifurcation not only explains natural phenomena,but also predicts future species changes in order to better protect biological systems.So Turing-Hopf bifurcation analysis of the reaction-diffusion equation is a kind of great practical significance.In this paper,the existence and stability of the positive steady states for two diffusive predator-prey models with homogeneous Neumann boundary and one-dimensional space are studied.By using a new two-parameter selection method,the existence of Hopf bifurcation and Turing bifurcation are considered.By analysing those bifurcation,the stable and unstable(including Turing unstable)regions of positive steady states and the existence of Turing-Hopf bifurcation are obtained.The normal form on the central manifold near Turing-Hopf bifurcation point of the two models is calculated.Numerical simulations are carried out,the diagram of constant equilibrium,nonconstant steady state,spatially homogeneous periodic solutions and spatially inhomogeneous periodic solutions is obtained,and verifys the theoretical analysis of this paper.