Bifurcation Dynamics Of A Class Of Predator-prey Reaction-diffusion Model

Predator-prey model is an vital model of population dynamical models.Questions related to the existence and stability of its equilibrium points and periodic solutions are quite important in the study of population dynamics,which have long been the focus of ecologists and mathematicians.This paper studies the stability of equilibrium points and bifurcation phenomena of a class of predator-prey reaction-diffusion model by linear stability theory,multi-time scale analysis and numerical simulation,which mainly focuses on the effect of habitat complexity.The dynamical behaviers of the system are investigated,such as the existence and stability of equilibrium points,Turing instability,steady state bifurcation and Hopf bifurcation.Firstly,consider the model without reaction diffusion.Calculate all the equilibrium points of the model,and we get a zero equilibrium point,a semi-trivial equilibrium and a positive equilibrium point respectively.According to linear analysis method,it is proved that the zero equilibrium point is unstable,the semi-trivial equilibrium branches the positive equilibrium under some condition,and there is Hopf bifurcation near the positive equilibrium point under another condition.Further,the model is simulated by numerical simulation.Secondly,the reaction-diffusion model with Neumann boundary conditions in one-dimensional and two-dimensional closed and bounded spatial domain is studied.It turns out that positive equilibrium point is locally asymptotic stable and there is never Turing instability.Next,the existence condition of steady state bifurcation is given.By multi-time scale analysis,bifurcation properties are further researched and some conclusions of the existence and stability of space inhomogeneous solution are got.In addition,the numerical simulation shows that Turing instability does not occur.