Stability Analysis Of A Delayed Reaction-Diffusion Predator-Prey Model With Ratio-Dependent Functional Response

In this paper,a delayed reaction-diffusion predator-prey model with ratio-dependent functional response is considered.We study the stability of the positive constant steady state and existence of Hopf bifurcation.The formula determining the direction and stability of Hopf bifurcation is derived by center manifold theorem and normal form theory.In addition,we obtain the sufficient condition for the global asymptotic stability of the constant steady state by the upper-lower solution method.This paper is organized as follows:In chapter 1,we state the background and recent research of the delayed reactiondiffusion predator-prey model with ratio dependent functional response.In chapter 2,the stability of the positive constant steady state and existence of Hopf bifurcation are established by analyzing the characteristic equation of the linearized system.In chapter 3,we derive the formula for the direction and stability of Hopf bifurcation through calculating the normal form on center manifold.In chapter 4,the sufficient condition for the global asymptotic stability of the positive constant steady state is established by successively constructing upper and lower solutions and monotone iteration method.In chapter 5,numerical simulations are carried out by MATLAB to check the validity of the theoretical results.