Dynamics Analysis Of A Class Of Reaction-Diffusion Predator-Prey System With Time Delay

In this thesis,a time-delay diffusion predator-prey model with Holling II-type functional function is studied.By analyzing the eigenvalue problem of the linearized system of the original system at the positive constant equilibrium,the local asymptotic stability and Hopf bifurcation for the positive equilibrium of the corresponding ordinary differential equations(ODEs)system and the positive constant equilibrium of the partial differential equations(PDEs)system are investigated.The main contents and structure of this thesis are as follows:Chapter 1 states the research background and current status of time-delay diffusion predator-prey models,and states the main contents of this paper.Chapter 2 considers the local asymptotic stability of the positive equilibrium and the existence of Hopf bifurcation for the ODE systems.Meanwhile,the theoretical results for the ODE systems are verified numerically by MATLAB software package.In Chapter 3,we first discuss the local asymptotic stability of the positive equilibrium solution of the diffusion system without the effect of time delay,and then the influence of the change of time delay on the stability of the positive equilibrium solution of the model is analyzed in detail,and it is found that the increase of time delay will lead to that the stable positive constant equilibrium becomes unstable though the emergence of Hopf bifurcation.Finally,the numerical simulation are given to check the obtained theoretical results.In Chapter 4,an explicit formula for determining the properties of Hopf bifurcation is given by using the normal form theory and central popular theorem of partial differential equations with delay.