Qualitative Analysis For A Diffusive Leslie-Gower Predator-Prey Model With Harvesting

In this essay,using the theories of bifurcation,the dynamics of a predator-prey model for the following Leslie-Gower type predator-prey model are investigated.This paper is structured as follows:Firstly,we discuss the existence and stability of the non-negative equilibrium points for the local system and the existence?bifurcation direction and stability of the Hopf bifurcation bifurcating from the positive equilibrium point.Secondly,treating the intrinsic growth rate of the predator as the bifurcation parameter,we discuss the effect of diffusion on the stability of the positive equilib-rium point,and the existence?stability and bifurcation directions of the spatially homogeneous and heterogeneous periodic solutions in the reaction-diffusion system.Numerical simulations are presented to verify and illustrate the theoretical results.Thirdly,treating the diffusion coefficient d2 as the bifurcation parameter and us-ing the Crandall-Rabinowitz bifurcation theory,we study the existence?bifurcation directions and stability of the local bifurcation solution bifurcating from the positive equilibrium.In addition,we prove that the local bifurcation curves can be extended to the global branches.Finally,by using the Leray-Schauder degree theory,the existence and non-existence of non-constant positive steady states of the reaction-diffusion model are abtained.