Bifurcation Analysis Of Diffusive Prey-predator System With Holling-Ⅲ Type Functional Response

This paper considers a predator-prey reaction-diffusion system with Holling-Ⅲ type functional response function and the homogeneous Neumann boundary condition is satisfied:(?)By analyzing the corresponding characteristic problems in detail,the stability of the normal equilibrium solution of the system and the existence of Hopf bifurcation are studied.Using the stability theory and central pop theorem of the reaction diffusion equation,the display formula for determining the properties of the Hopf bifurcating under spatial homogeneity and non-homogeneity is obtained.By estimating the nonnegative solution of the semilinear elliptic equation of the system a priori,the relevant theory of the steady-state bifurcation of the system is obtained.The research work in this paper is divided into the following four chapters:The first chapter briefly describes the research background,significance and current status of predator-prey reaction-diffusion system with Holling-Ⅲ functional response function,and points out the main research content and main results of this paper.The second chapter considers the ODE system corresponding to the predator-prey reaction-diffusion system with Holling-Ⅲ functional response function,and the stability of the positive equilibrium solution and the existence of limit rings of the model are obtained.The third chapter considers the Hopf bifurcation of the predator-prey reaction-diffusion system with Holling-Ⅲ type functional response function and the stability of the directional and branching periodic solutions of the Hopf bifurcation.By selecting λ as the branching parameter,the Hopf branching value of the model is found by using the Hopf branching principle.The stability and branching direction of the spatial homogeneous and spatial non-homogeneous periodic solutions of the reaction diffusion model are obtained by using the judgment gauge theory and the central manifold theorem.The forth chapter considers the steady-state branch of the predator-prey reactiondiffusion system with Holling-Ⅲ functional response function.By using priori estimation of the non-negative solution of the semilinear elliptic equation,the existence of steady-state branches of the system is obtained.