Effect Of Diffusion On The Stability Of A Predator-prey System With Non-monotonic Functional Response Function

This thesis considered a two-species predator-prey reaction-diffusion system with simplified Holling type-? functional response and subject to the homogeneous Neu-mann boundary condition of the form(?)In the absence of the spatial diffusion,the local asymptotic stability,the instabil-ity and the existence of Hopf bifurcation of the positive equilibria of the corresponding local system were analyzed in detail by means of the basic theory for dynamical sys-tems as c is the bifurcation parameter.Meanwhile,the global asymptotic stability of equilibria and the existence of saddle-node bifurcation were studied briefly.As well,by using the linearized method and the Hopf bifurcation theorem for semi-linear par-tial differential equations,the stability and the existence of Hopf bifurcation of the positive constant steady state of the diffusive system were considered by means of the distribution of roots in the complex plane of the associated eigenvalue problem.Next,the explicit formulas determining the direction of spatially homogeneous Hopf bifur-cation and the stability of the bifurcating periodic solutions were established accord-ing to the normal form method and the center manifold theorem of reaction-diffusion equations.By comparison,it was found that the spatial diffusion has no effect on the stability of the positive constant steady state and the existence and the properties of spatially homogeneous Hopf bifurcation.This paper was arranged as follows:The first chapter introduced the research background,significance and current situation of the two-species predator-prey reaction-diffusion system with Holling type-? functional response and pointed out the main work of this article.The second chapter introduced some basic knowledge,definition and lemma.In the third chapter,the stability and instability of the equilibria and the existence and direction of Hopf bifurcation of the local system were analyzed in detail by means of the basic theory for dynamical systems.To verify the obtained theoretical predic-tions,some examples and numerical simulations were also included by applying the numerical methods.In the fourth chapter,the effect of the spatial diffusion on the stability of the pos-itive constant steady state and the direction of Hopf bifurcation and stability of bifur-cating periodic solution were considered.To verify the obtained conclusions,some examples and numerical simulations were also included.