Dynamics Of Diffusive Predator-prey Models

Dynamical analysis of biological population models has been an important branch of mathematical ecology.More and more researchers have paid attention to the effects of the diffusion and spatial heterogeneity on the dynamics of biological population models due to the existence and important effects of diffusion and spatial heterogeneity.Based on the background of biological models,in this dissertation we investigate several classes of reaction diffusion systems in detail by means of Lyapunov-Schmidt reduction,center manifold reduction,norm form theory,local bifurcation theory,maximum principle,comparison principle,and so on.The main contents of this dissertation is organized as follows:Firstly,we present an analysis of the dynamics of a generalized reaction crossdiffusion system under Neumann boundary conditions.Existence and multiplicity of spatially nonhomogeneous/homogeneous steady-state solutions are investigated by means of Lyapunov-Schmidt reduction.Moreover,the linear stability and Hopf bifurcations of homogeneous steady-state solutions are described in detail.In particular,the Hopf bifurcation direction and the stability of bifurcating timeperiodic solutions are determined by using center manifold reduction and normal form theory.Finally,some of main results are illustrated by an application to a predator-prey model with Allee effect and one-dimensional spatial domain ? =(0,l?).Secondly,we investigate a diffusive predator-prey system with ratio-dependent predator influence and Neumann boundary conditions.Existence,nonexistence,and boundedness of positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation.Hopf and steady-state bifurcation analyses are carried out in detail.These results provide theoretical evidences to the complex spatio-temporal dynamics found by numerical simulation.Finally,we are concerned with a diffusive Leslie-Gower predator-prey model in heterogeneous environment.The global existence and boundedness of solutions are shown.By analyzing the sign of the principal eigenvalue corresponding to each semi-trivial solution,we obtain the linear stability and global stability of the semitrivial solution.The existence of positive steady state solution bifurcating from semi-trivial solution is obtained by using local bifurcation theory.The stability analysis of the positive steady state solution is investigated in detail.In addition,we explore the asymptotic profiles of the steady state solution for small and large diffusion rates.