The Properties Of Bifurcation Of Three Species Predator - Prey Model

The interactions between predator and prey in nature play a significant role in the rich biodiversity of species in complex ecosystems, therefore, the qualitative and quantitative analysis of the predator-prey are practical and theoretically important and become an important field in population biology.This article consists four chapters and mainly discusses the properties of solu-tion to three kinds of predator-prey model.In chapter 1, the background of predator-prey model is introduced and some research results are given there.In chapter 2, a prey-predator model with Holling II functional response whose predator growth rate is affected by the top predator subject to homogeneous Dirich-let boundary condition is studiedFirstly, the priori estimate of the steady-state solution to the model is gained through comparison principle, the condition of the existence of positive solution is given by the Leray-Schauder fixed-point index theory; secondly, the mortality rate e is treated as the bifurcation parameter, the equilibrium solution branch emanating from the semi-trivial solution (u*,0) is discussed by bifurcation theory, and the sufficient condition to the stability of local bifurcation solution is provided through the linear perturbation of eigenvalue theory; finally, the local bifurcation solution is extended to the global bifurcation solution.In chapter 3, a ratio-dependent predator-prey model with linear harvesting rate subject to homogeneous Neumann boundary condition is studied When the space is homogeneous and nonhomogeneous, the condition which can raise Hopf bifurcation is analysised, the normal form method and center manifold theorem are used to prove the direction of Hopf bifurcation which bifurcates from positive constant equilibrium is supercritical, spatially homogeneous periodic solutions are asymptotically stable.In chapter 4, a modified Leslie-Gower prey-predator model with Dirichlet bound-ary condition is discussedThe Lyapunov-Schmidt procedure is used to study the existence and asymptotic stability of bifurcation solution from double eigenvalue.