Analysis Of Bifurcations And Steady State For Some Ecological Systems

In this paper,we mainly investigate bifurcations and stability of several kinds ecological systems by applying the center manifold reduction,Hopf bifurcation theorem,operator spectral theory,maximum principle,Fixed point index theory and so on.This dissertation is divided into three chapters,the main contents are as follows:In chapter 1,we briefly introduce the research background and significance of the problems and the main work of this paper.In chapter 2,we discuss the dynamical properties of a three-species predator-prey model with delay.By analyzing the distribution of eigenvalues,we obtain sufficient conditions for the occurrence of Hopf bifurcations at the positive equilibrium point of the system.And the direction and stability of Hopf bifurcation for functional differential equations.Then steady state bifurcation is studied.Finally,by using the numerical simulation method to prove the results.In chapter 3,the dynamic properties of a predator-prey model with Beddington-De Angelis functional response are studied.The stability of the equilibrium point is proved by analyzing the distribution of the root of the characteristic equation of the system.Finally,the existence of the non-constant positive steady state solution of the system is obtained by using the operator spectral theory,maximum principle,fixed point index theory and the analysis technique.