A Class Of Reaction Diffusion Equations Of Coexistence State Analysis

Reaction-diffusion system has now been widely used in biological research, and it has a very important practical significance to analyze biological phenomena through mathematical models. In the past few decades, the classical Lotka-Volterra model has been studied extensively. This paper is based on the classic Lotka-Volterra model to study the coexistence state of following reaction-diffusion system by use of the nonlinear analysis and nonlinear PDE knowledge, especially the parabolic equations and the corresponding elliptic equations.It studied the existence, stability, uniqueness of positive steady-state solutions, and its global bifurcation. The mathematical theory include: upper and lower solution method, comparison principle, global bifurcation theory, stability theory, topological degree theory.This paper is divided into three chapters: The first chapter includs the foreword and prior knowledge. In chapter II, we study the existence of positive steady-state solutions of this model , it can be divided into three parts: In part I, using the maximum principle and upper and lower solution method, we give the estimate of solutions. In part II, sufficient conditions for the existance of positive steady-state solutions and its structure are obtained by using bifuration theory. In the last part ,we proved global structure of the coexistence solution and discussed the trend of solutions under different circumstances by using sobolev embedding theorem . In chapter III, by using of the perturbation theory, we discussed the stability of positive steady-state solutions.