Qualitative Analysis Of Two Types Of Reaction-diffusion Models

At present,scholars always use mathematical techniques and methods to solve problems in engineering,computational science,physics and biological science.In-filtration and connection between disciplines are increasing dramatically,which is reflected in mathematical biology.Reaction-diffusion equation is a bridge between mathematic and biological science,the qualitative analysis of its solutions is one of the hot topics studied by scholars.This paper studies two kinds of reaction-diffusion models which have biological meanings.In this thesis,the main contents are as follows:In chapter 1,we discuss the biological background and present status of devel-opment of predator-prey models and chemostat models,giving some related research works and results.In chapter 2,we study the predator-prey model with Michaelis-Menten type prey harvesting and refuge under homogeneous Neumann boundary conditions:Firstly,the asymptotic stability of positive constant solutions is obtained by means of linear stability theory.Secondly,a prior estimate of positive solutions is given,and the non-existence of non-constant positive solutions is proved through the energy integral property and Poincare inequality.Furthermore,the Leray-Schauder degree theory is used to prove that the model has at least a non-constant positive solution.Next,taking ? and d2 as bifurcation parameters,the structure of solutions bifurcated from the positive constant solutions is researched by local bifurcation theory,and bifurcation solutions are extended from local to global.Finally,Hopf bifurcation that occurs in a positive solution EB =(uB,vB)is investigated by using Hopf bifurcation theory.In chapter 3,we study the stability of a chemostat model with diffusion under homogeneous Neumann boundary conditions:Firstly,the upper and lower bounds of positive equilibriums are obtained by us-ing Harnack inequality and the maximum principle.Secondly,the local asymptotic stability of the positive equilibriums is studied by applying linear stability theory and spectral analysis method.Finally,we talk about the globally asymptotically stability of positive equilibriums by constructing a Lyapunov function and using the invariant sets theory.