Stability Analysis Of Two Types Of Reaction Diffusion Models

It is absolutely necessary to study reaction-diffusion equations in the research process of many subjects,such as biology,chemistry and physics.In this article,we explore the stability of solutions for following two kinds of reaction-diffusion systems,one kind is a predator-prey model with refuges and alternative prey the other kind is a HIV epidemic model with diffusionChapter 1 states the biological context,research progress of two kinds of reaction-diffusion models,and gives the arrangement of content in this article.In chapter 2,we study the bifurcation of predator-prey model under the ho-mogenous second boundary conditions by the bifurcation theory and eigenvalue perturbation theorem.Firstly,treating the carring capacity of the prey k as a bifur-cation parameter,the local bifurcation is obtained in the positive constant solution,the specific form and the condition for the local stability of bifurcation solutions are given.Furthermore,bifurcation solutions can be extended to infinite.In chapter 3,we investigate the stability of constant steady-state solutions for a HIV epidemic model with diffusion.Sufficient conditions for the local asymptot-ical stability of the equilibrium solutions are given by linearizd theory and Routh-Hurwitz criterion.Using the method of upper and lower solutions and its associated monotone iterations,it shows that the infection-free equilibrium is globally asymp-totically stable when the contact infection rate is small enough.