The Coexistence Of Two Types Of Reaction Diffusion Equation

So far, the Ecological research has made great achievements. And by setting a mathematical modal to study the ecosystem has become the preferred. This also contributed to the development of biomathematics. Recently, the population ecology and virus infection dynamics system are applied very widely. Many mathematicians and ecologists have great attention to the researches which have become the topic of extensive research in the ecology. At the same time, the job has also achieved good results.In this paper, dynamic nature of two reaction-diffusion equation is studied. One is the stationary patterns for a class of the prey-predator model with cross-diffusion:One is HBV virus model under homogeneous Neumann boundary conditionBy the maximum principle and Harnack inequality, the prior estimate to the positive solutions of the model is given. By using the integral property, the non-existence of the non-constant positive solutions is considered. The Leray-Schauder degree theory is utilized for discussing the existence of the non-constant positive solutions. It is shown that the infection free equilibrium is locally stable by Hurwitz theorem under some conditions. And it is also shown that the infection free equilib-rium is globally asymptotically stable by construct Lyapunov function under some conditions. The contets in this paper are as follows:In section1, the stationary patterns for a class of the prey-predator model with cross-diffusion are studied. This chapter is divided into there parts. First, by the maximum principle and Harnack inequality, the prior estimate to the positive solu-tions of the model is given. Second, the non-existence of the non-constant positive solutions is considered by using the integral property. Third, Leray-Schauder degree theory is utilized for discussing the existence of the non-constant positive solutions.In section2, we discuss the stability of dynamics of amend basic model of virus infection model under homogeneous Neumann boundary condition. This model is the introduction of new response function on the basis of original virus model.(?) is used to describe the growth rate of cell. Because of the geographical location is different, the virus will spread at a fast rate. Then in the model we add diffusibility. This chapter is divided into there parts:First, this paper gets the conditions which the infection free equilibrium is locally stable by Hurwitz theorem. Second, the prior estimate to the positive solutions of the model is given by the maximum principle and Harnack inequality. Third, the conditions that the infection free equilibrium is globally asymptotically are given by constructing Lyapunov function.