Dynamic Behavior Of Species Competition Model Under Rapid Diffusion

Reaction-diffusion equations have been an important part of ecological mathematics. Many mathematicians have studied this topic and they have got many interesting results. In this paper, we mainly study the effects of spatial heterogeneity and diffusion rate on population size. As an introduction, in the first chapter we introduce the background of reaction-diffusion equations and many results we have got in recent years, and we also introduce the main work and results of this paper. and some definitions and theorems we will use in the following chapters. In the second chapter, we mainly prove the monotonicity of discussed reaction-diffusion model in this paper and some definitions and theorems we will use in the following chapters. In the third chapter, we discuss in non-homogeneous space, when two competitive species diffusion rate tends to infinity, the dynamics of the two species. We will prove that when two species diffusion coefficients are large enough, the system has one globally asymptotically stable positive steady state solution.