Research Of Several Wuestions In Reaction-diffusion Equations

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 study the stabilities of several steady states, and we also discuss the effects of spatial heterogeneity and diffusion rate on single 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 well use in the following chapters. In the second chapter, we discuss the effects of spatial heterogeneity and diffusion rate on population size in single species disperse only by random diffusion model.In the third chapter, we discuss reaction-diffusion models that two species compete for the same resources but they have different dispersal strategies in a heterogenous environment. Well introduce the evolution of reaction-diffusion equations. Well also show that excessive strong biased movement along resource gradient is not beneficial for species.