Some Reaction-diffusion Equations In Spatially Heterogeneous Environments

In recent years,the application of reaction-diffusion models in biological mathematics has become more and more widespread,which has attracted the attention of scholars.By introducing a series of reaction-diffusion models,scholars have explored the effects of spatial variation,diffusion,advection,and competition on species survival,as well as attempted to study species evolution and corresponding population dynamics in spatial heterogeneous environments.After extensive and in-depth research,some new biological phenomena have been discovered and many interesting mathematical problems have been proposed.At the same time,many important research results have been established.In this dissertation,we mainly investigate the reaction-diffusion-advection model for two competing species in a spatially inhomogeneous but temporally constant environment.First,we expand the interspecific competition coefficients and the intrinsic growth rates,as well as discuss the stability of semi-trivial steady states and the coexistence of the two competing species under different dispersal strategies.Next,based on the correlation analysis of spatial heterogeneity,random dispersal rates,advection rates,and interspecific competition coefficients,the global dynamics of reaction-diffusion-advection system is described in detail by means of principal eigenvalue theory.Moreover,some dynamic behaviors such as the stability of semi-trivial steady states are obtained by varying dispersal strategies.The principal contents of this paper are as follows:The first chapter briefly introduce the research background and significance and expound the research status of the reaction-diffusion equation models.In the second chapter,the research model is illustrated concretely,and the basic concepts and related theorems of weak competition condition,semi-trivial steady states and coexistence states are presented.Furthermore,some basic properties of principal eigenvalue are summarized and demonstrated.In chapter 3,we debate the stability of the two semi-trivial steady states and the coexistence states of the system with different dispersal strategies when the intrinsic growth rates of the two competing species are the same and the interspecific competition coefficients satisfy the weak competition condition.The global dynamics of reaction-diffusionadvection system under a class of dispersal strategies is clearly depicted by means ofprincipal eigenvalue theory.For suitable resource functions,we find that the appropriate random dispersal rates and advection rates can make the system has a stable coexistence state.In chapter 4,on the one hand,in the case that the intrinsic growth rates of the two competing species are the same and the interspecific competition coefficients satisfy the weak competition condition,we consider the coexistence of two species with a class of dispersal strategies.On the other hand,in the situation that the intrinsic growth rates of the two species are different and the interspecific competition coefficients are 1,the stability of semi-trivial steady states are discussed with different dispersal strategies.At last,we summarize the main research results of this thesis and put forward the prospect of further research.