Srudy On The Dynamic Behavior Of Some Reaction Diffusion Models With Coupling Terms

In recent years,the rapid development of biomathematics has aroused the interest of many scholars,and the application of reaction-diffusion model in biomathematics is increasingly widespread.By introducing the corresponding reaction-diffusion model,scholars understand and study the influence of spatial environment,diffusion coefficien-t,competition coefficient,species coupling on species evolution,and try to explore the population dynamics behavior of the reaction-diffusion system of two competing species coupling in the spatial heterogeneous environment.After in-depth research,many interest-ing mathematical problems have been put forward and some new biological phenomena have been found.At the same time,a lot of classical results have been achieved.In this paper,we mainly study the dynamics of some reaction-diffusion systems in which two competing species have coupling terms in the environment of constant time and spatial heterogeneity.We assume that the population dynamics of two species are the same,the intrinsic growth rate is a positive nonconstant function,and the competition saturation is 1.When the diffusion coefficient and the interspecific competition coefficient are changed,the stability of the trivial equilibrium solution,the semi-trivial equilibrium solution and the coexistence of two species are studied by using the principal eigenvalue method,and the development process and trend of two species competitors with coupling terms are obtained.The main content of this paper is as follows:In the first chapter,this paper mainly introduces the background and significance of reaction-diffusion model,and expounds the development process of reaction-diffusion equation.In the second chapter,firstly,the concept of weak competition is introduced.Sec-ondly,the concept and properties of the coupled reaction-diffusion equation model and related eigenvalues are described in detail.In the third chapter,we mainly use the principal eigenvalue method to study the sta-bility and global asymptotic stability of two semi-trivial equilibrium solutions when the intrinsic growth rate of two competing species is the same and the competition coefficient between species satisfies the weak competition condition.In the fourth chapter,we mainly studies two competing species in a spatially hetero-geneous environment.On the one hand,when the interspecific competition coefficient of two species satisfies the weak competition condition and the diffusion rate is large e-nough or small enough,the two semi-trivial equilibrium solutions are unstable,then the two species will coexist;on the other hand,when the diffusion rate of two species be-longs to the middle value for the appropriate interspecific competition coefficient,one of the semi-trivial equilibrium solutions is stable and globally asymptotically stable.In the fifth chapter,this paper summarizes the research results and expresses the prospect of future research.