Study On The Dynamical Behavior Of Some Reaction-diffusion Equations

In recent years,the development of biological mathematics is very rapid and it has become one of the more active research fields in applied mathematics.Among them,the reaction diffusion equation has been widely applied in mathematical biology,by introducing the corresponding reaction diffusion model to understand and study the species diffusion in spatially heterogeneous environment and the corresponding population dynamical problem is one of the important research direction.Scholars have focused on the influence of diffusion rate,advection and the interspecific competitive coefficients on population stability and system coexistence,and many important results have been obtained.In this paper,we consider the reaction-diffusion-advection model for two competing species in a spatially varying,but temporally constant environment.Both species are assumed to have the same population dynamics but different dispersal strategies.The two species disperse by random diffusion and advection along environmental gradients,where the advection coefficients are different.By analysing the interspecific competition coefficients,random diffusion rate and diffusion strategy,we study the stability of semitrivial steady state and system coexistence.On the basis of previous work,we expand the interspecific competition coefficients from 1 to the range of(0,1).By analysing the related variables,we use eigenvalue method to study the population dynamics of each species,and some conclusions are similar to that of the previous literatures.The main arrangements of this paper are as follows:In chapter one,the generation,development,purpose and significance of the research on the reaction-diffusion-advection model are introduced.In the second chapter,we mainly introduce some basic concepts and related theorems about principal eigenvalue,semi-trivial steady state and system coexistence.In the third chapter,we firstly summarizes some conclusions when the random diffusion and interspecific competition coefficient are 1.Secondly,based on the original model,we expand the coefficient of inter-species competition to the range of(0,1)and introduce some relevant theorems and their proofs.In chapter four,we introduce some relevant conclusions when the random diffusion rates are greater than zero and the interspecific competition coefficients are 1.Then we expand the interspecific competition coefficient from 1 to the rang of(0,1)and give the proof of the theorem in this chapter.At last,we summarize the results of this paper and discuss the future research in chapter five.