A Class Of Heterogeneous Chemostat Models With Internal Competition

This paper mainly uses the theoretical knowledge of the reaction-diffusion equation,including the principle of minimum value,upper and lower solution,monotonic dynamic system theory,and local and global bifurcation theory,to analyze the dynamic properties of a class of chemostat models.As most of the species in the chemostat model are not only inter-species competition,but also internal competition.In order to consider the effect of internal competition on the dynamic behavior of the chemostat model,we study the following model:(?)The main contents and structure of this thesis are organized as follows:In chapter 1,the biological background and present research status of chemostat models are introduced firstly.After that,the research object of this paper,namely a kind of chemostat model with internal competition,is proposed.Finally,some preliminary knowledge needed in this paper and the main content of this paper is given.In chapter 2,the long time behavior of the chemostat model with internal competition is studied.First,long time behavior of the chemostat model is analyzed,hence the existence and uniform boundedness of the global solution of the system is obtained.Second,the long time behavior of the solution of the single species model is studied and the sufficient conditions of the extinction of the species are studied by using the maximum principle,comparison principle,Sobolev embedding theorem,egularization theory and a class of nonlinear eigenvalue problems.The results show that the species will extinct gradually when the maximum growth rate of the species is small.Finally,the sufficient conditions of competitive coexistence of two species are analyzed by using comparison principle,upper and lower solutions,uniform persistence analysis,nonlinear eigenvalue problem and other methods.In chapter 3,the global bifurcation of the steady state solution of the equilibrium equation is studied.First,the related lemmas needed to prove the local bifurcation and global bifurcation are given,including the prior estimation of the solutions and the determination of the eigenvalues of the linearization operators.Second,the existence of local bifurcation solution is obtained by treating the growth rate b of the species u2 as the bifurcation parameter.Finally,the local branch is extended by global bifurcation,then the global structure of this system is obtained.