Dynamic Analysis Of A Heterogeneous Chemostat Model With Internal Storage And External Inhibitor

The chemostat is a classical mathematical model describing the competitive phenomena in open ecosystems and bio-reactors.We mainly apply the comparison principle,a class of nonlinear eigenvalue problems,monotonic dynamical system theory,monotone method and topological degree theory to study the following un-stirred chemostat model with internal storage and external inhibitor:with boundary conditions and initial conditions?(x,0)=?0(x)?0,?=S,u,U,v,V,p,0?x?1.The main contents are as follows.In the first chapter,we describe the known results of chemostat models,and propose the model to be studied and state the required preliminary knowledge.In the second chapter,we mainly study the dynamic behavior of the system,which is roughly divided into four parts.Firstly,we give the relevant conclusions of the single species model.The results show that there is a critical diffusion coefficient,when the diffusion coefficient is smaller than the critical diffusion coefficient,the species survives.Conversely,the species goes to extinct.Secondly,we study the long-term behavior of the competition model.By com-parison principle and the maximum principle,we obtain the well-posedness of the model and the global existence and uniqueness of the classical solution.By means of a class of nonlinear eigenvalue problems,we study the asymptotic instability of the trivial and semi-trivial solutions of the system.It is found that it can be determined by the signs of the principal eigenvalues of the corresponding nonlinear eigenvalue problems.Furthermore,the uniform persistence of the system and the existence of the positive steady state solutions are obtained by monotonic dynamical system theory.Thirdly,the existence of positive steady state solutions is studied by the topo-logical degree theory and monotonic method.It turns out that there exists at least one positive steady state solution when the principal eigenvalues of the correspond-ing nonlinear eigenvalue problems are both positive or negative.Finally,the theoretical conclusions obtained in this paper are illustrated by nu-merical simulation technology.Moreover,the influence of parameters on the dynam-ic behavior of the system is discussed.The results show that competitive exclusion,coexistence,extinction or periodic coexistence may occur.