Study On A Non - Uniform Chemostat Model With External Suppressor And Mass

By means of the fixed point index theory, bifurcation theory and perturbation technique, we deal with an unstirred chemostat model with an external inhibitor:with boundary conditionsand initial conditionsWe studied the existence and stability of coexistence solutions, the longtime be-havior of time-dependent solutions and the effects of an external inhibitor on the multiplicity and stability of coexistence solutions.The main contents of this paper are as follows:In chapter1, the biological background and the recent work on this model are described in detail, and some preliminaries which are very useful in the forthcoming chapters are given.In chapter2, the properties of positive steady-state solutions of this model are considered. First of all, some elementary properties of semi-trivial solutions are given. Moreover, with the help of the bifurcation theory, we obtained the existence and the global structure of positive steady-state solutions. Finally, the local stability of the coexistence is also given. These results imply that the continuum of nontrivial solution finally connects with the branch{(q;u,v.p):q=0}.In chapter3, the effects of an external inhibitor are investigated. First, sufficient conditions for coexistence of stead-state are provided by the standard fixed-point index theory. Second, the effects of the inhibitor are considered in detail by means of the degree theory and perturbation technique. It turns out that the parameter μ, which represents the effect of the inhibitor, is sufficiently large, this model has only a unique asymptotically stable coexistence solution when the maximal growth rateIn chapter4, we determine the longtime behavior of the system. Asymptotic behavior of nonnegative solutions is given with the help of the super-sub solution method and the stability theory. What’s more, several sufficient conditions of the uniform persistence are obtained. It is shown that when the parameter a is less this model has no coexistence solutions; when the parameter the coexistence of the parabolic systems is uniformly persistent. Finally, we present some numerical simulations to verify and complement our theoretical analysis.