Coexistence State Analysis Of Two Types Of Chemostat Competition Models

By means of the maximum method, the super-sub solution theory, the bifurca-tion theory and the Lyapunov-Schmidt theory, two kinds of competition models in the chemostat are studied. One is a class of competition model between plasmid-bearing and plasmid-free organisms in the unstirred chemostat with metabolitesThe other one is a class of competition model with Ivlev functional response in the unstirred chemostatWe study the following problems of the steady-state system of the above chemo-stat, including the existence and stability of the semi-trivial solutions, the existence and stability of the coexistence solutions.The main contents are organized as follows:In chapter 1, we introduce the background and research results of competi-tion models in the chemostat, and some preliminaries which are very useful in the forthcoming chapters are given.In chapter 2, the existence of positive solutions of the steady-state system of a competition model between plasmid-bearing and plasmid-free organisms in the unstirred chemostat with metabolites is discussed. Firstly, some properties for steady-state solutions are gotten. Secondly, by using the local bifurcation theo-ry and regarding a as bifurcation parameter, the local bifurcation branch of positive steady-state solutions is constructed, and the trend of the global bifurcation solu-tions is analyzed. Finally, the local bifurcation branch can be extended to a global solution branch by using the global bifurcation theory, and the construction of the global bifurcation solutions is gotten.In chapter 3, the existence of the positive steady-state solutions of a competi-tion model with Ivlev functional response in the unstirred chemostat is discussed. Firstly, some estimates for steady-state solutions are established by the maximum principle. Secondly, the stability of semi-trivial solutions and trivial solution is dis-cussed. Moreover, by using the bifurcation theory and regarding b as bifurcation parameter, the local bifurcation branch of positive steady-state solutions is con-structed, which can be extended to a global solution branch. Finally, the existence and stability of bifurcation solutions that bifurcate from a double eigenvalue are studied by the Lyapunov-Schmidt theory.