Global Bifurcation of Coexistence State for a Competition Model in the Chemostat

We study the global bifurcation of coexistence solutions of a competition model in the unmixed chemostat with the Ivlev type response function.A priori estimates for coexistence solutions are established by the maximum principle.Necessary conditions for the existence of coexistence solutions are given by the eigenvalue theory and the upper and lower solution method.The local bifurcation branch of positive solutions is constructed by the local bifurcation theory,which can be extended to a global solution branch by using the global bifurcation theory.Moreover,the global solution branch connects the two semi-trivial solution branches of the model.From a biological point of view,two competitors can coexist when their maximal growth rates are within a certain region.