The Existence And Stability Of Bifurcation Solution Of A Simple Food Chain Model In The Chemostat

Chemostat is a kind of industrial reactor which may be consist of three connected vessels, it is not only used in the chemical reaction, but also used in microbial continuous culture, waste treatment, biology pharmacy and food processing etc. Using the chemostat continuous cultivation microorganism was already an important research method in a microbiology research, and an extremely important intermediary between the principle and application . Chemostat model has been widely applied to the study of the grow in different populations of microorganisms and their interactive law. It also has been applied to the forecast and management of the ecosystem in particular aquatic ecosystem, and control of the environmental pollution.In this article , in chapter one the historic background and current situation of Chemostat model are introduced. In chapter two the simple food chain model that the consuming rate parameter of prey to nutrient is a linear function and the consuming rate parameter of predator to prey is still for the constant is considered . The existence condition of periodic solutions is obtained by the bifurcation theorems from simple eigenvalues. The stability of periodic solutions is obtained by Crandall-Robinowitz stability theorems. In chapter three the simple food chain model that the yields are all linear functions is considered. The existence condition of periodic solutions is obtained by the bifurcation theorems from simple eigenvalues. In chapter four existenc of hopf bifurcation of the two-dimensional system has been proven with stable method. Finally, the prime work done in the paper is summarized and outlook is given.