Qualitative Analysis For A Simple Food Chain Model With Variable Yield In The Chemostat

Chemostat is a kind of industrial reactor; it is not only used in the chemical engineering, but also used in microbial continuous culture, waste treatment, biology pharmacy and food processing etc. Chemostat is a well controlled and well observed apparatus; the expected goals will be got by controlling some microorganisms' concentration or adjusting some parameters in the system, so it is very necessary to research Chemostat. Depending on mathematical methods to model, analyze, control, and optimize Chemostat system, which have very important significance in the designing and the application of Chemostat.When the mathematical model of continuous cultivation for microorganisms is discussed, the consuming rates for microorganisms are usually assumed to be constants. But it is not rational. The consuming rates for microorganisms should be concerned with the nutrient density. The greater the nutrient density is, the lower the consuming rate of prey, similarly, the greater the prey density is, the lower the consuming rate of predator. So the consuming rate is a non-decreasing function of the nutrient density.This thesis mainly investigates the simple food chain models with variable consuming rates in Chemostat. 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 parameters are all linear functions is considered. In chapter three, the simple food chain that the consuming rate parameter of prey to nutrient is a quadratic function and the consuming rate parameter of predator to prey is a linear function is considered. In chapter four, the simple food chain model that the yields are all quadratic functions is considered. By the qualitative theorem of ordinary differential equations the existence and the local stability of the equilibrium solutions are analyzed. The global stability of equilibrium points is studied by using a novel way of constructing a Lyapunov function.