Qualitative Analysis For The Chemostat Model With Variable Yield

Final state of biological mathematics model is the focus of the study, only the study of the model final state, can grasp the population evolved over time rule. People can according to the results of that, predict the final existence state of the population, thus make the relevant measures. Characterization of biological populations of final state, often with the global stability of equilibrium point, the problem of the existence of bifurcation, species survive continuously in several aspects. In the natural environment, because it has all kinds of competition between species, the problem of extinction or permanence is inevitable, and discusses the species permanence problems, especially how to make will be endangered species survive down, with profound realistic significance. The stability of the equilibrium and periodic solutions is the species survive reaction of thoughts, so has become the focus of scientific research workers.This paper first introduces study situation and the achievements of the Chemostat, based on this, mainly studies with varying consumption rate in a chemostat for bifurcation properties of food chain model; Mainly discusses when the consumption rate parameters are linear and nonlinear functions, the existence of Hopf bifurcation properties in the equilibrium point for the model, getting bifurcation existence and stability condition of the model; Then, studies solution structure of food chain model when promote the consumption rate parameters for the general nonlinear forms. This paper analyzes the existence, local stability and global stability of the equilibrium point of the model, the most important is to use a bifurcation existence theorem discussed the existence of hopf bifurcation and stability of periodic solution in the semi-trivial equilibrium point and the positive equilibrium point; Finally, analyzes the existence of hopf bifurcation in the positive equilibrium point of the chemostat predator-prey model (the consumption rate parameters for four function).