Chemostat System. Differential Equations To Describe The Prey Chain Qualitative Analysis

The research of the continuous culture of microorganism is an interdisciplinary scientific branch combining mathematics and microbiology, describing the continuous culture of microbes by establishing mathematical models using differential equations. Its objective focuses on the study of the equilibrium between nutrients and microbes. It has been widely applied to the study of the increase in different populations of micro-organisms and their interactive law. In addition, it has also been applied to the prediction and management of the marine ecosystem, and the control of the environmental pollution.This thesis mainly investigates the single food chain Chemostat models, analyzing the stability of equilibrium points, the existence and uniqueness of periodic solutions as well as the Hopf bifurcation theorem. In chapter 1, we introduce the historic background and current situation of the Chemostat model. In chapter 2, we discuss the single food chain Chemostat models with constant consumption rate. We first assume that the increase rate is in direct proportion, and then we discuss the model with quadratic saturated increase rate. We prove that there exist threshold points λ₁ and λ₂, and investigate the stability ofequilibrium points and the existence and uniqueness of the periodic solutions. In light of the existence of the ω limit set, we usually turn a problem about 3-dimension space to one about a 2-dimension plane. In chapter 3, we assume that the consumption rates are δ₁(s) = A + Bs and δ₂(x) = C + Dx, and the response function is Monod form. Thus wecan simulate the natural environment more effectively. By the qualitative theorem of ordinary differential equations we analyze the stability of the equilibrium points and discuss the problem of the bifurcation on the plane. In chapter 4, Numerical simulations of solutions and the corresponding graphs are also given to test the results mentioned above.