Hopf Bifurcation Of A Chemostat Model With Two Discrete Delays And Complementary Nutrients

In this paper, a detailed analysis is made on the model of the prey-predator system with two discrete time delays, some conclusions are drawn from the relevant mathematical theories.This paper is divided into four chapters.The first chapter is introduction. We introduces the research background, the main research contents and preliminary knowledge.In the second chapter, the equilibrium point and the Hopf bifurcation of the model with two discrete time delays and complementary nutrients is studied. Firstly, by constructing appropriate Liapunov functions, we make the four dimensional, two nutrients model reduced to two dimensional. At the same time, the model of this paper is obtained by considering the dependence of the microbial culture on the complementary nutrients. Next, when the model has two discrete delays, we devided into several cases to discuss the stability of the equilibrium point of the system and the existence of Hopf bifurcation.In the third chapter, we mainly study the direction of Hopf bifurcation and the stability of periodic solution. Using the P oincar?e normal form theory and the center manifold theorem,we derive some mathematical expression of deciding the direction of Hopf bifurcation and the stability of bifurcating periodic solutions.In the last chapter, some numerical simulations are given to verify the correctness of the conclusions obtained in this paper.