Hopf Bifurcation Of A Delayed Chemostat Model With General Monotone Response Functions

In this paper,a chemostat model with general monotone response functions and two discrete time delays was proposed to describe the dynamical behavior about predator-prey system.The article includes four chapters.In Chapter 1,we introduced the research background of this article,the main task and some important preliminaries.In Chapter 2,we studied the stability of equilibria and the existence of Hopf bifurcation.By analyzing the characteristic equation associated with the model,we obtained the conditions of the existence and stability of extinction equilibria and positive equilibrium.Choosing delays as bifurcation parameters,the existence of Hopf bifurcations was demonstrated.In Chapter 3,the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions were studied.By virtue of the P oincar(?) normal form theory and center manifold theorem,explicit formulas were derived to determine the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions.In Chapter 4,we performed some numerical simulations to illustrate the fidelity of theoretical results and explain the biological significance of model.