The Study Of Dynamics On Three Classes Of Turbidostat Models With Delay

In this paper, we investigate three kinds of turbidostat models with delay, Monod,Tissiet and Beddington-De Angelis functional response respectively. The time delay of digestion is the bifurcation parameter. By using some related theorems and methods,we discuss the stability of equilibrium, the properties of Hopf bifurcation, such as the existence, direction, stability and period of the bifurcating periodic solutions, the global stability of boundary equilibrium and permanence. Also, we present some examples and numerical simulation to illustrate the main results. The whole paper has been composed of four chapters.In the first chapter, we introduce the application background and research status of delay differential equations and culture of microorganism as well as the main works that we have done in this paper.In the second chapter, we consider a single-species turbidostat model with delay and Monod functional response. Using the theory of delay differential equations,center manifold theorem, normal form method and related lemmas, we discuss the stability of positive equilibrium, the existence of Hopf bifurcation, the direction,stability and period of the bifurcating periodic solutions. Moreover, we give an example and numerical simulation for the main results.In the third chapter, we establish a two-species turbidostat model with delay and Beddington-De Angelis functional response. With the same theories and methods as the second chapter, we study the stability of coexistence equilibrium and the same properties of Hopf bifurcation. Finally, we offer an example and numerical simulation to illustrate the main results.In the fourth chapter, we discuss a single-species turbidostat model with delay and Tissiet functional response. Using the theory of delay differential equations,comparison principle, Lyapunov-La Salle invariance principle and some related knowledge of limit set, we discuss the local stability of equilibriums, the existence of Hopf bifurcation, the global asymptotic stability of boundary equilibrium and the necessary and sufficient condition of permanence. At last, we present an example and numerical simulation for the results.