Lyapunov Functionals For The Stability Of Delayed Chemostat Model

Chemostat is an experimental device for continuous cultivation of microorgan-isms. It is also a very important model in mathematical biology. We establish and study two types of chemostat microbial culture differential equations model with time delay. The Lyapunov function method plays a vital role in the study of global dynamics. Using a direct and effective method of Lyapunov functionals, we obtain the global stability of equilibrium of two types of chemostat models. This paper is divided into four chapters.In the first chapter, we introduce the theory of chemostat, its background and current situation of chemostat models with time delay. And we state some basic theorems which will be used in this paper.In the second chapter, we establish a delayed microbial growth model. We analyze the existence of the microbial extinction equilibrium and microbial survival equilibrium of the system. The model is studied by using the method of Lyapunov functional and the LaSalle-type theorem. The global asymptotical stabilities of the microbial extinction equilibrium and microbial survival equilibrium are obtained. The proofs in related literatures are simplified and the conclusions are improved to not only for the Michaelis-Menten functional response of microbial nutrient utiliza-tion in previous studies, but also for general functional responses. In the third chapter, we use Lyapunov functions and invariance principle to study the global asymptotic behavior of a Bacteria and Phage chemostat model with time delay. We proved that the global asymptotical stabilities of the microbial extinction equilibrium and microbial survival equilibrium. The proofs in related literatures are simplified and the conclusions are improved to not only for the Monod functional response of microbial nutrient utilization in previous studies, but also for general functional responses.In the last chapter, we make a brief review of the above conclusions, and present the biological and practical meaning of these models. And we discuss some shortcomings of this paper and point out some questions and future work.