Permanence For Dynamical Systems In Biology

The research of Biomathematics dated from the Lotka-Volterra age, and the pub-lication of monographs such as May's"theoretical ecology", Smith's "ecological model" promoted the rapid development of Biomathematics in 1970's. The research's range of Biomathematics has been expended and applied in many domains such as population dy-namics, epidemic dynamics, pharmacokinetics, chemical reaction model, microbiological culture and so on. The research of permanence for biological populations, microorganisms and chemical reactants is the important aspect of the biomathematics'research. Based on the theory of functional differential equations, ordinary differential equations and im-pulsive differential equations, we study the kinetic properties of the population dynamics, microbial culture models and chemical reaction models.The dissertation has five chapters and the main results of the dissertation are sum-marized as follows:The first two chapters introduce the biological backgrounds of the ratio-dependent population model with stage structure, microbial culture model, the chemical reaction model and the relevant theories and concepts of functional differential equations, impulsive differential equations, respectively.In Chapter 3. the ratio-dependent population models with stage structure are con-sidered. We discuss a delayed ratio-dependent prcdator-prcy with prey dispersal and stage structure for predator in Section 3.1. Using the theory of functional differential equations, we obtain sufficient conditions for permanence of the system and sufficient condition for adult predators of the system tending to extinct. By Gaines and Mawhin's coincidence degree theory, we obtain sufficient conditions that the system exists posi-tive periodic solutions. The theoretical results are verified by numerical simulations. In Section 3.2, we consider a delayed ratio-dependent Holling-â…¢predator-prey system with stage-structured and impulsive stocking on prey and continuous harvesting on predator. Using the theory of functional differential equations and impulsive differential equations, we obtain sufficient conditions of the global attractivity of predator-extinction periodic solution, while the sufficient conditions are also sufficient conditions of the global attrac-tivity of predator-extinction periodic solution of the system without impulsive input. We also obtain sufficient conditions for permanence of the system.In Chapter 4, we discuss the application of state control impulsive equations in mi- crobiological culture. We establish a chemostat model with Bcddington-DcAnglis uptake function and impulsive state feedback control. By the qualitative theory of ordinary differ-ential equations, we sufficient conditions of the global asymptotic stability of the system without impulsive input. Using the relevant theory of impulsive differential equations, we obtain sufficient conditions for existence and stability of order one periodic solutions of impulsive system, and also analyze the existence of order two periodic solutions. The numerical simulation further confirms our theoretical results.In Chapter 5, we consider applications of impulsive differential equations in chemical reaction models. Section 5.1 studies a trimolecular response model with impulsive input. By qualitative theory of ordinary differential equations and a simple analysis of the system without impulsive input, sufficient conditions for global asymptotic stability are obtained. By the theory of impulsive differential equations, we obtain conditions for permanence of the trimolecular response model with impulsive input. By numerical analysis, we demonstrate complex phenomena such as limit cycles, periodic solutions, and chaos. In Section 5.2, we consider a irreversible three molecular reaction model with impulsive effect. By qualitative theory of ordinary differential equations, we obtain sufficient conditions that the system without impulsive input exists a limit cycle by a simple analysis of the system. Using the theory of impulsive differential equations, we obtain sufficient conditions for asymptotic stability of system with impulse effect. By bifurcation theory of impulsive differential equations, we obtain sufficient conditions that the system with impulse effect exists a stable positive periodic solutions. We further confirm our results and also show that the system has rich dynamic behaviors by numerical simulations.