Study And Application Of Periodic Variable Population Systems

Impulsive dynamical system is possible one of the most youngly and most attracting fields in main mathematical branches such as differential equations, dynamical systems, control theories etc. The theory of impulsive differential equations is not only richer than corresponding theory of differential equations but also represents a more natural framework for mathematical modelling of many real world phenomena. Theoretically we use a combined approach of discrete dynamics, continuous dynamics and impulsive dynamics to globally investigate dynamics behavior. At same time, impulsive differential equation provides us with many valuable research subjects. Many real world phenomena(births and deaths of population are seasonal or discrete) and human action (periodic exploitation of human for renewable resources) do exhibit impulsive effects, impulsive differential equations provide a natural description description of model of discrete perturbations. In this paper, based on an impulsive differential equations' theory, we introduce nonautonomous population dynamical models. The various dynamical behavior of the population models are globally studied and we carefully analyze the complexity of the given systems. In this process, we use some mathematical softwares, Maple and Matlabel.1. In chapter 2, we analyze a kind of impulsive differential equations (IDE), which are with the impulsive effect â–³x =-px,â–³y = b. Some very useful theorems about the periodic solutions and stabilities have been given. We can find that a kind of periodically time-dependent IDE can be changed into this kind state-dependent IDE. As applications, we want to study the maximum sustainable yields of single population models with periodically impulsive constant harvesting. Furthermore, we want to use these results to study the order-1 periodic solutions and stabilities of a single population model model with stage structure and with the mature being impulsively proportionally harvested and the immature being impulsively added with constant.2. In chapter 3, we introduce an autonomous and a nonautonomous harvesting systems under price law's control , modeling the dynamics of harvesting and renewable resource developing behaviors. Using the Dulac function can prove the autonomous system's positive equilibrium is globally asymptotical stable.Then, sufficient criteria are established for the existence of positive periodic solutions for the nonautonomoussystem. The method to the existence problem is based on the coincidence degree and the continuation theorem. Using Lyapunov functional, a set of sufficient conditions are obtained to guarantee the uniqueness and global stability of the nonautonomous system.3. In chapter 4, we construct a model of impulsive differential equations to describe the evolution of a population with normal and tumor cells which is acted by medicine. Competition among the two kinds of cells is considered. We prove in one case, the system has stable boundary periodic solution and in other case, it has globally asymptotical positive periodic solution and we can calculate its T-periodic average. Based on this studying, we propose a set of plans to determine therapeutic threshold and harmful threshold and combining two classical medicine distribution models , we give out the optimal injecting strategy.4. In chapter 5, we investigate a three species food chain with periodic impulsive effect on the top predator. Using the Floquet theory and small amplitude perturbation skills, we show that there exists a globally asymptotically stable mid-level predator-eradication periodic solution when the impulsive period is less than some critical value. Further ,we prove that the system is permanent if the impulsive period is larger than some critical value. Finally, numerical simulation shows that with increasing of the impulsive value, the system experiences a complex process of (l)cycles, (2) periodic doubling, (3)periodic halfing and (4) chaotic bands with periodic windows.5. In chapter 6, we introduce and study a model of a predator-prey system with Monod type functional response under periodic pulsed chemostat conditions, which contains with predator, prey, and periodically pulsed substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the impulsive system shows two kinds of bifurcations, whose are period-doubling and period-halfing.