Asymptotical Behavior Of The Solution Of Nonautonomous Species' Dynamical System

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 Matla-bel.In chapter 2, we study the nonautonomous three species dynamical system with delay. we get the ultimately bound for the positive solution of models we consider by estimate of inequality and constructing suitable persistence functions. Further, by using fixed point theorem, M-matrix theory, we obtain the existence and attractivity of positive periodic solution for the periodic system. Finally, by using the method of topological degree, we obtain the existence of positive periodic solution of three species predator-prey system with nonautonomous delay. We also discuss the effect of delay on permanence and periodic solutions of the system.In chapter 3, we investigate a classical periodic Lokta-Volterra two species predator-prey system with impulsive effects .This system exhibits three types of T-periodic component-wise nonnegative solutions: the solution (0,0), usually as the trivial one; those with one component vanishing, often known as the smei-trivial positive solution; and non-trivial periodic solutions, which are the solutions withboth components positive. We give boundedness of the system using comparison theorem of impulsive differential equation and also get some sufficient conditions for asymptotic stability of trivial periodic solution and semi-trivial periodic solutions. By using the theory of topological degree, we show the existence of positive periodic solution. Followed, we study competing system with diffusion, delay, functional response and impulsive effects. And we discuss the influence of diffusion and delay on periodic solution of the system.In chapter 4, the effects of impulses on the persistence and extinction of predator-prey systems are studied thoroughly. Firstly, we consider integrated pest management mathematical model with fixed moments, that is, a proportional periodic impulsive catching or poisoning for the pest population at fixed moments and a constant periodic releasing for the predator. Conditions for the global stability of boundary periodic solution and permanence of the system are obtained by using Floquet theory and comparison theorem of impulsive differential equation and analysis method. Conditions for the existence and local stability of positive periodic solution are establish via bifurcation theory of impulsive differential equation. Secondly, the functional response of Holling type IV is introduced to the predation and permanence and extinction of the system are studied. Since the system with functional response of Holling type IV may inherently oscillating, the effects of the amount of impulsively adding on the inherent oscillation are studied numerically. We discuss here many complexity phenomena are dominated impulse: quasi-periodic oscillating, periodic oscillating, periodic doubling cascade, chaos, per...