Some Results Of The Impulsive Theory On The Population Dynamical Systems

In the nature world, many species born and evolution have the character of impulse. People's captures on many recycled resources also have the effects of impulse. People control the nature through capture or complement, in order to protect the balance of ecological environment and keep the variety of species. Therefore, the research on population dynamical systems with impulsive effects is very important in both theories and application. In general, there is no species which can avoid the aggression of other species in the nature. Also, there is no species whose existence does not threat other species. To certain extent, if there are two species have this kind of conflict and the conflict is predominant, Lotka-Volterra ecosystems can be adopted to describe the conflict. As a result, the phenomenon of up and down of species and periodic circulation revealed by Lotka-Volterra models can always be discovered in nature. Thus, the research on permanence, existence of periodic solution and behavior of solutions of population dynamical systems with impulsive effects should be both typical and universal. In this article, we are going to apply the known theories and method of differential equation, functional differential equation and impulsive differential equation,such as Lyapunov function method,Razumikhin technique,the comparison theorem of impulsive differential equation,the continuation theorem of coincidence degree theory,the theorem of fixed point,the theories of matrix and impulsive differential inequality and so on, into the studies of impulsive population dynamical systems,in order to get its permanence, existence of periodic solution and the conditions of its stability. Analyzing the meaning and impact of impulse in the research system based on the practical background. Meanwhile, we will conduct numerical computation and numerical emulation in some examples,in order to illustrate and examine the conclusion. The permanence of n-species impulsive non-autonomous Lotka-Volterra system is studied in this paper. Under certain conditions, the impulsive non-autonomous system can be regarded as an impulsive autonomous system with perturbations. Based on the assumption of the small perturbance,sufficient conditions of permanence for above systems are obtained by employing the known conclusions and Lyapunov function method. Finally, the conclusion will be explained through example and numerical calculations. The existence of positive periodic solutions for n+m-dimension competition and predator-prey Lotka-Volterra ecosystem with delay and impulses are investigated. Firstly, by using Mawhin,s continuation theorem of coincidence degree theory, sufficient conditions are gained for existence of positive periodic solutions of above ecosystem. Furthermore, global asymptotical stability of positive periodic solutions is discussed when above ecosystem without delay by employing Lyapunov function method. Finally, an example of three species competition and predator-prey Lotka-Volterra model is given to illustrate our conclusions. Using developed theorem from continuation theorem of coincidence degree theory, sufficient conditions of existence of positive periodic solutions are obtained for neutral impulsive Lotka-Volterra model with delay. However, because of the behavior of solutions of above model may be interfered each other by impulses and time delay,which makes the study more complicated,there are more tasks should be conducted on the research of the stability of positive periodic solutions. The sufficient conditions of existence of positive periodic solutions and global asymptotic stability for an n-species cooperative Lotka-Volterra system with impulses are given by using the theories of matrix, the theorem of fixed point and impulsive differential inequality. Taking the example of two species cooperative Lotka-Volterra system with impulses,get the consistence of conclusion with theoretical analysis through numerical computation . Finally, the general differential systems with linear impulse and impulsive delay differential systems are investigated respectively. In the first case, it is shown that there always exists linear impulse such that differential systems are stable under certain conditions by the method of variation of parameters and Lyapunov function method. Especially, for autonomous systems the same conclusion is drawn under the more general conditions. The conclusion is illustrated by two models of autonomous and non-autonomous. Otherwise some sufficient conditions for uniform stability and asymptotic stability on the zero solution of impulsive delay differential equations are obtained by using Lyapunov function method and Razumikhin technique. In general,our conclusions are less conservative than the existing conclusions. An example is given to examine the conclusions.