Application And Research Of Predation System In Pest Control

In nature,changes in the state of many systems are not only related to the current state,but also rely on the time in the past.Therefore,for this kind of systems,the delay differential equations is more reasonable.On the other hand,many evolutionary processes are characterized by the fact that they undergo an abrupt change of state at certain moments.These processes are subject to short-term perturbations of negligible duration compared with duration of the process.Consequently,it is natural to assume that these perturabations are instantaneous,that is,in the form of an impluse.As a result,instead of being continuous,the system would be semi-continuous.Now,it is more reasonable to set up impulsive differential equations to solve the problem.Based on the above background,this paper studies the biological systems with impulsive effects and time delay:1 ? A class of non-autonomous predator-prey system with Beddington-De Angelis functional response(later reffered to as B-D functional response)and Allee effect are studied.By using the comparison theorem and the differential inequality theorem,the sufficient conditions for uniform persistence of the system are obtained.For corresponding periodic system,globally asymptotic stability of the periodic solution is obtained by Browder's fixed point theory and constructing a suitable Lyapunov function.Fially,some numerical examples and simulations are given for the theoretical results by Matlab software,and the influence of parameter change on the persistence of system is analyzed.2?A predator-prey system with B-D functional response and Allee effect are studied where impulsive effects are also considered in the model.By using small parameters perturbation skills and comparison theorem for the implulsive equations,the globally asymptotical stability of the prey-eradication periodic solution and the persistence for the system are obtained.Finally,some numerical and some simulations are presented by Matlab software to support our theoretical results.3?Based on the fact that the system changes in the environment,we established a nonautonomous predator-prey system with B-D functional response and Allee effect where impulsive effects are also considered in the model.Applying the comparison theorem of differential equations and the differential comparison inequality of impulsive,the conditions for the persistence and asymptotical stability of system solutions are discussed.4?Based on the state of the system varies with time,a non-autonomous predator-prey system with B-D functional response and Allee effect are studied where impulsive effects and time delays are also considered in the mode.Using the techniques of differential inequality,differential mean-value theorem,integral inequality and so on,the sufficient conditions for the persistence of the system are obtained.Then,by constructing Lyapunov function,we obtain that almost periodic solutions of uniformly asymptotical stability of system exist certain conditions.