Periodic Solutions Of Several Impulsive Population Dynamics Models

Mathematical ecology is one of the important research contents of Biomathematics.Differential equation is its commonly used research tool,and the system state is continuous in time.However,any organism in nature will be affected by various instantaneous actions,which makes the population dynamic system variable or growth law change suddenly.It leads to the failure of traditional continuous systems.Impulsive differential equation is a very suitable modeling tool for the situation that population is affected by human beings or other natural conditions,and relevant research has made a lot of significant achievements.However,due to the discontinuity of impulsive equation,its research is more difficult than differential equation,and there are many problems to be solved.This paper mainly considers several kinds of population dynamical systems with impulsive effects.In this paper,research is done on the dynamic properties of the model after applying the impulsive disturbance at a fixed time,especially the influence of the impulsive on the periodic solution of the system.Besides,the periodic solution of the state feedback impulsive system is studied,by using the Conley index.The paper is divided into four chapters.First,a predator-prey model with impulsive effect and Beddington DeAngelis functional response function is established.The new model can be applied to the situation that the continuous model can not deal with,such as regular pesticide spraying,regular capture,regular stocking,etc.by applying the fixed time impulsive disturbance,thereby gaining a wider applicability.The existence of periodic solution of the model is proved,and the sufficient conditions for the existence of periodic solution are given.Next,the age-specific integrated control model of the beetle is studied.The palm leaf beetles(Brontispa longissima)is one of the main pests of Palmae.According to the characteristic that two kinds of parasitic wasps attack the beetles in different age stages,this study establishes the age-specific structures of eggs,larvae,pupae and adults of the beetles.The control model of periodic spraying and releasing of natural enemies with Holling ?functional response function was established.Similar to the method in Chapter 2,the sufficient conditions for the existence of periodic solution of the system are obtained and verified by numerical simulation.The results show that when the number of parasitic wasps released and the dose of drug sprayed are in reasonable control,the damage of the beetles could be effectively controlled.Last,the periodic solutions of the semi-continuous dynamical system are studied by using the Conley index theory.The semi-continuous dynamical system is generated by the state feedback impulsive differential equation.In some modeling cases,it is more practical than the impulsive differential equation at a fixed time.By using the Conley index of the hyperbolic equilibrium point of the discrete dynamical system,a new method systems is established,for determining the existence and stability of periodic solutions of impulsive semi-continuous dynamical.Based on the analysis of a classical state feedback impulsive differential equation model,the method is proved valid.Nevertheless,as the information in Conley index is weak,the new method is still inconvenient in use,which is an important direction for further research.Finally,this paper is summerized and envisions the future research.