Stability Of Lotka-Volterra Models With Impulsive Perturbations

The development and utilization of ecological resources can cause short-term rapid changes in their values.For example,farmers should spray pesticide or launch its natu-ral enemies regularly to kill pest.In order to describe these discontinuous evolutionary processes,we have t.o build some appropriate impulsive models.Impulsive differential e-quations are used to describe and forecast the natural phenomena of ecology,information science,neural network,and economics with instantaneous disturbance in the process of natural growth.The theory of impulsive differential equations is now being recognized as having much richer content than the corresponding theory of differential equations with-out impulse perturbations,also it represents a more natural framework for mathematical modeling of many real-world phenomena.The research of dynamic behaviors of impulsive differential equations,such as the boundedness of the solution,the permanence,stability,and extinction of the system,as well as the existence of the almost periodic solution,has an important significance both in theory and reality.In this thesis,we consider five classes of impulsive differential equations,and the results are summarized as follows.First,we consider an impulsive autonomous logistic model.We obtain the bound-edness of solutions of this model and discuss the stability and extinction of the model.For the model without impulse,our result is in accordance with the existing ones for the traditional logistic model.We further consider the corresponding impulsive delay logistic model.By applying the impulsive comparison theorem and constructing some suitable Lyapunov functions:we obtain sufficient conditions ensuring the permanence,stability and extinct.ion of the delay model.All our results show tha.t the delay is harmless for the permanence of the model,while the impulse perturbation plays an important role in the permanence and stability of the model.Secondly,we propose an N-species Lotka-Volterra competitive system with nonlinear impulse perturbations.We first consider the corresponding autonomous logistic model.We respectively discuss the permanence and stability of the solutions,the existence of a global-ly stable equilibrium,and the extinction of the mode,under different values of the impulse perturbations.It is shown that the theory of differential equations with nonlinear impulse is richer than that with linear impulse perturbations.We then apply the obtained result-s t.o investigate an.N-species Lotka-Volterra competitive system with nonlinear impulse perturbations.Using the impulsive comparison theorem,we obtain sufficient conditions ensuring the stability and extinction of the system.Our results complement and improve the existing ones in[4,44].Also,the influence of the nonlinear impulse perturbations on the dynamic behaviors of the system is discussedThirdly,we study an impulsive Lotka-Volterra competitive system with infinite de-lays.Based on the results obtained in Chapter 1,we obtain some sufficient conditions guaranteeing the stability and extinction of the system.For the logistic type impulsive equation with infinite delays,our results improve those of Yang.Wang and Shen[107]by removing a stronger condition,and can be used to describe and forecast the natural phenomena more accurately.For the corresponding nona.utonomous two-species impulsive competitive system without infinite delays,we discuss its permanence,stability,and ex-tinction of the models,which weaken the assumptions and complement,the results of Liu and Wang[71]Fourthly,we study an impulsive Lotka-Volterra competitive system with pure infinite delays.We first consider the corresponding logistic model and obtain some sufficient conditions guaranteeing its permanence and stability.For the Lotka-Volterra system,using the impulsive comparison theorem,we discuss its permanence,as well as its extinction under two different sets of sufficient conditions.Our result indicates that the stability of the system depends not only on the rates of growth and competition,but also on the impulse perturbations.For the system without impulse perturbations,our result generalizes that of Oca and Perez[79]Finally,we investigate an impulsive model of plankton allelopathy with delays.By use of the impulsive comparison theorem,we obtain sufficient conditions which respectively guarantee the permanence,stability and extinction of the system.Our result generalizes the corresponding one of Abbas,Sen and Banerjee[1].Meanwhile,by employing the hull theory of impulsive almost periodic system,we investigate the existence of the almost periodic solution.For the system without delays,our results weaken the assumptions of He,Chen and Li[41].