The Qualitative Analysis Of Two Kinds Of Predator-prey Models With State Dependent Impulsive Effects

Abstract:Integrated pest management has always been one of the hot topics of bio-mathematics. With the development of bio-mathematics, the models for the problem about the integrated pest management, which is from the ordinary differential equation to impulsive differential ones at fixed time and to the impulsive differential systems with state-dependent impulsive effects now. Under this background, this paper mainly study the dynamic qualitative analysis of two kind of predator-prey models with state-dependent impulsive effects, and the main works as following:1) Firstly, this paper mainly discuss the existence of periodic solution of a Holling Ⅱ type predator-prey model with state-dependent impulsive effects. According to the different time of spraying insecticide and releasing enemies, we build an impulsive differential system with two impulsive thresholds and corresponding Poincare maps, which turns the problem of existence and stability of the periodic solutions of the system to the problem of the existence and stability of the fixed point of the Poincare maps. We obtain the conditions for the existence and stability of semi-trivial solution by the means of dynamic analysis and Poincare maps, and discuss the sufficient condition of the existence of positive order-1periodic solution under different cases. We get the picture and time series picture of the semi-trivial solution and the positive periodic solution by Matlab, respectively, which prove the validity of the conclusions gave by this paper.2) Secondly, we introduce the impulsive differential system with state-dependent impulsive effects in [69] and the corresponding conclusions, confirm the rationality of the system, and point out some mistakes of the theorems. We have proved that the positive periodic solution neither does not existed nor exists but is unstable if the biological impulsive effect doesn’t occur, obtained the correct conditions of the existence and stability of positive periodic solution by Poincare maps and first-integral method, and verified the accuracy of the gave conclusions by numerical simulation.