Analysis To Several Predator-Prey Systems With Impulsive Effect

Impulsive differential equation mainly describes the states of the system chang-ing very rapidly in a short time or instantaneously, which has been widely used in physics, economics, pharmacokinetics, and space technology and other fields. In re-cent years, population dynamical system with impulsive effect has been thoroughly studied such as releasing immature fish and harvesting mature fish in fisheries man-agement, planting and harvesting in plant protection research and spraying pes-ticides and releasing natural enemies in agricultural production, which are strong evidence of human behavior impacting and controlling on the ecological environment and show that impulsive effect has a high theoretical value and broad application prospects in the development and management of the ecological resources.Based on biological control and chemical control strategies on pest and consid-ering the impact of spraying pesticides on natural enemies, firstly, a predator-prey system with stage-structured pest and natural enemy with Beddington-DeAngelis functional response and spraying pesticides and releasing predators at different im-pulsive times is studied. Using the Floquet theory and Comparison theorem, suf-ficient conditions of the globally asymptotically stable of pest-extinction periodic solution and the permanence of the system are derived. Finally, Matlab numerical simulations are presented to illustrate the feasibility of our theoretical results. The results provide a theoretical guide for the pest management.Secondly, a delayed predator-prey system with stage-structured prey and con-tinuous harvesting on mature prey and impulsive harvesting on predator is discussed. Applying the theory of delay differential equation and the comparison theorem of impulsive differential equation, we obtain the sufficient conditions of the global at-tractiveness of prey-extinction periodic solution and the permanence of the system. Finally, numerical simulations verify the theoretical results.As the Leslie-Gower predator-prey model is more realistic than the traditional Lotka-Volterra predator-prey model and has rich dynamic natures, based on the model in the third chapter, a delayed predator-prey system with stage-structured prey and continuous harvesting on mature prey and impulsive releasing and harvest-ing on predator at different impulsive times is studied. Employing the comparison theorem and the theory of delay differential equation, sufficient conditions of the global attractiveness of prey-extinction periodic solution and the permanence of the system are obtained. Furthermore, numerical simulations show the theoretical results obtained.Owing to seasonal effects of weather, temperature, food supply, mating habits and other resource of physical environmental quantities, as an improved model of the third chapter, a predator-prey non-autonomous system with delayed stage-structured prey and impulsive harvesting on predator is investigated in the last chapter. Based on the comparison theorem and non-autonomous theory of impul-sive delay differential equations, the sufficient conditions of the global attractiveness of predator-extinction periodic solution and the permanence of the system are ob-tained. In addition, we prove the existence of positive periodic solution by virtue of the Mawhin's continuation theorem.