Analysis Of Stability And Bifurcation In Some Predator-prey System With Time Delays

In this paper, we established two mathematical models of Predator-Prey system, researched and analyzed the dynamics of these models through the mathematical method of dynamics. Finally, numerical simulations are consistent with the conclusions which we obtained in this paper and these results have the better biological significance.This article includes the following four chapters.In the first chapter, the background, the current situation and significance of Predator-Prey system are briefly reviewed. And the same time, some mathematical models of Predator-Prey are given. Finally, we introduced the mainly Predator-Prey models which researched in this paper.In the second chapter, we considered a three species Predator-Prey system with two time delays. Based on the parameter of time delay, by constructing differential equation, we investigated the existence of Hopf bifurcation and the stability of the positive equilibrium. Finally, we used some numerical simulations to illustrate the obtained results.In the third chapter, we introduced two delays to food chain model, the two delays are the predator population’s hurt time and prey predator need a certain time to translate into its own growth, so established an improved Predator-Prey mathematical model with two delays. The following we investigated the existence of Hopf bifurcation and the stability of the positive equilibrium. By using the center manifold argument and normal form theory, the character of Hopf bifurcation are researched. Finally we use some numerical simulations to illustrate the obtained results.In the forth chapter, we summarized this article.