The Study On Stability And Bifurcation Problems Of Several Predator-Prey Systems

The dynamics of predator-prey systems has been favored by both biologists and mathemati-cians since the well-known Lotka-Volterra model was brought forward. The research achievementsalso emerge in endlessly and has been applied extensively in many fields such as chemistry, bi-ology, economics etc. In this thesis we consider mainly the stability and bifurcation problemsof predator-prey systems, have a sufcient condition for the global asymptotic stability of thepositive equilibrium. And we analyse a delayed predator-prey difusion system with difusionand functional response, discusses the existence of Hopf bifurcation and the Bogdanov-Takenssingularity.This dissertation have five chapters.In the first chapter, we expound the main work background of Lotka-Volterra system, Lotka-Volterra system which applied in competitive web sites and enterprise competition background,and expound the main results of this thesis.In the second chapter, we consider a delayed predator-prey system. And we obtain thesufcient condition for the global asymptotic stability of the positive equilibrium by using thezooming method, which improves some known results.In the third chapter, a ratio-dependent predator-prey system has been discussed. And thedegenerate equilibrium of the system has been discussed in order to obtain all possible phaseportraits for its perturbations.we study a delayed predator-prey difusion system with functional response, and analysethe stability of the equilibrium by using the qualitative analysis of the diferential equations. Wealso discusses the existence of Hopf bifurcation and the influence that the difusion impact onthe Bogdanov-Takens singularity.In the fourth chapter, we consider a delayed predator-prey difusion system with homoge-neous Neumann boundary condition. By studying the impact of the time delay on the stability ofthe model, we show that Hopf bifurcations can occur when we take the delay τ as the bifurcationparameter. Finally, we also considered the efect of the difusion on bifurcated periodic solution.Chapter fifth we make a summary and outlook to the contents of this dissertation.