The Study Of Predator-prey Models With Discrete And Distrib-Uted Delay

In this paper, we mainly discuss the dynamics of predator-prey models with discrete anddistributed delay. We obtain some results by use of related mathematical theories. There arethree chapters in this paper.In chapter1, we introduce the research background, the main results and some importantpreliminaries, of this article.In chapter2, we mainly discuss the dynamic of a predator-prey model with discrete anddistributed delays. By analyzing the characteristic equation of the linearized system of the orig-inal system at the positive equilibrium, we obtain the occuring conditions for Hopf bifurcation.By using the normal form approach and center manifold theory, we study the direction andstability of Hopf bifurcation.In chapter3, we mainly discuss a stage-structured predator-prey model with discrete anddistributed delay. There is stage structure and discrete delay in prey population; and distributeddelay in predator population. By the same methods as in chapter2, we obtain the occuringconditions for Hopf bifurcation. In addition, the direction and stability of Hopf bifurcations isalso studied.