A Class Of Delay And Ratio-dependent Holling Type Iii Predator - Prey System Stability And Hopf Bifurcation

The dynamic relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology. Although the predator-prey theory has seen much progress, many long standing mathematical and ecological problems remain open.In this paper, a delayed ratio-dependent predator-prey system of holling-III is considered. We explicitly derive a domain of attraction for the positive equilibrium of system by using Liapunov method. We also investigate Hopf bifurcation by using Hassard method. Finally, some numerical simulations are made for system.Our paper are separated into seven chapters. The first part presents develops of predator-prey systems and gives the model that is investigated in our paper; Chapter 2 presents the results on boundedness of solutions and permanence of system; In chapter 3, we explicitly derive a domain of attraction for the positive equilibrium of system by using Liapunov method; The local stability and the conditions that Hopf bifurcation occurs are obtained in chapter 4; In chapter 5, some explicit fomulae of the direction of Hopf bifurcation and the stability of bifurcating periodic solutions on the center manifold are determined,using normal form and center manifold introduced by Hassard et al; In chapter 6, some numerical simulations are performed to illustrate the analytical results found; Our paper ends with a brief conclusion.