With The Hopf Bifurcation, Delay-leslie, The System Of Holling Type Iv

Ecology is an important discipline aimed at studying the relationship between living beings and the environment.In recent years,much attention has been paid to biological model by many biologicians and mathematicians.In this paper,a HollingIV-Leslie predator-prey system with delay is investigated.Existence and local stability of the equilibrium is studied.Hopf bifurcation's existence is discussed.The general formulae of Hopf bifurcation direction and stability for periodic solution is given by applying Hassard's method.The paper includes four chapter .In the first chapter, we introduces the study of the development process and its respective categories.In the second chapter,the basic methods and basic theorems which will be applied to the paper are presented.In the three chapter,we introduce the background of the system.The behavior of equilibrium is studied,the conditions of existence for Hopf bifurcation is obtained. We calculate the direction of Hopf bifurcation and stability for periodic solution.In the four chapter,we give a specific example by applying of the formulae,numerical simulation for the specific model is given.Lastly,we summarize the whole paper and point out some future research direction.