Hopf Bifurcation With Time Delay Ratio Dependent Predator - Prey System

In this paper, a ratio-dependent predator-prey system with delay is investigated, the stability of the two equilibria is obtained. The general formula for bifurcation direction of Hopf bifurcation is calculated.The delay bound τ₀ of the delay differential equation at the positive equilibrium point is given. With the suit parameter condition, τ₀ will be the bifurcation value of Hopf bifurcation, τ₀ is obtained concretely.A brief arrange of the organization of the paper as follows.The first chapter introduces the charactericteristic equation for differential equations with delays and some fundamental definitions and theories as well as elementary tools.In chapter 2,A ratio-dependent predator-prey system with delay is investigated using the theory listed in thapter 1.The conditions of the stability are obtained . The conditions are brief and practical algebraic criterions .Furthermore,we get the delay bound .The general formula for bifurcation direction is calculated,and the estimation formula of the period for periodic soluttion is discussed using Hassard method.In chapter 3,given an example,we test the frontal formulae...