Stability And Hopf Bifurcation Of A Predator-prey Model With Time Delay

In this paper,the stability and Hopf bifurcation of a predator-prey model with time delay are studied.The paper consists of three parts.In Section 1,a diffusive predator-prey model with time delay is introduced.In Section 2,a predator-prey model with uniform spatial distribution and time delay is discussed.Firstly,the local asymptotic stability of the positive equilibrium is analyzed by using linearization method and the existence of Hopf bifurcation is investigated by choosing time delay as the bifurcation parameter.Next,the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are studied by applying the center manifold theory and normal form method.In Section 3,a predator-prey model with nonuniform spatial distribution and time delay is discussed.Firstly,the local asymptotic stability of the positive equilibrium is considered.Then,the existence of Hopf bifurcation is discussed by choosing time delay as the bifurcation parameter.Finally,the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are studied by applying the center manifold theory and normal form method.