Dynamical Properties Of Two Classes Of Predator-prey Systems With Two Time Delays

Predator-prey model is an important model in the dynamic of population and is paid close attentions.Combine with the Lyapunov method,comparison theorem and persistence theory,here we study the dynamical properties of two classes of predator-prey model with two time delays.Firstly,we propose and investigate a predator-prey system with modified Leslie-Gower and two time delays are investigated.By analyzing the corresponding characteristic equations,the local stability of a positive equilibrium and two boundary equilibria of the system is discussed,respectively.By choosing the two delays as the bifurcation parameter,the existence of Hopf bifurcation with respect to both delays are established.By means of the persistence theory on infinite dimensional systems,it is proven that the system is permanent if the positive equilibrium is feasible.By using the comparison theorem and the Fluctuation Lemma,respectively,sufficient conditions are obtained for the global stability of the positive equilibrium and one of the boundary equilibria of the proposed system.Numerical simulations are carried out to illustrate the main results.Moreover,based on the previous studies,we propose and study a predator-prey system with two delays and stage structure for the prey is investigated.By analyzing the corresponding characteristic equations,the local stability of a positive equilibrium and two boundary equilibria of the system is discussed,respectively.By choosing the two delays as the bifurcation parameter,the existence of Hopf bifurcation with respect to both delays are established.By means of the persistence theory on infinite dimensional systems,it is proven that the system is permanent if the positive equilibrium is feasible.By using the comparison theorem and iterative analysis method,sufficient conditions are obtained forthe global stability of the positive equilibrium and one of the boundary equilibria of the proposed system.Numerical simulations are carried out to illustrate the main results.