Qualitative Analysis Of Two Classes Of Predator-Prey Systems With Time Delays

In the nature,there are many interactions between populations,which promote the development of the whole nature.The relationship of predator-prey is one of the basic relationships among species,which plays an important roles.In this paper,we mainly research the existence,extinction and oscillation in predator-prey model.Firstly,a predator-prey model with stage structure and a time delay for the prey and Beddington-DeAngelis functional response for the predator is investigated.The positivity and boundedness of the solution are discussed.By the method of characteristic,the local stability of the equilibria of the model are discussed,respectively.By using the persistence theory on infinite dimensional systems,it is proven that the system is permanent if the positive equilibrium exists.By applying the comparison principle of differential equation and iterative method,respectively,sufficient conditions are obtained for the global stability of the boundary equilibria and the positive equilibrium.Numerical simulations are carried out to illustrate the main results.Next,a predator-prey model with two time delays and competitive interference is investigated.Above all,we analyze the model without delays.By calculating the corresponding characteristic roots of the Jacobian matrix,the locally stability of non-negative equilibria of the system is discussed,respectively.By using Bendixon-Dulac criterion,it is proven that the system does not have any limit cycle under a certain condition and therefore the sufficient condition is obtained for the globally asymptotically stable of the positive equilibrium.Then we study the model with time delays.It is mainly to study the effect of delays on the stability of positive equilibrium.By choosing the two delays as the bifurcation parameter,the existence of Hopf bifurcation with respect to both delays are established.Numerical simulations are carried out to illustrate the main results.