Hopf Bifurcation In Predator Systems With Time Delays And Functional Response

Give the current research about staged-structured system with HollingⅢfunctional response and time delay, rarely takes into account all of the elements about stage structure, time delay and functional response. On the other hand, in the course of studies about three-specie food chain system with time delays, seldom consider the impact of functional response on ecosystem. Consequently, in order to embody the interaction relation among populations more completely and accurately by mathematical model, the strategies of this paper are as follows:A stage-structured predator-prey system with HollingⅢfunctional response and time delay is investigated. In this system, we consider immature and mature individuals of the prey population are divided by a fixed age. The immature predators don't have the ability of preying prey. Sufficient conditions about the asymptotic stability of the positive equilibrium are derived by using characteristic value method and Hurwitz criterion. Further, the Hopf bifurcation theory is used to deal with the existence of Hopf bifurcation. The direction of Hopf bifurcation and the stability of period solutions bifurcating from Hopf bifurcations are derived by using normal form theory and center manifold argument. Finally, numerical simulations are carried out to illustrate these conclusions.A food chain model with time delays and HollingⅡfunctional response is studied. The existence of positive equilibrium is obtained. Sufficient conditions about the permanent persistent survival of the system and the global asymptotic stability of the positive solutions are derived by using comparison principle and constructing Liapunov functional. Through studying the existence of solution of the polynomial equation, sufficient conditions for the existence of the unique positive equilibrium are established. Through Hurwitz criterion, sufficient conditions for the local asymptotic stability of the positive equilibrium are given. Through Hopf bifurcation theorems, the existence of Hopf bifurcation on the positive equilibrium is discussed. Finally, numerical simulations are carried out to illustrate these conclusions.