The Stability Of Predator Prey System With Time Delay And The Hopf Branch Research

Recently, according to extensive research on the population dynamics model at homeand abroad. Moreover, as time-delay was introduced , it greatly enriched the research contentof the population dynamics, and promoted population dynamic development. This paperconsiders the time-delay predator-prey model with Holling type functional response.Chapter II study Holling-IV type predator-prey model, and discuss the boundednessof the solution of system and the existence of equilibrium point. By the Routh-Hurwitzcriterion the su?cient condition of locally asymptotical stability equilibrium point couldbe obtained. As the introduction of two discrete-time delays, important results could beobtained, that is, the stability of the boundary equilibrium point asτ1 increases, variesfrom stable to unstable, and may happen Hopf bifurcation. With regard to the stabilityswitches of positive equilibrium point, This paper analyses two equal delays, and get thesmall amplitude periodic solutions.Chapter III study the age structure and double-delay predator-prey model. Basedon some comparison arguments, sharp threshold conditions which are both necessary andsu?cient for the global stability of the equilibrium point of predator extinction could beobtained. Finally, the variation of predator stage structure could a?ect the existence of theinterior equilibrium point and drive the predator into extinction by changing the maturation(through-stage) time delay.Chapter IV study the dynamic behavior of age structure predator-prey model under thein?uence of seasonal growth. The combined e?ect of delay and seasonality on the dynamicsof the system could be obtained. A variety of complex population dynamics including chaos,quasi-periodicity, and periodic resonance could be obtained. The degree of complexity in thesystem increases with increasing the magnitude of delay, or frequency of seasonal variation.