Weak Resonant Double-Hopf Bifurcation Of The Ratio-dependent Predator-prey System With Both Propagation Time Delays

With the development of science and technology, ecological problem has increasingly become intertwined with human beings. People employ biological models to reflect the ecological law of biology. Actually in a large number of systems, we should also consider the effect of time delay, and the models with time delay are more reasonable. The thesis further discovers that the models with two time delays are possible in the real ecology. The thesis is devoted to a ratio-dependent predator-prey model with Holling-II type functional response functions, which has two propagation time delays,through selecting parameters and two time delays,the stability and biodiversity of ecosystem can be achieved.The rea-sons of selecting the system are introduced as follows:firstly, the predator-prey model with Holling-II type is a very representative ecology model,and the model is more reasonable with the effect of both propagation time delays; secondly under the certain parameter values, weak resonant double-Hopf bifurcation can appear in the system,and there are complex dynamical behaviors near the double Hopf bifurcation points, the system has the research value related to keeping biodiversity and ecosystem stability. In this thesis, by changing the parameters, it was found that, if selecting parameters and taking r1and τ1as the bifurcation parameters, some double Hopf bifurcation can be discovered where r1indi-cates the intrinsic growth rate of prey and τ1means the propagation delay. The system is reduced to a four dimensional center manifold, and the normal form equations are obtained by employing the method of normal form. After analysis the equations, the dynamics near weak-resonant double-Hopf bifurcation are unfolded and classified. Near the bifurcation point,it was found that the stable equilibrium, the periodic solution,co-existing periodic solutions and so on can occur in the original system. Lastly the theoretical analysis of the outcomes is verified by employing the nonlinear dynamical software-WinPP. The conclu-sions are shown as follows:(l)Under some certain parameter conditions, the prey has critical influence on the dynam-ics of the predator-prey system; (2)The growth of the prey has important influence on the dynamics of the whole system, and is also the key of the system evolution;(3)Not only the intrinsic growth rate r1, but. also the propagation delay τ1have the great influence on the dynamics of the whole system;(4)Near the double Hopf bifurcation points, the system can maintain biodiversity and stability, so the population will not become extinct, the system will not collapse.