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Bogdanov-Takens Bifurcation in a Delayed Michaelis-Menten Type Ratio-Dependent Predator-Prey System with Prey Harvesting
Yunxian Dai,Ping Yang,Zhiliang Luo,Yiping Lin
Keywords:Delayed ratio-dependent predator-prey model, Michaelis-Menten Type, prey harvesting, Bogdanov-Takens bifurcation
Abstract:
      In this paper, we study a delayed Michaelis-Menten Type ratio-dependent predator-prey model with prey harvesting. By considering the characteristic equation associated with the nonhyperbolic equilibrium, the critical value of the parameters for the Bogdanov-Takens bifurcation is obtained. The conditions for the characteristic equation having negative real parts are discussed. Using the normal form theory of Bogdanov-Takens bifurcation for retarded functional differential equations, the corresponding normal form restricted to the associated tow-dimensional center manifold is calculated and the versal unfolding is considered. The parameter conditions for saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation are obtained. Numerical simulations are given to support the analytic results.