Hopf Bifurcation for a Predator-Prey System with θ Logistic Growth

Abstract

In predator-prey population model, the density capacity restricts the growth rate of prey population. Prey species is assumed with θ logistic growth rate, which is θ exponent dependent. The generalized Hopf point is found, hence the Bautin bifurcation with limit cycle hysteresis phenomena is observed, which brings forth the limit point cycle appearing on Bautin bifurcation line. The homoclinc bifurcation line starts from Bogdanov-Takens point and tangents to saddle-node bifurcation line. The homoclinic orbits and near dynamics are simulated, which manifests the complex dynamics of system.