Bogdanov-Takens bifurcation analysis of a delayed predator-prey system with double Allee effect

The Bogdanov-Takens (B-T) bifurcation of a delayed predator-prey system with double Allee effect in prey are studied in this paper. According to the existence conditions of B-T bifurcation, we give the associated generic unfolding, and derive the normal forms of the B-T bifurcation of the model at its interior equilibria by generalizing and using the normal form theory and center manifold theorem for delay differential equations. By analyzing the topologically equivalent normal form system, one find that the Allee effect and delay can lead to varies dynamic behaviors, which is believed to be beneficial for understanding the potential mathematical mechanism that driving population dynamics.