Complicate dynamics of a discrete predator-prey model with double Allee effect

In this paper, a discrete predator-prey model with double Allee effect is discussed. We first simplify the corresponding continuous predator-prey model, and use the semidiscretization method to obtain a new discrete model. Next, the existence and local stability of nonnegative fixed points of the new discrete model are studied by using a key lemma. Then, by using the center manifold theorem and bifurcation theory, the sufficient conditions for the occurrences of transcritical bifurcation and Neimark-Sacker bifurcation and the stability of closed orbit bifurcated are obtained. Finally, the numerical simulations are presented, which not only verify the existence of Neimark-Sacker bifurcation but also reveal some new dynamic phenomena of this model.