Bifurcations and Control in a Discrete Predator–Prey Model with Strong Allee Effect

Abstract

In this paper, the bifurcations and the control of a discrete predator–prey model with strong Allee effect on the prey are investigated. This shows that the model undergoes a supercritical Neimark–Sacker bifurcation. Meanwhile, the explicit approximate expression of the stable closed invariant curve caused by the Neimark–Sacker bifurcation is given. 1:3 strong resonance is investigated through approximation by a flow, and the bifurcation curves around 1:3 resonance are obtained. Moreover, for the sake of regulating the stability of the biological system, we extend the hybrid control strategy to control the Neimark–Sacker bifurcation and 1:3 strong resonance. The theoretical analyses are validated by numerical simulations and are explained from the biological point of view.