Bifurcation and hybrid control in a discrete predator–prey model with Holling type-IV functional response

Abstract

In this paper we investigate the 1:1 resonance and the hybrid control in a discrete predator–prey model with Holling-IV functional response, which is derived from a 2-dimensional continuous one of Gause type. When the parameters satisfy some conditions, this discrete model has a positive fixed point, which has a double eigenvalue 1 with geometric multiplicity 1. With the Picard iteration and the time-one map, this discrete one is converted into an ordinary differential system. It is shown that a Bogdanov–Takens bifurcation for this ordinary differential system happens by the bifurcation theory. This implies that this discrete model undergoes a Neimark–Sacker bifurcation and a homoclinic bifurcation. The stability of its fixed point is obtained. Then, the hybrid control strategy is applied to control the stability of this fixed point. Finally, the local phase portraits of these systems are also simulated by the Matlab software.