Bifurcation analysis in a discrete predator–prey model with herd behaviour and group defense

Abstract

In this paper, we utilize the semi-discretization method to construct a discrete model from a continuous predator-prey model with herd behaviour and group defense. Specifically, some new results for the transcritical bifurcation, the period-doubling bifurcation, and the Neimark-Sacker bifurcation are derived by using the center manifold theorem and bifurcation theory. Novelty includes a smooth transition from individual behaviour (low number of prey) to herd behaviour (large number of prey). Our results not only formulate simpler forms for the existence conditions of these bifurcations, but also clearly present the conditions for the direction and stability of the bifurcated closed orbits. Numerical simulations are also given to illustrate the existence of the derived Neimark-Sacker bifurcation.

Keywords

• discrete predator-prey system with herd behaviour and group defense
• semi-discretization method
• transcritical bifurcation
• period-doubling bifurcation
• neimark-sacker bifurcation