Transcritical bifurcation and Neimark-Sacker bifurcation of a discrete predator-prey model with herd behaviour and square root functional response

Abstract

ABSTRACTIn this paper, a discrete predator-prey model incorporating herd behaviour and square root response function is deduced from its continuous version by the semi-discretization method. Firstly, the existence and local stability of fixed points of the system are studied by applying a key lemma. Secondly, by employing the centre manifold theorem and bifurcation theory, the conditions for the occurrences of the transcritical bifurcation and Neimark-Sacker bifurcation are obtained. Not only that but the direction and stability conditions of the bifurcated closed orbits are also clearly shown. Finally, numerical simulations are also given to confirm the existence of Neimark-Sacker bifurcation.

Keywords

• Predator-prey system with herd behaviour
• square root functional response
• semi-discretization method
• transcritical bifurcation
• Neimark-Sacker bifurcation