Bifurcations in a modified Leslie–Gower predator–prey discrete model with Michaelis–Menten prey harvesting

Abstract

In this paper, a modified Leslie–Gower predator–prey discrete model with Michaelis–Menten type prey harvesting is investigated. It is shown that the model exhibits several bifurcations of codimension 1 viz. Neimark–Sacker bifurcation, transcritical bifurcation and flip bifurcation on varying one parameter. Bifurcation theory and center manifold theory are used to establish the conditions for the existence of these bifurcations. Moreover, existence of Bogdanov–Takens bifurcation of codimension 2 (i.e. two parameters must be varied for the occurrence of bifurcation) is exhibited. The non-degeneracy conditions are determined for occurrence of Bogdanov–Takens bifurcation. The extensive numerical simulation is performed to demonstrate the analytical findings. The system exhibits periodic solutions including flip bifurcation and Neimark–Sacker bifurcation followed by the wide range of dense chaos.