Bifurcation analysis of modified Leslie–Gower predator–prey model with Michaelis–Menten type prey harvesting

In the present paper we discuss bifurcation analysis of a modified Leslie–Gower prey–predator model in the presence of nonlinear harvesting in prey. We give a detailed mathematical analysis of the model to describe some significant results that may arise from the interaction of biological resources. The model displays a complex dynamics in the prey–predator plane. The permanence, stability and bifurcation (saddle–node bifurcation, transcritical, Hopf–Andronov and Bogdanov–Takens) of this model are discussed. We have analyzed the effect of prey harvesting and growth rate of predator on the proposed model by considering them as bifurcation parameters as they are important from the ecological point of view. The local existence and stability of the limit cycle emerging through Hopf bifurcation is given. The emergence of homoclinic loops has been shown through simulation when the limit cycle arising though Hopf bifurcation collides with a saddle point. This work reflects that the feasible upper bound of the rate of harvesting for the coexistence of the species can be guaranteed. Numerical simulations using MATLAB are carried out to demonstrate the results obtained.