Bifurcation analysis of a modified Leslie–Gower predator–prey model with Beddington–DeAngelis functional response and strong Allee effect

Abstract

The paper is concerned with a modified Leslie–Gower delayed predator–prey system where the growth of prey population is governed by Allee effect and the predator population consumes the prey according to Beddington–DeAngelis type functional response. The situation of bi-stability and existence of two interior equilibrium points for the proposed model system are addressed. The stability of the steady state together with its dependence on the magnitude of time delay has been obtained. The conditions that guarantee the occurrence of the Hopf bifurcation in presence of delay are demonstrated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. It is shown that time delay is incapable of avoiding the situation of extinction of the prey species. Finally, some numerical simulations have been carried out in order to validate the assumptions of the model.