Impact of Allee effect in Leslie-Gower model with increasing functional response

Abstract

This paper presents a study of dynamic behavior and bifurcation analysis of a predator–prey system with the functional response proposed by Cosner et al. (Theor Popul Biol 56:65–75, 1999) and Allee effect in prey population. The functional response used is specific in compare with the conventional functional responses according to its monotonicity for both prey and predator density, and moreover it increases as predator density increase. This function response describes a behavioral mechanism which a group of predators foraging in linear formation, contacts and then hunts gathering around the herd or a school of prey. Mainly, our aim is to demonstrate the impact of strong and weak Allee effect on the system dynamics. Mathematically our analysis primarily focuses on the stability of coexisting equilibrium points and all possible bifurcations that the system may exhibit. Actually, we consider the existence of equilibria and analyze their stability. The possibility of extinction of both populations is also considered, by studying dynamics of the system near the origin. The bifurcation of the system will be analyzed, including the occurrence of saddle–node bifurcation, Hopf and degenerate Hopf bifurcation, and Bogdanov–Takens bifurcation. The theoretical results are verified by numerical simulations. We observe the bi-stability and tri-stability, so that we further discuss the basins of attraction in all possible cases of existence of multiple attractors.