Dynamics of a modified Leslie-Gower predator-prey model with Allee effect on the prey and a generalist predator

TL;DR

This paper analyzes a predator-prey model incorporating Allee effects, functional responses, and generalist predators, revealing multiple equilibria, bifurcations, and stability phenomena through theoretical analysis and numerical simulations.

Contribution

It introduces a modified Leslie-Gower predator-prey model with Allee effects and generalist predators, exploring complex dynamics and bifurcations not previously detailed.

Findings

Strong Allee effect leads to at most two positive equilibria, with one always saddle.

Weak Allee effect allows up to three positive equilibria, with two potentially stable.

Numerical simulations demonstrate the impact of Allee effects and alternative food sources on stability.

Abstract

A predator-prey model with functional response Holling type II, Allee effect in the prey and a generalist predator is considered. It is shown that the model with strong Allee effect has at most two positive equilibrium point in the first quadrant, one is always a saddle point and the other exhibits multi-stability phenomenon since the equilibrium point can be stable or unstable. While the model with weak Allee effect has at most three positive equilibrium point in the first quadrant, one is always a saddle point and the other two can be stable or unstable node. In addition, when the parameters vary in a small neighbourhood of system parameters the model undergoes to different bifurcations, such as saddle-node, Hopf and Bogadonov-Takens bifurcations. Moreover, numerical simulation is used to illustrate the impact in the stability of positive equilibrium point(s) by adding an Allee effect and an alternative food source for predators.