Dynamics of a Leslie–Gower predator–prey model with Holling type II functional response, Allee effect and a generalist predator

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 points 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. The model with weak Allee effect has at most three positive equilibrium points 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 different bifurcations, such as saddle–node, Hopf and Bogdanov–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 sources for predators.