Abstract

The model formulated by Patrick H. Leslie in 1948 is an alternative, which does not follow the Lotka–Volterra model scheme. Its main feature is that the equation for predators is the logistic-type growth function. Thus it is a model assuming implicitly the competition among predators. In this chapter, the dynamic of a modified Leslie–Gower type predator-prey model is analyzed using a topologically equivalent system. The main properties of the system are established, considering two important issues: (i) the predators capture an alternative food when the quantity of prey is scarce, and (ii) the prey population is affected by an Allee effect. Necessary and sufficient conditions for the existence and local stability of equilibria are determined. Important properties are also proven as the existence of i) a homoclinic curve, ii) two concentric limit cycles surrounding a positive equilibrium point, and iii) a non-infinitesimal limit cycle, generated by the breaking of the homoclinic curve. If the predators are generalists, the dynamics of the model change enough from the model that takes into account predator specialists. This is evidenced by the appearance of additional equilibrium points and the previously described homoclinic orbit. Some numerical simulations are depicted as examples to reinforce the analytical results.