Dynamics of a Class of Leslie–Gower Predation Models with a Non-Differentiable Functional Response

Abstract

The main peculiarity of the Leslie–Gower type models is the predator growth equation is the logistic type, in which the environmental carrying capacity is proportional to the prey population size. This assumption implies the predators are specialists. Considering that the predator is generalist, the environmental carrying capacity is modified adding a positive constant. In this work, the two simple classes of Leslie–Gower type predator-prey models are analyzed, considering a non-usual functional response, called Rosenzweig or power functional responses, being its main feature that is non-differentiable over the vertical axis. Just as Volterra predator-prey model, when the Rosenzweig functional response is incorporated, the systems describing the models have distinctive properties from the original one; moreover, differences between them are established. One of the main properties proved is the existence of a wide set of parameter values for which a separatrix curve, dividing the phase plane in two complementary sectors. Trajectories with initial conditions upper this curve have the origin or a point over the vertical axis as their $\omega$-limit. Meanwhile those trajectories with initial conditions under this curve can have a positive equilibrium point, or a limit cycle or a heteroclinic curve as their $\omega$-limit. The marked differences between the two cases studied shows as a little change in the mathematical expressions to describe the models can produce rich dynamics. In other words, little perturbations over the functions representing predator interactions have significant consequences on the behavior of the solutions, without change the general structure in the classical systems.