On a class of predator-prey models of Gause type with Allee effect and a square-root functional response

A predator-prey model of Gause type is an extension of the classicalLotka-Volterra predator-prey model. In this work, we study a predator-prey model of Gause type, where the prey growth rate is subject to anAllee effect and the action of the predator over the prey is given bya square-root functional response, which is non-differentiable at they-axis. This kind of functional response appropriately models systems inwhich the prey have a strong herd structure, as the predators mostlyinteract with the prey on the boundary of the herd. Because of thesquare root term in the functional response, studying the behavior ofthe solutions near the origin is more subtle and interesting than otherstandard models.Our study is divided into two parts: the local classification of the equi-librium points, and the behavior of the solutions in certain invariantset when the model has a strong Allee effect. In one our main resultswe prove, for a wide choice of parameters, that the solutions in certaininvariant set approach to they-axis. Moreover, for a certain choice ofparameters, we show the existence of a separatrix curve dividing the in-variant set in two regions, where in one region any solution approachesthey-axis and in the other there is a globally asymptotically stable equi-librium point. We also give conditions on the parameters to ensure theexistence of a center-type equilibrium, and show the existence of a Hopfbifurcation.