Dynamic Analysis of a Predator–Prey Model of Gause Type with Allee Effect and Non-Lipschitzian Hyperbolic-Type Functional Response

In this work, we study a predator–prey model of Gause type, in which the prey growth rate is subject to an Allee effect and the action of the predator over the prey is determined by a generalized hyperbolic-type functional response, which is neither differentiable nor locally Lipschitz at the predator axis. This kind of functional response is an extension of the so-called square root functional response, used to model systems in which the prey have a strong herd structure. We study the behavior of the solutions in the first quadrant and the existence of limit cycles. We prove that, for a wide choice of parameters, the solutions arrive at the predator axis in finite time. We also characterize the existence of an equilibrium point and, when it exists, we provide necessary and sufficient conditions for it to be a center-type equilibrium. In fact, we show that the set of parameters that yield a center-type equilibrium, is the graph of a function with an open domain. We also prove that any center-type equilibrium is stable and it always possesses a supercritical Hopf bifurcation. In particular, we guarantee the existence of a unique limit cycle, for small perturbations of the system.