Population dynamics in a Leslie-Gower predator-prey model with proportional prey refuge at low densities

Abstract

In this paper we propose a mathematical Leslie-Gower predator-prey model, in which the prey takes refuge from the predator when its population size is below a critical threshold, the functional response of the predator is represented by a Holling II function, and the growth of the prey in the absence of the predator is subject to a semi-saturation parameter that affects its birth curve. Since the model is composed of two vector fields, its qualitative analysis includes, in addition to the determination of the number and stability of the equilibria for each vector field and belonging to the biological sense set, the study of the dynamics in the trajectories close to the dividing curve of the two vector fields in order to determine possible pseudo-equilibria. As a result, if the proposed model has a single inner equilibrium, then there is the possibility of having between one or at least two limit cycles, coexisting or not in both vector fields and around the inner equilibrium. Likewise, the model has a stable pseudo-equilibrium which may be surrounded by at least two limit cycle.