Population dynamics in a Leslie-Gower predator-prey model with predator harvesting at high densities

In this paper, we propose a Leslie-Gower predator-prey model in which the predator can only be captured when its population size exceeds a critical value; the mean growth rate of the prey in the absence of the predator is subject to a semi-saturation rate that affects its birth curve, and the interaction between the two species is defined by a Holling II predation functional with alternative food for the predator. Since the proposed model is equivalent to a Filippov system, its mathematical analysis leads to a local study of the equilibria in each vector field corresponding to the proposed model, in addition to the study of the stability of its pseudo-equilibria located on the curve separating the two vector fields. In particular, the model could have between one and three pseudo-equilibria and at least one limit cycle surrounding one or two inner equilibria, locally unstable points.