Impact of prey refuge in a discontinuous Leslie-Gower model with harvesting and alternative food for predators and linear functional response

Abstract

Since in nature there are species that take refuge, totally or proportionally, from the predator if the prey population size exceeds a threshold value, and become available again to the predator, with linear predator functional response, if its population size is higher than the threshold value, in this work we show all the changes in the dynamics presented by two discontinuous Leslie–Gower predator–prey models, assuming harvesting and alternative food for predators and total protection, or a constant proportional refuge of prey from being consumed by the predator when its population size is below the threshold value. The conditions on their parameters to determine the dynamics of each discontinuous model are developed by means of a bifurcation analysis, with respect to the threshold value of the prey population size and the collection rate for predators. It is concluded that for certain conditions on their parameters, the prey population size could reach a convergence equal or higher than its threshold value when considering the discontinuous model with a mechanism to protect prey from being consumed by the predator, as opposed to the discontinuous model with a constant proportion of prey refuges above the threshold value, whose convergence could be lower, equal or higher than the threshold value.