Bifurcations of a Leslie-Gower prey-predator system with ratio-dependent Holling IV functional response and prey harvesting

Abstract

The dynamics of a Leslie-Gower prey-predator system with ratio-dependent Holling IV functional response and constant harvesting rate of prey are taken into account. The results developed in this article reveal far richer dynamics compared with the system without harvesting. We first make qualitative and bifurcation analysis of the system without harvesting and show that the system has a weak focus of multiplicity at most 2, at which a Hopf bifurcation occurs. However, the system with harvesting has four nonhyperbolic equilibria for some parameter values, such as two saddle-node, a cusp, and a weak focus of multiplicity at most 4, and exhibits two saddle-node bifurcations, a Bogdanov-Takens bifurcation of codimension 2, and a Hopf bifurcation. It reveals that there exist some critical harvesting values such that the species are in danger of extinction when the harvesting rate is greater than the critical values, which indicates that the dynamics of the system are sensitive to the constant prey harvesting. Moreover, numerical simulations are presented to illustrate our theoretical results.