Phase portraits of predator--prey systems with harvesting rates

Abstract

We investigate positive equilibria and phase portraits of predator-prey systems with constant harvesting ratesarising in ecology. These systems are generalizations of the well--known predator--prey systemswith Beddington--DeAngelis functional responses. We seek the ranges of the five parameters involved for whichthe equilibria of the systems to be positive and obtainall the positive equilibria of the systems.We prove that these positive equilibria are saddles, topological saddles, nodes, saddle-nodes, foci, centers, or cusps by providing suitable ranges of the five parameters. These results show how the harvesting rate and another parameter used in the Beddington--DeAngelis functional responses affect the dynamical behaviors of these systems. In particular, if the harvesting rate is larger than $1/4$ or the parameter just mentioned is too large, then the mutual extinction occurs.