# American Institute of Mathematical Sciences

March  2012, 32(3): 901-933. doi: 10.3934/dcds.2012.32.901

## Phase portraits of predator--prey systems with harvesting rates

 1 Department of Mathematics, Ryerson University, Toronto, Ontario, M5B 2K3, Canada, Canada

Received  October 2010 Revised  April 2011 Published  October 2011

We investigate positive equilibria and phase portraits of predator-prey systems with constant harvesting rates arising in ecology. These systems are generalizations of the well--known predator--prey systems with Beddington--DeAngelis functional responses. We seek the ranges of the five parameters involved for which the equilibria of the systems to be positive and obtain all 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.
Citation: K. Q. Lan, C. R. Zhu. Phase portraits of predator--prey systems with harvesting rates. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 901-933. doi: 10.3934/dcds.2012.32.901
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