Phase portraits, Hopf bifurcations and limit cyclesof Leslie-Gower predator-prey systems with harvesting rates

Abstract

The dynamics of Leslie-Gower predator-prey models with constant harvesting ratesare investigated. The ranges of the parameters involved in the systems are given under whichthe equilibria of the systems are positive.The phase portraits near these positive equilibria are studied.It is proved that the positive equilibria on the $x$-axis are saddle-nodes, saddles or unstable nodesdepending on the choices of the parameters involved while the interior positive equilibria in the first quadrantare saddles, stable or unstable nodes, foci, centers, saddle-nodes or cusps.It is shown that there are two saddle-node bifurcations and by computingthe Liapunov numbers and determining its signs, the supercritical or subcritical Hopf bifurcationsand limit cycles for the weak centers are obtained.