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

The dynamics of Leslie-Gower predator-prey models with constant harvesting rates are investigated. The ranges of the parameters involved in the systems are given under which the 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 nodes depending on the choices of the parameters involved while the interior positive equilibria in the first quadrant are saddles, stable or unstable nodes, foci, centers, saddle-nodes or cusps. It is shown that there are two saddle-node bifurcations and by computing the Liapunov numbers and determining its signs, the supercritical or subcritical Hopf bifurcations and limit cycles for the weak centers are obtained.