Journal of Nonlinear Functional Analysis

We study the phase portraits and Hopf bifurcations of the ratio-dependent Holling type III models with  constant harvesting rates. We provide the ranges of the parameters involved under which all the equilibria be positive. We show that these positive equilibria can be saddles, topological saddles, saddle-nodes, stable (or unstable) nodes or focuses, weak centers or cusps, depending on the choices of the parameters. We compute the first Lyapunov number to obtain the subcritical and supercritical Hopf bifurcations of the models.