The ratio-dependent predator-prey model exhibits rich interesting dynamics due to the singularity of the origin. The objective of this paper is to study the dynamical properties of the ratio-dependent predator-prey model with nonzero constant rate harvesting. For this model, the origin is not an equilibrium. It is shown that numerous kinds of bifurcation occur for the model, such as the saddle-node bifurcation, the subcritical and supercritical Hopf bifurcations, Bogdanov - Takens bifurcation, the homoclinic bifurcation, and the heteroclinic bifurcation, as the values of parameters of the model vary. Hence, there are different parameter values for which the model has a limit cycle, or a homoclinic loop, or a heteroclinic orbit, or a separatrix connecting a saddle and a saddle-node. These results reveal far richer dynamics compared to the model with no harvesting.
Xiao, D., & Jennings, L. (2005). Bifurcations of a Ratio-dependent Predator-prey System with Constant Rate Harvesting. SIAM Journal on Applied Mathematics, 65(3), 737-753. https://doi.org/10.1137/S0036139903428719