The authors consider the following ratio-dependent predator-prey model with nonzero constant predator harvesting
where all parameters and are positive constants; represents the rate of predator harvesting. It is shown that the model has at most two equilibria in the first quadrant and can exhibit numerous kinds of bifurcation phenomena, including the bifurcation of cusp type of codimension 2 (i.e., Bogdanov-Takens bifurcation), the subcritical and supercritical Hopf bifurcation. These results show far richer dynamics compared to the model without harvesting and different dynamics compared to the model with nonzero constant rate of prey harvesting developed by D. Xiao and L. S. Jennings [SIAM Appl. Math. 65, 737–753 (2005; Zbl 1094.34024)]. Biologically, it is shown that nonzero constant rate of predator harvesting can prevent mutual extinction as a possible outcome of the predator-prey interaction.