The authors investigate a predator-prey system that represents a generalization of Holling-Tanner models and was first proposed by Hanski et al.. It contains five parameters and Holling terms of type two and three. In the positive quadrant, there exist at most three interior and three boundary equilibria. First, the type of these equilibria is examined. Next, a saddle bifurcation of codimension two is established for the case when all boundary equilibria collapse. The main objective of the paper is a detailed proof of a generic Bogdanov-Takens bifurcation of codimension three in the neighborhood of those parameter values for which only one interior equilibrium exists. Moreover, numerical simulations also yield parameter regions with zero, one, and two limit cycles, respectively.