The authors investigate a planar mapping that, by discretization, originates from a predator-prey differential system whose righthand side is composed of cubic terms as well as a rational term expressing a satiation effect of the predator. Including the step size of the discretization, the mapping depends on six parameters. Using the center manifold at the unique positive fixed point, and choosing the step size as the bifurcation parameter, lengthy conditions for period doubling and Hopf bifurcations are established. Extensive numerical calculations for different choices of parameters complete these results, yielding a series of bifurcation diagrams and phase portraits with corresponding Lyapunov exponents, where alternately also the death rate of the predator plays the role of bifurcation parameter. The results show a variety of interesting dynamic behavior, including invariant cycles, periodic orbits of various periods, period doubling and halving, sudden onset and disappearance of chaotic dynamics, and intermittency.