  In this paper, we study the dynamics of a discrete Kolmogorov predator-prey
model with Ricker-type prey growth. We give the sufficient and necessary
condition to guarantee the existence and uniqueness of the positive fixed
point. Using the center manifold theory, we prove that the period-doubling
bifurcations can occur at the positive fixed point. Furthermore, our numerical
simulations reveal that the model can exhibit cascades of period-doubling
bifurcations leading to chaos, which is a significant difference from the
behavior of continuous predator-prey models. Despite the complexities of the
model dynamics, we are able to provide a criterion for the global stability of
the positive fixed point by using a geometric analysis of the nullclines.
