<p>This work investigates the dynamical properties of a discrete-time predator–prey system obtained from a continuous model using the piecewise constant argument method. We study the existence and stability of fixed points. It is shown that the system undergoes period-doubling and Neimark-Sacker bifurcations at the positive fixed point. Further numerical examples confirm these analytical findings and elucidate how variations in parameter values may result in transitions from stability to chaos and even species extinction. The findings emphasize the increased intricacy of discrete-time models in comparison with continuous-time models. Specifically, the discrete-time system not only undergoes bifurcations but also includes chaotic dynamics, underlining the subtle and multifaceted nature of ecological interactions in discrete time. This work emphasizes the need to use discrete models to accurately represent a wider variety of dynamic behaviors in ecological systems, such as bifurcations and chaos, which are often ignored in continuous-time models. For better knowledge of ecological dynamics and the correct modeling of actual ecological interactions, these insights are essential.</p>