<p>This study incorporates the Beddington–DeAngelis functional response and prey refuges into a prey–predator model. With this incorporation, we provide conditions for the positivity and boundedness of the solutions. Observing the dynamics of the model through fixed points, we derived various conditions that confirm different topological classifications at these fixed points. We also depict the graphs to verify the existence of the calculated topological types numerically. We have derived different conditions that show how the prey refuges impact the stability of all the fixed points. For the positive fixed point, we observed the consistent behavior that the greater the number of prey refuges, the more stabilizing the model. In contrast, the fewer prey refuges cause instability in the model. Additionally, the complexities in the dynamics of the model are observed through bifurcation theory. The theoretical and numerical results confirm that the two types of bifurcation patterns cause complexities in the model, i.e., the Neimark-Sacker and period-doubling bifurcations. These bifurcations result in the chaoticness of the dynamics of the model. The chaoticness of the dynamics is confirmed through Marotto’s sense and Lyapunov exponent. The two-parameter Lyapunov exponent plots affirm that the bifurcation causes chaos in the model. We employ a simple hybrid control technique to regulate this unpredictable behavior due to chaos.</p>