<p>In this study, we extend the Beddington–DeAngelis predator–prey model to rigorously analyze interactions under three critical ecological factors: fear effects, dynamic prey refuges, and supplementary food for predators. Our model incorporates a functional response where refuge effectiveness is inversely related to predator density, providing a more realistic representation of prey behavior. We investigate the model’s fundamental properties, including population persistence, boundedness, and the local and global stability of its equilibrium points. Our analysis reveals a rich array of dynamical behaviors, including subcritical Hopf, transcritical, and saddle-node bifurcations, which delineate transitions in population stability. Crucially, the model sheds light on the paradox of enrichment, demonstrating how increased prey resources can lead to system instability, a phenomenon often observed in real ecosystems. Through extensive numerical simulations, we corroborate all theoretical results, underscoring the ecological relevance of considering fear, dynamic refuge, and supplementary food in understanding predator–prey dynamics and their implications for ecosystem management</p>