Mathematical Analysis of a Delayed Predator–Prey System with Prey Refuge: Bifurcations and Stability Thresholds

This study investigates the complex interplay between prey refuge and time delay in a predator–prey system governed by a Holling Type II functional response. Building upon the work of Elmojtaba et al. [22], we establish that while increasing environmental carrying capacity triggers a "Paradox of Enrichment" via Hopf bifurcation and discrete time delays induce destabilizing limit-cycle oscillations, the utilization of a constant prey refuge acts as a vital regulatory "buffer." Analytical proofs and numerical simulations demonstrate that the system transitions from predator extinction to stable coexistence through a transcritical bifurcation, and that increasing the refuge constant can effectively counteract the destabilizing effects of both enrichment and gestation lags. Ultimately, our findings suggest that maintaining physical refuges is essential for ecosystem resilience, as they expand the region of asymptotic stability and prevent species extinction in otherwise oscillatory environments.