Stability and Bifurcation in a Delayed Predator–Prey Model with Environmental Stress and Dynamic Carrying Capacity

We develop a delayed predator–prey model that integrates nonlinear environmental stress and a dynamic carrying capacity into the prey’s growth function. The model introduces a discrete time delay in the predator’s response, capturing ecological lags such as gestation or behavioral adaptation. Unlike previous studies, our framework couples environmental degradation and biological delay, two destabilizing forces often treated in isolation, to examine their combined impact on ecosystem dynamics. We derive analytical conditions for the local and global stability of equilibrium states and identify critical Hopf bifurcation thresholds as functions of the delay and environmental stress level. The model reveals how interactions between these parameters govern transitions between extinction, stable coexistence, and sustained oscillations. Notably, we extend the classical Wangersky–Cunningham framework by incorporating feedback-regulated carrying capacity. Numerical simulations validate the theoretical predictions and map resilience boundaries that highlight tipping points in ecological regimes. The results have practical implications for biodiversity conservation and adaptive ecosystem management, especially under anthropogenic stress and delayed biological feedbacks.