Stability and Bifurcation in a Predator–Prey Model with Strong Allee Effects and Gestation Delay

This work develops and analyzes a predator–prey system in which prey growth is regulated by a generalized strong Allee function, enabling a broad representation of Allee-type mechanisms within a unified mathematical framework. The predator population incorporates a discrete gestation delay, capturing realistic time-lagged reproductive responses. After establishing positivity and boundedness of solutions, we characterize all equilibria and examine their local stability, demonstrating how the generalized Allee function significantly shapes coexistence conditions and modifies stability thresholds. The system exhibits both transcritical and Hopf bifurcations. For the non-delayed model, we derive the Hopf bifurcation threshold and compute the first Lyapunov coefficient, thereby determining whether the bifurcation is supercritical or subcritical and establishing the stability of the emerging periodic orbits. For the delayed system, we obtain the delay-induced Hopf threshold and show that increasing gestation delay can destabilize an otherwise stable coexistence state and generate sustained oscillations. Analytical results are supported by numerical simulations illustrating extinction, persistence, bistability, stable limit cycles, and delay-driven oscillatory dynamics. Overall, the study reveals the combined influence of predator mortality, predation intensity, Allee strength, and reproductive delay on the qualitative behavior of predator–prey interactions.