Dynamical analysis of a delayed stage-structured predator-prey model with fear effect and prey refuge

This paper investigates a delayed stage-structured predator-prey model that incorporates prey refuge, fear effect, and intraspecific protection among predators. Different predation mechanisms are assigned to predator life stages: mature predators follow a Crowley-Martin functional response, while immature predators obey a Holling type II response. The analysis proves positivity, boundedness, and global stability of the coexistence equilibrium through a Lyapunov approach. The study reveals two major dynamical findings. First, predator maturation can trigger a transcritical bifurcation, determining whether predators persist or become extinct. Second, the delayed fear effect can induce a Hopf bifurcation, generating persistent population oscillations and destabilizing the ecosystem. Numerical simulations further show that prey refuge and protective behavior among predators promote species coexistence and enhance ecosystem resilience, whereas excessive fear delay may lead to instability or extinction. These results demonstrate that behavioral responses and stage structure jointly play a crucial role in maintaining ecological stability, providing new biological insight into predator-prey coexistence and conservation management.