Asian Basic and Applied Research Journal

This study investigates the direction and stability of gestation-delay-induced Hopf bifurcation in a stage-structured prey–predator system incorporating predator-induced fear and herd behaviour. The model divides the prey population into juvenile and adult stages and describes predator recruitment using a gestation delay. A square-root functional response is employed to account for the protective effect of prey aggregation, while a predator-dependent fear term is incorporated into prey reproduction. Positivity and boundedness of solutions are established, and the extinction, predator-free and coexistence equilibria are derived. Local stability of the coexistence equilibrium in the absence of delay is investigated using the Routh–Hurwitz criterion. For positive gestation delay, a transcendental characteristic equation is derived and separated into real and imaginary components. Elimination of the delay yields a cubic characteristic polynomial in , whose positive root provides the Hopf frequency and corresponding critical delay. Numerical simulations confirm the existence of a positive root of the characteristic polynomial and demonstrate that the critical gestation delay increases with the fear coefficient over the parameter range considered. Center-manifold and normal-form analysis further shows that the parameter set considered produces a supercritical, orbitally stable Hopf bifurcation, with stable periodic solutions emerging as the gestation delay crosses the critical threshold. The findings demonstrate that predator-induced fear can delay the onset of delay-induced instability, while gestation delay remains a significant mechanism for generating population oscillations. The study provides a mathematical framework for examining the combined effects of behavioural responses, herd defence and reproductive time lags on predator–prey population dynamics.