Hopf Bifurcation and Stability Switching in a Dual‐Delay Predator–Prey System With Schooling Modified Predation and Sigmoidal Harvesting

ABSTRACT

Predator–prey models incorporating nonlinear predation, interspecific competition, harvesting, and reproductive delays provide important tools for understanding the mechanisms governing ecological stability and oscillatory dynamics. However, the combined effects of schooling‐modified Holling type‐IV predation, predator competition, biomass‐dependent harvesting, and multiple gestation delays remain insufficiently explored in one‐prey–two‐predator systems. To address this gap, we propose and analyze a nonlinear one‐prey–two‐predator model that integrates these mechanisms within a unified framework. For the non‐delayed system, the existence and uniqueness of solutions, positivity, boundedness, persistence, and coexistence are established, followed by an analysis of the stability of the interior equilibrium. The delayed model is then considered to represent gestation‐dependent predator reproduction, with the well‐posedness and boundedness of nonnegative solutions established before undertaking the stability and bifurcation analysis. The resulting characteristic equation is analyzed by treating the delays as bifurcation parameters, and the analysis demonstrates that critical delays can destabilize the coexistence equilibrium and generate Hopf bifurcations. For parameter set , Hopf bifurcations occur at when , at for , and at for . For parameter set , critical values for , for and for are obtained. The transversality conditions are satisfied at these critical values, while center‐manifold and normal‐form analyses show that the Hopf bifurcations considered are supercritical and generate asymptotically stable periodic solutions. Numerical simulations further reveal stable coexistence, damped and sustained oscillations, stability transitions, and, for sufficiently large second delays under , more complex aperiodic dynamics. These results demonstrate that gestation delays act as critical regulators of predator–prey stability, while nonlinear predation and harvesting modify the thresholds at which ecological oscillations emerge. The findings provide quantitative insights into how reproductive delays and harvesting‐related mechanisms influence the persistence and stability of interacting populations, with implications for ecological management and conservation.