Hopf bifurcation and stability in a fractional-order age-structured predator-prey model with Allee effect and dual delays

This paper investigates the dynamical behavior and Hopf bifurcation phenomena in a fractional-order age-structured predator-prey system incorporating a strong Allee effect and dual discrete delays. The model captures ecological memory through Caputo derivatives and time-lagged maturation and predation processes. By employing fractional integrated semigroup theory and the Laplace transform of fertility kernels, we establish the existence, uniqueness, and positivity of solutions, followed by a rigorous linearized stability analysis around the interior equilibrium. Analytical conditions for Hopf bifurcation onset are derived in terms of fractional order α and delay parameters τ1, τ2. Numerical simulations, based on the Adams-Bashforth-Moulton predictor-corrector scheme, confirm the theoretical results and illustrate complex transitions from periodic to chaotic oscillations as fractional memory weakens or delay increases. The results reveal that fractional damping stabilizes the system, while dual delays act as bifurcation drivers leading to quasiperiodicity and chaos. This study provides a unified theoretical framework linking memory, time delay, and nonlinear ecological feedback, offering new insights into the control of chaotic dynamics in fractional biological systems.