Stability and Bifurcation in an Age-Structured Predator–Prey System with Beddington–Deangelis Response, Constant Prey Harvesting, and Two Delays

This work investigates the long-term dynamics of a novel predator-prey system that explicitly accounts for age structure, Beddington-Deangelis functional response, constant-yield prey harvesting, and two distinct time delays. The first delay, τ1 denotes the predator's maturation period; the second, τ2 reflects the gestation delay between prey consumption and predator reproduction. Firstly, we recast the system into an abstract non-densely defined Cauchy problem. Then the existence of solutions and the uniqueness of positive equilibrium are discussed. By analyzing the distribution of the corresponding eigenvalues, the rigorous establishment of asymptotic stability for the boundary equilibria are achieved. We further present a detailed Hopf bifurcation analysis that simultaneously incorporates both time-delay parameters, employing stability-switching curves to clarify how varying delays reshape the system's dynamic behavior. Lastly, the practical implications of these theoretical findings are demonstrated through several numerical examples.