Hopf bifurcations and period-doubling cascade in a two-delay predator–prey model with Holling type II response

We investigate the dynamics of a predator–prey model with Holling type II functional response, and two ecologically distinct delays: a prey maturation delay (tau _1) acting through density-dependent self-regulation, and a predator conversion delay (tau _2) acting through delayed recruitment. After establishing well-posedness, positivity, and uniform boundedness of solutions, we perform a detailed numerical bifurcation analysis using DDE-Biftool. As (tau _1) increases, the coexistence equilibrium undergoes a sequence of Hopf bifurcations whose criticality alternates between subcritical and supercritical, with the transition organised by generalized Hopf (Bautin) bifurcations in the ((tau _1,tau _2))-plane. Continuation of the emerging periodic branch reveals a fold of periodic orbits that produces bistability between coexistence and large-amplitude cycles, and a complete period-doubling cascade through period-2, period-4, period-8, period-16, and period-32 orbits. The Floquet multiplier at each detected period-doubling bifurcation equals (-1) to numerical accuracy, and the spacing of successive doublings yields empirical Feigenbaum ratios that converge toward the universal value (delta approx 4.6692). Aitken (Delta ^2) extrapolation places the accumulation point of the cascade at (tau _1^{infty }approx 18.64), beyond which periodic continuation no longer resolves the dynamics and direct numerical integration is consistent with aperiodic oscillation. These results show that the combination of maturation and conversion delays in a biologically interpretable predator–prey model can generate stability switching, bistability, and a classical Feigenbaum route to complex temporal variability.