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

Abstract 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.