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 τ 1 acting through density-dependent self-regulation, and a predator conversion delay τ 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 τ 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 (τ 1,τ 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 δ ≈ 4.6692 . Aitken Δ 2 extrapolation places the accumulation point of the cascade at τ 1∞ ≈ 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.
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BibTeX
@article{alawfi2026hopf,
title = {{Hopf bifurcations and period-doubling cascade in a two-delay predator–prey model with Holling type II response}},
author = {Asim Alawfi},
journal = {Scientific Reports},
year = {2026},
month = oct,
doi = {10.1038/s41598-026-74386-7},
url = {https://www.nature.com/articles/s41598-026-74386-7},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Hopf bifurcations and period-doubling cascade in a two-delay predator–prey model with Holling type II response
AU - Asim Alawfi
PY - 2026
DA - 2026/10/04
JO - Scientific Reports
DO - 10.1038/s41598-026-74386-7
UR - https://www.nature.com/articles/s41598-026-74386-7
AB - We investigate the dynamics of a predator–prey model with Holling type II functional response, and two ecologically distinct delays: a prey maturation delay τ _1 acting through density-dependent self-regulation, and a predator conversion delay τ _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 τ _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 (τ _1,τ _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 δ ≈ 4.6692 . Aitken Δ ^2 extrapolation places the accumulation point of the cascade at τ _1^∞ ≈ 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.
SN - 2045-2322
ER -