The trace field on the prey nullcline: criticality of planar Hopf bifurcations on the admissible branch
Abstract
Along the prey nullcline y=g(x) of a planar system, the Jacobian entry satisfies J11=P g', where P=−f1y|γ. Under P>0 and predator self-damping, the Hopf trace condition forces g'(x∗)>0: critical points of g are spectral barriers, so the Hopf locus lies on ascending branches. This geometric localization determines where oscillatory instability can occur; we determine how the bifurcation unfolds there. Let D=f1x+f2y and τ=D|γ. Using nullcline coordinates (u=x,v=y−g(x)) and Hadamard's lemma gives u̇=−vH, v̇=W, with H(u,0)=P. The mixed jet of W is determined by the jets of τ and ν=f2|γ, yielding a closed five-term formula for the first Lyapunov coefficient ℓ1 in trace-field form, valid for predator-dependent functional responses. Three distinct reductions of the cubic jet are established. In general planar Hopf bifurcations, eigenvector pairing gives ∂ℓ1/∂ f1yyy=∂ℓ1/∂ f2xxx=0. Straightening the nullcline removes two further slots, while within the Gause class the reparametrization φ↦ g removes p'''. They have different scopes. For predator-dependent responses, comparison with the y-affine truncation of the $3$-jet at fixed linear part gives ℓ1=ℓ1aff+Δ, where Δ=Λ(f1xyy+ω0−2⟨∂x f,∇ D⟩ f1yy) and Λ>0. On the Hopf locus, ⟨∂x f,∇ D⟩=J11τ'+(ω02/P)Dy. Thus the same quantity J11=Pg' that localizes the bifurcation at first order reappears at third order as the weight of the tangential trace derivative. The bridge identity is independent of eigenvector normalization, while ℓ1,ℓ1aff,Δ,Λ scale by |c|2 under q↦ cq; their signs and ratios are unchanged.
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BibTeX
@misc{chanlopez2026trace,
title = {{The trace field on the prey nullcline: criticality of planar Hopf bifurcations on the admissible branch}},
author = {E. Chan-López},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.22047},
eprint = {2608.22047},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22047},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The trace field on the prey nullcline: criticality of planar Hopf bifurcations on the admissible branch
AU - E. Chan-López
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2608.22047
UR - https://arxiv.org/abs/2608.22047
AB - Along the prey nullcline y=g(x) of a planar system, the Jacobian entry satisfies J_11=P g', where P=−f_1y|_γ. Under P>0 and predator self-damping, the Hopf trace condition forces g'(x^∗)>0: critical points of g are spectral barriers, so the Hopf locus lies on ascending branches. This geometric localization determines where oscillatory instability can occur; we determine how the bifurcation unfolds there. Let D=f_1x+f_2y and τ=D|_γ. Using nullcline coordinates (u=x,v=y−g(x)) and Hadamard's lemma gives u̇=−vH, v̇=W, with H(u,0)=P. The mixed jet of W is determined by the jets of τ and ν=f_2|_γ, yielding a closed five-term formula for the first Lyapunov coefficient ℓ_1 in trace-field form, valid for predator-dependent functional responses. Three distinct reductions of the cubic jet are established. In general planar Hopf bifurcations, eigenvector pairing gives ∂ℓ_1/∂ f_1yyy=∂ℓ_1/∂ f_2xxx=0. Straightening the nullcline removes two further slots, while within the Gause class the reparametrization φ↦ g removes p'''. They have different scopes. For predator-dependent responses, comparison with the y-affine truncation of the $3$-jet at fixed linear part gives ℓ_1=ℓ_1^aff+Δ, where Δ=Λ(f_1xyy+ω_0^−2⟨∂_x f,∇ D⟩ f_1yy) and Λ>0. On the Hopf locus, ⟨∂_x f,∇ D⟩=J_11τ'+(ω_0^2/P)D_y. Thus the same quantity J_11=Pg' that localizes the bifurcation at first order reappears at third order as the weight of the tangential trace derivative. The bridge identity is independent of eigenvector normalization, while ℓ_1,ℓ_1^aff,Δ,Λ scale by |c|^2 under q↦ cq; their signs and ratios are unchanged.
ER -