{
  "abstract": "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 $g'(x^\\ast)>0$: 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 $\\dot u=-vH$, $\\dot v=W$, 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 $\\ell_1$ 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 $\\partial\\ell_1/\\partial f_{1yyy}=\\partial\\ell_1/\\partial f_{2xxx}=0$. Straightening the nullcline removes two further slots, while within the Gause class the reparametrization $φ\\mapsto g$ 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 $\\ell_1=\\ell_1^{\\mathrm{aff}}+Δ$, where $Δ=Λ\\bigl(f_{1xyy}+ω_0^{-2}\\langle\\partial_x f,\\nabla D\\rangle f_{1yy}\\bigr)$ and $Λ>0$. On the Hopf locus, $\\langle\\partial_x f,\\nabla D\\rangle=J_{11}τ'+(ω_0^2/P)D_y$. 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 $\\ell_1,\\ell_1^{\\mathrm{aff}},Δ,Λ$ scale by $|c|^2$ under $q\\mapsto cq$; their signs and ratios are unchanged.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.22047",
  "author_total": 1,
  "authors": [
    {
      "name": "E. Chan-López"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3djkv1bxagpmh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.22047",
  "fulltext_url": null,
  "id": "3djkv1bxagpmh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.22047",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2739|Mathematical and Theoretical Epidemiology and Ecology Models"
    ]
  ],
  "openalex": "W7204170205",
  "pdf_url": "https://arxiv.org/pdf/2608.22047",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3djkv1bxagpmh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2608.22047",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.22047"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The trace field on the prey nullcline: criticality of planar Hopf bifurcations on the admissible branch",
  "title_text": "The trace field on the prey nullcline: criticality of planar Hopf bifurcations on the admissible branch",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/3djkv1bxagpmh",
  "venue": "arXiv",
  "versions": []
}