{
  "abstract": "For birth--death processes with a strong Allee effect, long survival near a positive stable equilibrium does not by itself determine the principal spectral gap. We identify its large-population limit when the birth and death rates at population size $n$ are $n\\widetildeλ(n/K)$ and $n\\widetildeμ(n/K)$, where $K$ is the population scale and $\\widetildeλ,\\widetildeμ$ are the per-capita rates. Write $V(x)=x(\\widetildeλ(x)-\\widetildeμ(x))$ for the deterministic drift, with unstable threshold $x_1$ and positive stable equilibrium $x_2$. For the two smallest eigenvalues $ρ_1(K)<ρ_2(K)$ of the negative generator killed at extinction, we prove \\[ \\lim_{K\\to\\infty}\\bigl(ρ_2(K)-ρ_1(K)\\bigr) =\\min\\{\\widetildeμ(0)-\\widetildeλ(0),V'(x_1),-V'(x_2)\\}, \\] so the limit depends on the linearization rates at all three equilibria. Each term can be the unique minimum even when $x_1$ and $x_2$ are fixed. The proof combines a frozen boundary model with two local oscillator limits through discrete Ismagilov--Morgan--Simon (IMS) localization. A uniform comparison of the global and stable local ground states controls the orthogonality constraint in the variational lower bound; projected local trial vectors give the matching upper bound. The result determines the limiting $L^2$ spectral gap and optimal Poincaré constant of the $Q$-process, which describes conditioning on indefinite survival. Combined with principal-eigenvalue asymptotics from a companion work, it separates an exponentially growing quasi-stationary mean extinction time from a spectral relaxation time with a finite positive limit.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31808",
  "author_total": 1,
  "authors": [
    {
      "name": "Dun Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1ek6gq2pf40t1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31808",
  "fulltext_url": null,
  "id": "1ek6gq2pf40t1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31808",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7214702738",
  "pdf_url": "https://arxiv.org/pdf/2609.31808",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1ek6gq2pf40t1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.31808",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31808"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The principal spectral gap for birth--death processes with a strong Allee effect",
  "title_text": "The principal spectral gap for birth--death processes with a strong Allee effect",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/1ek6gq2pf40t1",
  "venue": "arXiv",
  "versions": []
}