{
  "abstract": "We study a trait-structured population model incorporating mutation, selection and density-dependent regulation on a countably infinite trait space. The underlying stochastic process is a continuous-time Markov chain in which individuals reproduce at a trait-independent rate, offspring traits are determined by a mutation kernel on $\\mathbb Z$, mortality depends on trait, and births are progressively suppressed as the population approaches a fixed population ceiling. Using results for density-dependent Markov population processes with countably many types, we derive a deterministic approximation in the form of an infinite system of nonlinear differential equations. We establish existence and positive invariance of solutions, and investigate the equilibrium structure of the deterministic system. A fundamental distinction emerges between bounded and confining mortality profiles. When mortality remains bounded, mutation may continually transport mass through the trait space and a stationary trait distribution need not exist. In contrast, when mortality increases without bound as the absolute value of the trait index becomes large, the operator $L=D^{-1}P$, where $P$ and $D$ govern mutation and mortality, is compact. By combining compactness with Kre\\uın-Rutman theory for compact positive operators on Banach lattices, we show that $L$ has an algebraically simple principal eigenvalue with a strictly positive eigenvector, and we derive a threshold condition for the existence of a non-zero equilibrium. In this regime the equilibrium is unique, and its trait distribution is determined by the principal eigenvector of $L$. Numerical experiments support the theoretical results and illustrate the contrasting behaviours associated with bounded and confining mortality profiles.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02086",
  "author_total": 1,
  "authors": [
    {
      "name": "Phil. Pollett"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f2y0byynrybqk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02086",
  "fulltext_url": null,
  "id": "f2y0byynrybqk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02086",
  "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.PR"
    ],
    [
      "arxiv",
      "q-bio.PE"
    ],
    [
      "openalex",
      "2739|Mathematical and Theoretical Epidemiology and Ecology Models"
    ]
  ],
  "openalex": "W7207582826",
  "pdf_url": "https://arxiv.org/pdf/2609.02086",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f2y0byynrybqk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02086",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02086"
    }
  ],
  "subjects": [
    {
      "field": "biology",
      "origin": "arxiv:q-bio.PE",
      "subfield": "evolutionary-biology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Mutation--selection balance on an infinite trait space: confinement, drift and equilibrium",
  "title_text": "Mutation--selection balance on an infinite trait space: confinement, drift and equilibrium",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:21Z",
  "url": "https://research.draconitic.com/paper/f2y0byynrybqk",
  "venue": "arXiv",
  "versions": []
}