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Genealogies under logistic growth

Ruairi Garrett, Julio Ernesto Nava Trejo

TL;DR

This work identifies the genealogies of a logistic branching process with large carrying capacity by constructing a modified lookdown representation and applying stochastic averaging across three offspring-tail regimes. It proves that the ancestral partitions converge, under appropriate time rescalings, to Kingman’s coalescent (finite-variance case), the Beta$(2-\alpha,\alpha)$-coalescent (alpha-stable tail), or Bolthausen–Sznitman coalescent (Neveu-type heavy tail), with explicit effective population sizes and Lambda measures. A Brownian-spatial extension on the torus shows convergence to the Brownian spatial $\Lambda$-coalescent, linking forward-time type-frequency convergence (to $\Lambda$-Fleming–Viot) to backward-time genealogies via a pathwise duality framework. The results extend prior forward-in-time convergence (F25) to full genealogical structure, offering a rigorous bridge between reproduction-competition dynamics and coalescent processes. This advances understanding of how the tail of offspring-distribution shapes genealogies in populations under logistic constraints and provides tools for studying spatially structured populations under competition.

Abstract

We derive the asymptotic behaviour of the genealogy of a logistic branching process in the setting where the equilibrium population size is large. In three regimes on the tail of the offspring distribution we recover the Kingman, $\text{Beta}(2-α, α)$ and Bolthausen-Sznitman coalescents as a scaling parameter governing the population size is taken to infinity, the deduction going via the convergence in distribution of a modified lookdown construction. This resolves a question asked in arxiv:2501.16837 who studied the same population process forwards in time and showed convergence of the type frequency process to the corresponding $Λ$-Fleming-Viot process in each regime.

Genealogies under logistic growth

TL;DR

This work identifies the genealogies of a logistic branching process with large carrying capacity by constructing a modified lookdown representation and applying stochastic averaging across three offspring-tail regimes. It proves that the ancestral partitions converge, under appropriate time rescalings, to Kingman’s coalescent (finite-variance case), the Beta-coalescent (alpha-stable tail), or Bolthausen–Sznitman coalescent (Neveu-type heavy tail), with explicit effective population sizes and Lambda measures. A Brownian-spatial extension on the torus shows convergence to the Brownian spatial -coalescent, linking forward-time type-frequency convergence (to -Fleming–Viot) to backward-time genealogies via a pathwise duality framework. The results extend prior forward-in-time convergence (F25) to full genealogical structure, offering a rigorous bridge between reproduction-competition dynamics and coalescent processes. This advances understanding of how the tail of offspring-distribution shapes genealogies in populations under logistic constraints and provides tools for studying spatially structured populations under competition.

Abstract

We derive the asymptotic behaviour of the genealogy of a logistic branching process in the setting where the equilibrium population size is large. In three regimes on the tail of the offspring distribution we recover the Kingman, and Bolthausen-Sznitman coalescents as a scaling parameter governing the population size is taken to infinity, the deduction going via the convergence in distribution of a modified lookdown construction. This resolves a question asked in arxiv:2501.16837 who studied the same population process forwards in time and showed convergence of the type frequency process to the corresponding -Fleming-Viot process in each regime.

Paper Structure

This paper contains 18 sections, 15 theorems, 92 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Logistic branching and $(\Lambda, \mathcal{Q})$-Fleming-Viot processes
  3. Coalescents, spatial coalescents and Brownian spatial $\Lambda$-coalescents
  4. Main results
  5. Related work and alternative approaches
  6. Outline of the article
  7. Modified lookdown construction
  8. Modified lookdown construction for the prelimit
  9. Limiting lookdown process
  10. Convergence of the lookdown processes
  11. Finite variance regime: Kingman's coalescent
  12. Beta coalescent regime
  13. Neveu’s branching process with logistic competition: Bolthausen-Sznitman coalescent
  14. Deduction of Theorem \ref{['Thm:raphael']} and Theorem \ref{['thm:main']}
  15. Appendix
  16. ...and 3 more sections

Key Result

Theorem 1.3

(F25, Theorems 2, 3, 4) Specialise to the case $\mathcal{Q}_K = 0$ (no mutation).

Figures (1)

  • Figure 1: In \ref{['fig:sub1']}, individuals born during the birth event from the second individual are placed in the second, third, forth and fifth level, the rest of individuals are pushed upwards retaining their original order. In \ref{['fig:sub2']} the individual in the top level is removed from the population.

Theorems & Definitions (34)

  • Definition 1.1: Logistic branching process with neutral mutation
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 1.5: Spatial coalescent, Brownian spatial coalescent
  • Definition 1.6: Ancestral partition process, spatial ancestral partition process
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 2.1
  • ...and 24 more