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Moments of density-dependent branching processes and their genealogy

Mathilde André, Félix Foutel-Rodier, Emmanuel Schertzer

TL;DR

This work develops a moment-based framework to study genealogies in density-dependent branching processes, where population size fluctuates around a stable equilibrium instead of being fixed. By encoding genealogies as planar forests and employing a spine-based exponential change of measure, the authors derive a recursive many-to-few formula for penalized planar moments, which reduces higher-order genealogical information to lower-order moments. Under a finite second moment assumption and near-equilibrium scaling, they prove that the genealogy of a uniform sample converges to Kingman’s coalescent with rate $q(1)m_2(1)$ as the carrying capacity $K$ grows, with the population density exhibiting Brownian fluctuations around the equilibrium. The methodology extends coalescent limit techniques to density-dependent ecosystems and provides a robust approach that can potentially be adapted to structured populations and more complex ecological interactions.

Abstract

A density-dependent branching process is a particle system in which individuals reproduce independently, but in a way that depends on the current population size. This feature can model a wide range of ecological interactions at the cost of breaking the branching property. We propose a general approach for studying the genealogy of these models based on moments. Building on a recent work of Bansaye, we show how to compute recursively these moments in a similar spirit to the many-to-few formula in the theory of branching processes. These formulas enable one to deduce the convergence of the genealogy by studying the population density, for which stochastic calculus techniques are available. As a first application of these ideas, we consider a density-dependent branching process started close to a stable equilibrium of the ecological dynamics. We show that, under a finite second moment assumption, its genealogy converges to Kingman's coalescent when the carrying capacity of the population goes to infinity.

Moments of density-dependent branching processes and their genealogy

TL;DR

This work develops a moment-based framework to study genealogies in density-dependent branching processes, where population size fluctuates around a stable equilibrium instead of being fixed. By encoding genealogies as planar forests and employing a spine-based exponential change of measure, the authors derive a recursive many-to-few formula for penalized planar moments, which reduces higher-order genealogical information to lower-order moments. Under a finite second moment assumption and near-equilibrium scaling, they prove that the genealogy of a uniform sample converges to Kingman’s coalescent with rate as the carrying capacity grows, with the population density exhibiting Brownian fluctuations around the equilibrium. The methodology extends coalescent limit techniques to density-dependent ecosystems and provides a robust approach that can potentially be adapted to structured populations and more complex ecological interactions.

Abstract

A density-dependent branching process is a particle system in which individuals reproduce independently, but in a way that depends on the current population size. This feature can model a wide range of ecological interactions at the cost of breaking the branching property. We propose a general approach for studying the genealogy of these models based on moments. Building on a recent work of Bansaye, we show how to compute recursively these moments in a similar spirit to the many-to-few formula in the theory of branching processes. These formulas enable one to deduce the convergence of the genealogy by studying the population density, for which stochastic calculus techniques are available. As a first application of these ideas, we consider a density-dependent branching process started close to a stable equilibrium of the ecological dynamics. We show that, under a finite second moment assumption, its genealogy converges to Kingman's coalescent when the carrying capacity of the population goes to infinity.

Paper Structure

This paper contains 16 sections, 13 theorems, 121 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Model and main result
  4. Proof heuristics and outline
  5. The moments of density-dependent branching processes
  6. Encoding genealogies
  7. Martingale change of measure
  8. The moment formula
  9. Stochastic analysis of the population size
  10. The population density under Pk
  11. Convergence to the equilibrium density
  12. Convergence to Brownian motion
  13. Estimation of the Feynman--Kac term
  14. Convergence of the genealogy
  15. Limit of the penalized moments
  16. ...and 1 more sections

Key Result

Theorem 1.1

Suppose that hyp:equilibrium and hyp:moments hold and let $t > 0$, $k\geqslant 1$.

Figures (1)

  • Figure 1: In blue: illustration of a planar coalescent $\Pi\in \mathbb{G}^9$ and $b=4$ branching points generated from the sequence of mergers $\bm c=((4,3),(3,2),(4,3),(2,2))$ and $\bm\tau = ((\tau_j)_{1\leqslant j\leqslant 4})$. The planar coalescent $\Theta_{\tau_1}(\Pi)$ obtained from pruning at the first coalescing point is pictured in red. According to \ref{['eq:pruning:coalescent']}, $\Theta_{\tau_1}(\Pi)\in\mathbb{G}^{6}$ and is generated from $(\bm c', \bm\tau')= (( c_2, c_3, c_4),~(\tau_2-\tau_1, \tau_3-\tau_1, \tau_4-\tau_1))$

Theorems & Definitions (30)

  • Theorem 1.1: Convergence of the genealogies
  • Remark 1.2
  • Definition 2.1
  • Lemma 2.2: Martingale change of measure
  • proof : Proof of \ref{['lem:martingale_change_measure_kspine']}
  • Theorem 2.3: Induction for the planar moment measures
  • Remark 2.4: Many-to-few
  • Remark 2.5
  • proof : Proof of \ref{['thm:recursion_moment']}
  • Proposition 3.1: Convergence of the population density
  • ...and 20 more