Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Structured coalescents, coagulation equations and multi-type branching processes

Fernando Cordero, Sophia-Marie Mellis, Emmanuel Schertzer

TL;DR

The paper links a multi-type structured coalescent to multi-dimensional coagulation equations under two sampling regimes: critical (N_K ~ K) and large (N_K >> K). By scaling time by 1/K and appropriately scaling block configurations, the empirical-measure process converges to a d-dimensional coagulation equation, which admits a stochastic representation via multi-type branching processes (CSBP/ Feller diffusion) in the respective regimes. In the critical case, the limit solves a discrete coagulation equation with migration, while in the large-sample regime the limit is a continuous coagulation equation whose solution is represented by the entrance law of a multi-type diffusion, ensuring uniqueness. The work also develops a robust generator-convergence framework, moment bounds, and tightness arguments, and lays out conjectures for the site-frequency spectrum in fast-migration, large-sample settings, linking genealogies to deterministic mass-transport equations.

Abstract

Consider a structured population consisting of $d$ colonies, with migration rates proportional to a positive parameter $K$. We sample $N_K$ individuals, distributed evenly across the $d$ colonies, and trace their ancestral lineages backward in time. Within each colony, we assume that any pair of ancestral lineages coalesces at a constant rate, as in Kingman's coalescent. We identify each ancestral lineage with the set, or block, of its sampled descendants, and we encode the state of the system using a $d$-dimensional vector of empirical measures; the $i$-th component records the blocks present in colony $i$ together with the initial locations of the lineages composing each block. We are interested in the asymptotic behavior of the process of empirical measures as $K \to \infty$. We consider two regimes: the critical sampling regime, where $N_K \sim K$, and the large-sample regime, where $N_K \gg K$. After an appropriate time rescaling, we show that the process of empirical measures converges to the solution of a $d$-dimensional coagulation equation. In the critical sampling regime, the solution can be represented in terms of a multi-type branching process. In the large-sample regime, the solution can be represented in terms of the entrance law of a multi-type continuous-state branching process.

Structured coalescents, coagulation equations and multi-type branching processes

TL;DR

The paper links a multi-type structured coalescent to multi-dimensional coagulation equations under two sampling regimes: critical (N_K ~ K) and large (N_K >> K). By scaling time by 1/K and appropriately scaling block configurations, the empirical-measure process converges to a d-dimensional coagulation equation, which admits a stochastic representation via multi-type branching processes (CSBP/ Feller diffusion) in the respective regimes. In the critical case, the limit solves a discrete coagulation equation with migration, while in the large-sample regime the limit is a continuous coagulation equation whose solution is represented by the entrance law of a multi-type diffusion, ensuring uniqueness. The work also develops a robust generator-convergence framework, moment bounds, and tightness arguments, and lays out conjectures for the site-frequency spectrum in fast-migration, large-sample settings, linking genealogies to deterministic mass-transport equations.

Abstract

Consider a structured population consisting of colonies, with migration rates proportional to a positive parameter . We sample individuals, distributed evenly across the colonies, and trace their ancestral lineages backward in time. Within each colony, we assume that any pair of ancestral lineages coalesces at a constant rate, as in Kingman's coalescent. We identify each ancestral lineage with the set, or block, of its sampled descendants, and we encode the state of the system using a -dimensional vector of empirical measures; the -th component records the blocks present in colony together with the initial locations of the lineages composing each block. We are interested in the asymptotic behavior of the process of empirical measures as . We consider two regimes: the critical sampling regime, where , and the large-sample regime, where . After an appropriate time rescaling, we show that the process of empirical measures converges to the solution of a -dimensional coagulation equation. In the critical sampling regime, the solution can be represented in terms of a multi-type branching process. In the large-sample regime, the solution can be represented in terms of the entrance law of a multi-type continuous-state branching process.
Paper Structure (29 sections, 20 theorems, 163 equations, 6 figures)

This paper contains 29 sections, 20 theorems, 163 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Model and main results
  3. The model
  4. The empirical measure
  5. The one-dimensional case (the critical sampling regime)
  6. Convergence
  7. Stochastic representation
  8. Conjectures on the site-frequency spectrum
  9. Proof strategy for the main results
  10. Generator convergence and moment bounds
  11. The action of the generator
  12. Convergence of the generators
  13. Moment bounds
  14. Comparison results
  15. The coupling
  16. ...and 14 more sections

Key Result

Theorem 2.4

Assume that $\gamma_K \to c$ as $K \to \infty$. If Assumption assu1 holds, then $(\mu_i^{K})_{i \in [d]}$ converges weakly, as $K \to \infty$, to the solution of the $d$-dimensional discrete coagulation equation for all $t \ge 0$, $\boldsymbol{n} \in \mathbb{N}_0^d \setminus \{\boldsymbol{0}\}$ and $i \in [d]$, where $(\boldsymbol{e}_i)_{i \in [d]}$ denotes the canonical basis of $\mathbb{R}^d$.

Figures (6)

  • Figure 1: An illustration of the structured coalescent for $d=3$, $N_{K}=10$. Blocks are classified according to their colors, which code for the different colonies. A change in color represents a migration event. Specifically, multi-colored squares indicate a migration from colony $i$ to colony $j$, where the upper color corresponds to the origin colony $i$, and the lower color to the destination colony $j$.
  • Figure 2: An illustration of the effect of the time scaling $t\mapsto t/K$.
  • Figure 3: The coalescent with mutations for $d=3$, $N_{K}=10$. Black circles indicate mutations occurring at constant rate $\theta$ per block.
  • Figure 4: An illustration of the two types of transitions. Left: coalescence (the block color remains the same while the resulting configuration is the sum of the configurations of the two merging blocks). Right: migration (the block color changes while the configuration remains the same).
  • Figure 5: An illustration of the two scenarios. Left: mono-chromatic scenario (configurations at time $\varepsilon/K$: $(5,0,0)$, $(11,0,0)$, $(4,0,0)$ in colony $1$, $(0,5,0)$, $(0,7,0)$, $(0,4,0)$, $(0,7,0)$ in colony $2$, $(0,0,7)$, $(0,0,9)$, $(0,0,7)$ in colony $3$). Right: poly-chromatic scenario (configurations at time $\varepsilon/K$: $(11,0,0)$, $(0,6,0)$, $(0,2,7)$ in colony $1$, $(5,7,0)$, $(0,0,3)$ in colony $2$, $(4,8,0)$, $(0,0, 13)$ in colony $3$).
  • ...and 1 more figures

Theorems & Definitions (49)

  • Definition 2.1: Colored partition
  • Example 2.2
  • Theorem 2.4: Critical sampling
  • Theorem 2.5: Large sampling
  • Remark 2.6
  • Theorem 2.7
  • Remark 2.8
  • Theorem 2.9
  • Remark 2.10
  • Remark 2.11: Stochastic representation at equilibrium
  • ...and 39 more