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Lookdown construction for a Moran seed-bank model

Maria Clara Fittipaldi, Adrián González Casanova, Julio Ernesto Nava

TL;DR

The paper develops a Markovian lookdown framework for a Moran model with a seed-bank, allowing the active/dormant population proportions to vary while keeping the total population fixed. It proves that the SB-lookdown empirical measure matches the SB-Moran model's empirical measure via the Markov Mapping theorem and derives the ancestry structure through a seed-bank coalescent embedded in the lookdown construction. The results characterize the time to the most recent common ancestor (TMRCA) as asymptotically governed by the largest inactivity period, with a Gumbel limit after centering, and show that the fixation time of a single beneficial mutant, conditioned to invade, shares the same limiting Gumbel distribution and scales as $\ln(N)$. Collectively, these findings connect seed-bank dynamics to explicit coalescent and fixation-time laws, enabling precise asymptotics for genealogies and selective sweeps in seed-bank populations.

Abstract

We present a lookdown construction for a Moran seed-bank model with variable active and inactive population sizes and we show that the empirical measure of our model coincides with that of the Seed-Bank-Moran Model with latency of Greven, den Hollander and Oomen, 2022. Furthermore, we prove that the time to the most recent common ancestor, starting from $N$ individuals with stationary distribution over its state (active or inactive), has the same asymptotic order as the largest inactivity period. We then obtain an asymptotic distribution of the TMRCA, and use this result to find the first order of the asymptotic distribution of the fixation time of a single beneficial mutant conditioned to invade the whole population, which surprisingly is of order $\ln(N)$.

Lookdown construction for a Moran seed-bank model

TL;DR

The paper develops a Markovian lookdown framework for a Moran model with a seed-bank, allowing the active/dormant population proportions to vary while keeping the total population fixed. It proves that the SB-lookdown empirical measure matches the SB-Moran model's empirical measure via the Markov Mapping theorem and derives the ancestry structure through a seed-bank coalescent embedded in the lookdown construction. The results characterize the time to the most recent common ancestor (TMRCA) as asymptotically governed by the largest inactivity period, with a Gumbel limit after centering, and show that the fixation time of a single beneficial mutant, conditioned to invade, shares the same limiting Gumbel distribution and scales as . Collectively, these findings connect seed-bank dynamics to explicit coalescent and fixation-time laws, enabling precise asymptotics for genealogies and selective sweeps in seed-bank populations.

Abstract

We present a lookdown construction for a Moran seed-bank model with variable active and inactive population sizes and we show that the empirical measure of our model coincides with that of the Seed-Bank-Moran Model with latency of Greven, den Hollander and Oomen, 2022. Furthermore, we prove that the time to the most recent common ancestor, starting from individuals with stationary distribution over its state (active or inactive), has the same asymptotic order as the largest inactivity period. We then obtain an asymptotic distribution of the TMRCA, and use this result to find the first order of the asymptotic distribution of the fixation time of a single beneficial mutant conditioned to invade the whole population, which surprisingly is of order .
Paper Structure (9 sections, 5 theorems, 45 equations, 1 figure)

This paper contains 9 sections, 5 theorems, 45 equations, 1 figure.

Table of Contents

  1. Seed-Bank lookdown model
  2. Seed-Bank Lookdown construction
  3. Equality in law of the SB-lookdown and SB-Moran Models
  4. Proof of Theorem \ref{['thm:emp_eq']}.
  5. Time to the most recent common ancestor
  6. SB-coalescent process as the ancestry process of a SB-lookdown particle system
  7. Asymptotic distribution of the time to the most recent common ancestor
  8. Proof of Theorem \ref{['thm:line']}
  9. Fixation time of a type in the SB-lookdown

Key Result

Theorem 1.1

The laws of empirical measures associated with the SB-Moran model and the SB-lookdown model coincide on $D_{E\times S}[0,\infty)$.

Figures (1)

  • Figure 1: An illustration of the $5$-particle lookdown model and the ancestry process of a sample of individuals in levels $\{2,\ldots,5\}$ at time T. Straight lines indicates activity periods meanwhile dashed ones indicates the dormancy periods.

Theorems & Definitions (10)

  • Remark 1.1
  • Theorem 1.1
  • Definition 2.1: The Seed-bank $k$-coalescent
  • Proposition 2.1: First coalition time with k lower levels.
  • proof
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.1
  • Corollary 3.0.1