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$Λ$-Seed-Bank-Wright-Fisher process conditioned on fixation

María Clara Fittipaldi, Adrián González Casanova, Julio Ernesto Nava Trejo

TL;DR

We address conditioning the $Λ$-Seed-Bank-Wright-Fisher process on eventual fixation in a setting with seed banks and skewed offspring. Our approach combines a lookdown construction with sampling duality to implement a change of measure that yields a conditioned process, and to recover its genealogical structure. The main result shows that the conditioned process is a $(Λ,ξ,\mathrm{M})$-Seed-Bank-Wright-Fisher process driven by a switching environment $ξ$ and a coordinated mutation mechanism with $\mathrm{M}({0})=a$ and $\mathrm{M}_0(dy)=Λ_0(dy)/y$. The resulting genealogy is a structured $Λ$-coalescent with coordinated mutations dictated by the switching environment, enabling a pathwise construction and retrospective population genealogy under seed-bank and skewed reproduction dynamics.

Abstract

We investigate the $Λ$-Seed-Bank-Wright-Fisher process, a model describing allele frequency dynamics in populations exhibiting both skewed offspring distributions and dormancy. By performing a change of measure, we condition this process on the eventual fixation of a specified genetic type. The resulting process is again a $Λ$-Wright-Fisher process with a seed bank, but now features coordinated mutations driven by a random switching environment. Our analysis relies on two key techniques: the lookdown construction and sampling duality. These tools provide a pathwise construction of the conditioned process while preserving a means to recover the conditioned population genealogy. The resulting genealogy corresponds to a structured $Λ$-coalescent with coordinated mutations determined by the switching environment.

$Λ$-Seed-Bank-Wright-Fisher process conditioned on fixation

TL;DR

We address conditioning the -Seed-Bank-Wright-Fisher process on eventual fixation in a setting with seed banks and skewed offspring. Our approach combines a lookdown construction with sampling duality to implement a change of measure that yields a conditioned process, and to recover its genealogical structure. The main result shows that the conditioned process is a -Seed-Bank-Wright-Fisher process driven by a switching environment and a coordinated mutation mechanism with and . The resulting genealogy is a structured -coalescent with coordinated mutations dictated by the switching environment, enabling a pathwise construction and retrospective population genealogy under seed-bank and skewed reproduction dynamics.

Abstract

We investigate the -Seed-Bank-Wright-Fisher process, a model describing allele frequency dynamics in populations exhibiting both skewed offspring distributions and dormancy. By performing a change of measure, we condition this process on the eventual fixation of a specified genetic type. The resulting process is again a -Wright-Fisher process with a seed bank, but now features coordinated mutations driven by a random switching environment. Our analysis relies on two key techniques: the lookdown construction and sampling duality. These tools provide a pathwise construction of the conditioned process while preserving a means to recover the conditioned population genealogy. The resulting genealogy corresponds to a structured -coalescent with coordinated mutations determined by the switching environment.

Paper Structure

This paper contains 13 sections, 12 theorems, 72 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Moran model and Lookdown construction
  3. $N$-$\left(\Lambda,\xi,\mathrm{M}\right)$-Seed-Bank-Moran model and its Poissonian construction
  4. Frequency processes
  5. Ancestry process
  6. Lookdown construction for $N$-$(\Lambda,\xi,\mathrm{M})$-Seed-Bank-Moran model
  7. Ordered model
  8. Convergence results
  9. Conditioned process
  10. Proofs
  11. Proof Theorem \ref{['thm:Strong_sol']}
  12. Proof of Theorem \ref{['thm:weak_conv']}
  13. Proof of Theorem \ref{['thm:fixation']}

Key Result

Theorem 1

Let $\mathbf{Z}\coloneqq \left\{\mathbf{Z}(t)\right\}_{t\geq 0}$ be a $\Lambda$-Seed-Bank-Wright-Fisher process, define then there exist a continuous time Markov chain $\xi$ on $\{0,1\}$, with $q_{01}=\alpha$ and $q_{10}=\sigma$ such that $\widetilde{\mathbf{Z}}$ distributed as a $(\Lambda,\xi,\mathrm{M})$-Seed-Bank-Wright-Fisher process with switching environment $\xi$ and mutation measure given

Figures (2)

  • Figure 1: $N$-$(\Lambda,S,\mathrm{M})$-Moran Model. Solid lines represent periods of activity, while dashed lines indicate periods of dormancy. Single arrows represent small reproduction events, whereas multiple arrows indicate large reproduction events. The environment marks the times during which mutations are possible ($\xi_t = 1$). Individual mutations are marked with a cross, and coordinated mutations are indicated by squares.
  • Figure 2: Illustration of the particle system in Lemma \ref{['lem_fixation']}. The state process of the first level becomes the environment process, given by $\xi_t =\mathbbm{1}_{\left\{X_1^{N,2}(t) = \heartsuit\right\}}$. Reproduction events involving the first level became mutations, either single or multiple.

Theorems & Definitions (27)

  • Theorem 1
  • Remark 1
  • Definition 1: $k$-$(\Lambda,\xi,\mathrm{M})$-Seed-Bank Coalescent
  • Proposition 1
  • proof
  • Theorem 2
  • Lemma 3
  • proof
  • proof : Proof of Theorem \ref{['thm:ld_exch']}.
  • Remark 2
  • ...and 17 more