$\Lambda$-Seed-Bank-Wright-Fisher process conditioned on fixation

TL;DR

This paper studies a complex population genetics model incorporating seed banks and skewed offspring distributions, conditioned on fixation, revealing a structured genealogy with coordinated mutations.

Contribution

It introduces a conditioned $ ext{Lambda}$-Seed-Bank-Wright-Fisher process with a novel switching environment, using lookdown construction and sampling duality techniques.

Findings

The conditioned process remains a $ ext{Lambda}$-Wright-Fisher with seed bank.

Genealogy corresponds to a structured $ ext{Lambda}$-coalescent with mutations.

Provides a pathwise construction of the conditioned process.

Abstract

We investigate the $\Lambda$-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 $\Lambda$-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 $\Lambda$-coalescent with coordinated mutations determined by the switching environment.