# A dual process for the coupled Wright-Fisher diffusion

## TL;DR

This paper introduces a dual ancestral process for the coupled Wright-Fisher diffusion, capturing multi-locus, multi-allelic genetic interactions with explicit formulas for special cases.

## Contribution

It derives a dual ancestral process for the coupled Wright-Fisher diffusion, incorporating pairwise interactions and providing explicit stationary and transition rates in special cases.

## Key findings

- Derived the dual ancestral process with coalescence, mutation, and branching events.
- Explicit stationary density and transition rates for two loci with selection and mutation.
- Extended the ancestral selection graph framework to coupled multi-locus models.

## Abstract

The coupled Wright-Fisher diffusion is a multi-dimensional Wright-Fisher diffusion for multi-locus and multi-allelic genetic frequencies, expressed as the strong solution to a system of stochastic differential equations that are coupled in the drift, where the pairwise interaction among loci is modelled by an inter-locus selection. In this paper, an ancestral process, which is dual to the coupled Wright-Fisher diffusion, is derived. The dual process corresponds to the block counting process of coupled ancestral selection graphs, one for each locus. Jumps of the dual process arise from coalescence, mutation, single-branching, which occur at one locus at the time, and double-branching, which occur simultaneously at two loci. The coalescence and mutation rates have the typical structure of the transition rates of the Kingman coalescent process. The single-branching rate not only contains the one-locus selection parameters in a form that generalises the rates of an ancestral selection graph, but it also contains the two-locus selection parameters to include the effect of the pairwise interaction on the single loci. The double-branching rate reflects the particular structure of pairwise selection interactions of the coupled Wright-Fisher diffusion. Moreover, in the special case of two loci, two alleles, with selection and parent independent mutation, the stationary density for the coupled Wright-Fisher diffusion and the transition rates of the dual process are obtained in an explicit form.