The common ancestor type distribution of a $\Lambda$-Wright-Fisher process with selection and mutation

Anton Wakolbinger,Ellen Baake,Ute Lenz

doi:10.1214/16-ecp16

Abstract

Using graphical methods based on a `lookdown' and pruned version of the {\em ancestral selection graph}, we obtain a representation of the type distribution of the ancestor in a two-type Wright-Fisher population with mutation and selection, conditional on the overall type frequency in the old population. This extends results from Lenz, Kluth, Baake, and Wakolbinger (Theor. Pop. Biol., 103 (2015), 27-37) to the case of heavy-tailed offspring, directed by a reproduction measure $\Lambda$. The representation is in terms of the equilibrium tail probabilities of the line-counting process $L$ of the graph. We identify a strong pathwise Siegmund dual of $L$, and characterise the equilibrium tail probabilities of $L$ in terms of hitting probabilities of the dual process.

Highlights

Using graphical methods based on a ‘lookdown’ and pruned version of the ancestral selection graph, we obtain a representation of the type distribution of the ancestor in a two-type Wright-Fisher population with mutation and selection, conditional on the overall type frequency in the old population
We consider a Wright-Fisher process with two-way mutation and selection
In a previous paper [17], we have presented a graphical construction, termed the pruned lookdown ancestral selection graph (p-LD-ASG), which allows us to identify the common ancestor of a population in the distant past, and to represent its type distribution

Summary

Introduction

We consider a Wright-Fisher process with two-way mutation and selection. This is a classical model of mathematical population genetics, which describes the evolution, forward in time, of the type composition of a population with two types. In a previous paper [17], we have presented a graphical construction, termed the pruned lookdown ancestral selection graph (p-LD-ASG), which allows us to identify the common ancestor of a population in the distant past, and to represent its type distribution. This construction keeps track of the collection of all potential ancestral lines of an individual. We will consider a natural generalisation, the so-called Λ-Wright-Fisher processes These include reproduction events where a fraction z > 0 of the population is replaced by the offspring of a single individual; this leads to multiple merger events in the ancestral process. Once the dual is identified, it leads to the tail probabilities of L with little effort

Model and main result

The pruned lookdown-Λ-ancestral selection graph

An application of Siegmund duality

Tail probabilities and hitting probabilities

Flights and their duals

A Siegmund dual for the process L