The Moran model with selection: Fixation probabilities, ancestral lines, and an alternative particle representation

TL;DR

This paper introduces a new labelled particle representation of the Moran model with selection, providing insights into fixation probabilities and the role of selective events using elementary methods.

Contribution

The authors propose the labelled Moran model, a novel particle representation that preserves type frequency distribution and offers new analytical insights into fixation dynamics.

Findings

Recovered fixation probabilities using the new model

Gained detailed understanding of selective event roles

Provided a simpler analytical approach

Abstract

We reconsider the Moran model in continuous time with population size $N$, two allelic types, and selection. We introduce a new particle representation, which we call the labelled Moran model, and which has the same distribution of type frequencies as the original Moran model, provided the initial values are chosen appropriately. In the new model, individuals are labelled $1,2, \dots, N$; neutral resampling events may take place between arbitrary labels, whereas selective events only occur in the direction of increasing labels. With the help of elementary methods only, we not only recover fixation probabilities, but also obtain detailed insight into the number and nature of the selective events that play a role in the fixation process forward in time.