The common ancestor process revisited

TL;DR

This paper revisits the ancestral line process in the Moran model with mutation and selection, providing a new characterization of the stationary type distribution through fixation probabilities, extending previous work in a finite population setting.

Contribution

It offers a novel characterization of the ancestral line's stationary distribution using fixation probabilities, extending prior results to finite populations.

Findings

Extended previous results to finite populations.

Provided a new particle-based perspective on ancestral lineages.

Characterized the stationary distribution via fixation probabilities.

Abstract

We consider the Moran model in continuous time with two types, mutation, and selection. We concentrate on the ancestral line and its stationary type distribution. Building on work by Fearnhead (J. Appl. Prob. 39 (2002), 38-54) and Taylor (Electron. J. Probab. 12 (2007), 808-847), we characterise this distribution via the fixation probability of the offspring of all individuals of favourable type (regardless of the offspring's types). We concentrate on a finite population and stay with the resulting discrete setting all the way through. This way, we extend previous results and gain new insight into the underlying particle picture.