Common ancestor type distribution: a Moran model and its deterministic limit

TL;DR

This paper analyzes the distribution of common ancestor types in a two-type Moran model with mutation and selection, deriving explicit formulas and their limits as population size grows infinitely large.

Contribution

It provides explicit combinatorial formulas for the common ancestor type distribution and connects finite models to their deterministic limits via ancestral selection graph pruning.

Findings

Explicit formulas for finite population distributions

Convergence to an explicit limiting function

Connection between finite and infinite population models

Abstract

We study the common ancestor type distribution in a $2$-type Moran model with population size $N$, mutation and selection, and in the deterministic limit regime arising in the former when $N$ tends to infinity, without any rescaling of parameters or time. In the finite case, we express the common ancestor type distribution as a weighted sum of combinatorial terms, and we show that the latter converges to an explicit function. Next, we recover the previous results through pruning of the ancestral selection graph (ASG). The notions of relevant ASG, finite and asymptotic pruned lookdown ASG permit to achieve this task.