On mutations in the branching model for multitype populations

TL;DR

This paper studies the mutation structure in multitype branching forests, showing it forms a branching forest itself, and analyzes its properties, distribution, and asymptotic behavior in both discrete and continuous time models.

Contribution

It demonstrates that the mutation forest retains a branching structure and provides detailed distributional and asymptotic analyses, including in continuous time with edge lengths.

Findings

The mutation forest is itself a branching forest.

Explicit progeny distribution for the mutation forest.

Asymptotic behavior of mutation emergence times.

Abstract

The forest of mutations associated to a multitype branching forest is obtained by merging together all vertices of its clusters and by preserving connections between them. We first show that the forest of mutations of any mulitype branching forest is itself a branching forest. Then we give its progeny distribution and describe some of its crucial properties in terms the initial progeny distribution. We also obtain the limiting behaviour of the number of mutations both when the total number of individuals tends to infinity and when the number of roots tends to infinity. The continuous time case is then investigated by considering multitype branching forests with edge lengths. When mutations are non reversible, we give a representation of their emergence times which allows us to describe the asymptotic behaviour of the latters, when the ratios of successive mutation rates tend to 0.