Williams' decomposition of the L\'evy continuous random tree and simultaneous extinction probability for populations with neutral mutations

TL;DR

This paper analyzes the genealogy of populations with neutral mutations using continuous state branching processes, providing a Williams' decomposition and a formula for the probability of simultaneous extinction.

Contribution

It introduces a Williams' decomposition for the genealogy of populations with neutral mutations and derives a closed-form expression for simultaneous extinction probability.

Findings

Williams' decomposition of the genealogy is established.

A closed formula for simultaneous extinction probability is provided.

The evolution of populations is modeled with continuous state branching processes.

Abstract

We consider an initial Eve-population and a population of neutral mutants, such that the total population dies out in finite time. We describe the evolution of the Eve-population and the total population with continuous state branching processes, and the neutral mutation procedure can be seen as an immigration process with intensity proportional to the size of the population. First we establish a Williams' decomposition of the genealogy of the total population given by a continuous random tree, according to the ancestral lineage of the last individual alive. This allows us give a closed formula for the probability of simultaneous extinction of the Eve-population and the total population.