Labelled partitions in action: recombination, selection, mutation, and more

TL;DR

This paper models the evolution of large populations under recombination, selection, and mutation using measure-valued ODEs and introduces a stochastic dual process with labelled partitions, providing recursive solutions and unifying previous results.

Contribution

It introduces a novel labelled partitioning process with Markovian labels to represent solutions, extending and unifying existing models of evolutionary dynamics.

Findings

Stochastic representation of population evolution models

Recursive solution formula for single-crossover case

Unified framework for selection, recombination, and mutation equations

Abstract

In this paper, we consider the evolution of an (infinitely large) population under recombination and additional evolutionary forces, modelled by a measure-valued ordinary differential equation. We provide a stochastic representation for the solution of this model via duality to a new labelled partitioning process with Markovian labels. In the special case of single-crossover, this leads to a recursive solution formula. This extends (and unifies) previous results on the selection-recombination equation. As a concrete example, we consider the selection-mutation-recombination equation.