A probabilistic view on the deterministic mutation-selection equation: dynamics, equilibria, and ancestry via individual lines of descent

TL;DR

This paper offers a probabilistic perspective on the deterministic mutation-selection equation, linking its solutions and equilibria to ancestral processes and genealogical structures, enhancing understanding of error thresholds and type distributions.

Contribution

It introduces ancestral process representations for the deterministic model, connecting solutions to genealogical structures and characterizing equilibrium distributions.

Findings

Representation of solutions via killed ancestral selection graph

Genealogical interpretation of error threshold bifurcations

Explicit characterization of ancestral type distribution at equilibrium

Abstract

We reconsider the deterministic haploid mutation-selection equation with two types. This is an ordinary differential equation that describes the type distribution (forward in time) in a population of infinite size. This paper establishes ancestral (random) structures inherent in this deterministic model. In a first step, we obtain a representation of the deterministic equation's solution (and, in particular, of its equilibrium) in terms of an ancestral process called the killed ancestral selection graph. This representation allows one to understand the bifurcations related to the error threshold phenomenon from a genealogical point of view. Next, we characterise the ancestral type distribution by means of the pruned lookdown ancestral selection graph and study its properties at equilibrium. We also provide an alternative characterisation in terms of a piecewise-deterministic Markov process. Throughout, emphasis is on the underlying dualities as well as on explicit results.