Explicit solutions for replicator-mutator equations: extinction vs. acceleration

TL;DR

This paper derives explicit solutions for replicator-mutator equations in evolutionary genetics, revealing how initial data tails influence extinction, acceleration, and long-term behavior, with implications for biological modeling.

Contribution

It provides explicit solutions by transforming the equations into heat equations and analyzes the impact of initial data tails on solution behavior.

Findings

Solutions can be global, extinct, or undefined depending on initial data tails.

Solutions exhibit acceleration and often converge to Gaussian profiles.

The results clarify biological implications of mutation effects in populations.

Abstract

We consider a class of nonlocal reaction-diffusion problems, referred to as replicator-mutator equations in evolutionary genetics. By using explicit changes of unknown function, we show that they are equivalent to the heat equation and, therefore, compute their solution explicitly. Based on this, we then prove that, in the case of beneficial mutations in asexual populations, solutions dramatically depend on the tails of the initial data: they can be global, become extinct in finite time or, even, be defined for no positive time. In the former case, we prove that solutions are accelerating, and in many cases converge for large time to some universal Gaussian profile. This sheds light on the biological relevance of such models.