Evolutionary branching via replicator-mutator equations

TL;DR

This paper analyzes replicator-mutator equations in evolutionary genetics, providing explicit solutions via spectral analysis, and investigates conditions for evolutionary branching through theoretical and numerical methods.

Contribution

It introduces explicit solutions for the equations using Schrödinger spectral elements and rigorously explores conditions for evolutionary branching.

Findings

Long-term behavior determined by principal eigenfunction

Conditions for evolutionary branching depend on fitness and mutation rate

Provides new spectral estimates for non-local reaction-diffusion problems

Abstract

We consider a class of non-local reaction-diffusion problems, referred to as replicator-mutator equations in evolutionary genetics. For a confining fitness function, we prove well-posedness and write the solution explicitly, via some underlying Schr\"odinger spectral elements (for which we provide new and non-standard estimates). As a consequence, the long time behaviour is determined by the principal eigenfunction or ground state. Based on this, we discuss (rigorously and via numerical explorations) the conditions on the fitness function and the mutation rate for evolutionary branching to occur.