Confining integro-differential equations originating from evolutionary biology: ground states and long time dynamics

TL;DR

This paper analyzes nonlinear mutation-selection models in evolutionary biology, establishing conditions for long-term behavior, existence of principal eigenfunctions, and spectral gap quantification, with new results for symmetric mutations.

Contribution

It introduces a milder condition for the existence of a principal eigenfunction and provides the first quantification of the spectral gap for symmetric mutation models.

Findings

Long-term behavior is determined by the principal eigenfunction.

A new sufficient condition for eigenfunction existence is proposed.

Spectral gap quantification is achieved for symmetric mutations.

Abstract

We consider nonlinear mutation selection models, known as replicator-mutator equations in evolutionary biology. They involve a nonlocal mutation kernel and a confining fitness potential. We prove that the long time behaviour of the Cauchy problem is determined by the principal eigenelement of the underlying linear operator. The novelties compared to the literature on these models are about the case of symmetric mutations: we propose a new milder sufficient condition for the existence of a principal eigenfunction, and we provide what is to our knowledge the first quantification of the spectral gap. We also recover existing results in the non-symmetric case, through a new approach.