A general multi-scale description of metastable adaptive motion across fitness valleys

TL;DR

This paper develops a multi-scale framework to describe metastable adaptive motion across fitness valleys in stochastic models, bridging previous works and characterizing transitions between stable states on various time scales.

Contribution

It introduces a meta graph framework capturing ESCs and metastable transitions, providing a detailed multi-scale description of adaptive dynamics in finite trait graphs.

Findings

Characterizes metastable transitions between ESCs.

Develops a meta graph for multi-scale jump chain analysis.

Proves convergence of population process to Markov jump processes.

Abstract

We consider a stochastic individual-based model of adaptive dynamics on a finite trait graph $G=(V,E)$. The evolution is driven by a linear birth rate, a density dependent logistic death rate an the possibility of mutations along the (possibly directed) edges in $E$. We study the limit of small mutation rates for a simultaneously diverging population size. Closing the gap between the works of Bovier, Coquille and Smadi (2019) and Coquille, Kraut and Smadi (2021), we give a precise description of transitions between evolutionary stable conditions (ESC), where multiple mutations are needed to cross a valley in the fitness landscape. The system shows a metastable behaviour on several divergent time scales associated to a degree of stability. We develop the framework of a meta graph that is constituted of ESCs and possible metastable transitions between those. This allows for a concise description of the multi-scale jump chain arising from concatenating several jumps. Finally, for each of the various time scale, we prove the convergence of the population process to a Markov jump process visiting only ESCs of sufficiently high stability.