Branching with selection and mutation II: Mutant fitness of Gumbel type

TL;DR

This paper analyzes a branching process with selection and mutation, focusing on Gumbel-type fitness distributions, and demonstrates the emergence of a Gaussian traveling wave in the empirical fitness distribution.

Contribution

It provides a detailed analysis of mutant fitness growth in Gumbel-type distributions and introduces a simplified model showing Gaussian traveling wave behavior.

Findings

Population growth is characterized for Gumbel-type fitness distributions.

Empirical fitness distribution forms a Gaussian traveling wave.

Simplified model accurately captures key dynamics of the original process.

Abstract

We study a model of a branching process subject to selection, modeled by giving each family an individual fitness acting as a branching rate, and mutation, modeled by resampling the fitness of a proportion of offspring in each generation. For two large classes of fitness distributions of Gumbel type we determine the growth of the population, almost surely on survival. We then study the empirical fitness distribution in a simplified model, which is numerically indistinguishable from the original model, and show the emergence of a Gaussian travelling wave.