Branching with selection and mutation I: Mutant fitness of Fr\'echet type

TL;DR

This paper analyzes stochastic models of evolving populations with mutation and selection, focusing on mutant fitness distributions in the Fréchet domain, and rigorously characterizes their superexponential growth rates.

Contribution

It provides a rigorous proof of the superexponential growth rate in models with mutant fitnesses sampled from Fréchet domain distributions.

Findings

Superexponential growth rate established

Heuristic and numerical analysis supports theoretical results

Mutant fitness distribution influences growth dynamics

Abstract

We investigate two stochastic models of a growing population subject to selection and mutation. In our models each individual carries a fitness which determines its mean offspring number. Many of these offspring inherit their parent's fitness, but some are mutants and obtain a fitness randomly sampled from a distribution in the domain of attraction of the Fr\'echet distribution. We give a rigorous proof for the precise rate of superexponential growth of these stochastic processes and support the argument by a heuristic and numerical study of the mechanism underlying this growth.