An evolution model with uncountably many alleles

TL;DR

This paper introduces a mathematical model of evolution with an uncountably infinite set of genomes, analyzing its long-term behavior and phase transitions as population size grows, with implications for understanding complex genetic systems.

Contribution

It develops a framework for studying evolution models with uncountably many alleles, including limiting theorems and phase transition analysis under different breeding process assumptions.

Findings

Stationary distribution characterized for large populations

Phase transitions depend on the scaling parameter 

Limit theorems established for different breeding process models

Abstract

We study a class of evolution models, where the breeding process involves an arbitrary exchangeable process, allowing for mutations to appear. The population size $n$ is fixed, hence after breeding, selection is applied. Individuals are characterized by their genome, picked inside a set $X$ (which may be uncountable), and there is a fitness associated to each genome. Being less fit implies a higher chance of being discarded in the selection process. The stationary distribution of the process can be described and studied. We are interested in the asymptotic behavior of this stationary distribution as $n$ goes to infinity. Choosing a parameter $\lambda>0$ to tune the scaling of the fitness when $n$ grows, we prove limiting theorems both for the case when the breeding process does not depend on $n$, and for the case when it is given by a Dirichlet process prior. In both cases, the limit exhibits phase transitions depending on the parameter $\lambda