Scaling limits of a model for selection at two scales

TL;DR

This paper analyzes a stochastic model of population selection at two scales, proving convergence to a differential equation or a measure-valued process, revealing how opposing selection pressures influence population dynamics.

Contribution

It introduces a novel stochastic ball-and-urn model for dual-scale selection and establishes its convergence to both a nonlinear PDE and a Fleming-Viot process.

Findings

Fixed points of the PDE are Beta distributions.

Stability of fixed points depends on parameter λ.

Model captures opposing selection pressures at different scales.

Abstract

The dynamics of a population undergoing selection is a central topic in evolutionary biology. This question is particularly intriguing in the case where selective forces act in opposing directions at two population scales. For example, a fast-replicating virus strain outcompetes slower-replicating strains at the within-host scale. However, if the fast-replicating strain causes host morbidity and is less frequently transmitted, it can be outcompeted by slower-replicating strains at the between-host scale. Here we consider a stochastic ball-and-urn process which models this type of phenomenon. We prove the weak convergence of this process under two natural scalings. The first scaling leads to a deterministic nonlinear integro-partial differential equation on the interval $[0,1]$ with dependence on a single parameter, $\lambda$. We show that the fixed points of this differential equation are Beta distributions and that their stability depends on $\lambda$ and the behavior of the initial data around $1$. The second scaling leads to a measure-valued Fleming-Viot process, an infinite dimensional stochastic process that is frequently associated with a population genetics.