# Fixation in Fluctuating Populations

## TL;DR

This paper studies how endogenous population fluctuations influence consensus times in a voter model with birth-death dynamics, revealing that increased fluctuation rates accelerate fixation.

## Contribution

It introduces a model combining voter dynamics with birth-death processes and analyzes how population fluctuations affect consensus times and fixation probabilities.

## Key findings

- Consensus times scale as t^{-3} in the limit of high birth-death rates.
- Probability of population size at consensus scales as n^{-3}.
- Higher birth-death rates lead to faster fixation.

## Abstract

We investigate the dynamics of the voter model in which the population itself changes endogenously via the birth-death process. There are two species of voters, labeled A and B, and the population of each species can grow or shrink by the birth-death process at equal rates $b$. Individuals of opposite species also undergo voter model dynamics in which an AB pair can equiprobably become AA or BB with rate $v$---neutral evolution. In the limit $b/v\to\infty$, the distribution of consensus times varies as $t^{-3}$ and the probability that the population size equals $n$ at the moment of consensus varies as $n^{-3}$. As the birth/death rate $b$ is increased, fixation occurs more more quickly; that is, population fluctuations promote consensus.