Rare mutations in the spatial Lambda-Fleming-Viot model in a fluctuating environment and SuperBrownian Motion

TL;DR

This paper studies the behavior of rare mutations under the Lambda-Fleming-Viot model with fluctuating selection, showing convergence to diffusion processes and extending results to spatial populations using novel lookdown constructions.

Contribution

It introduces a new application of lookdown constructions to analyze rare mutations in spatial Lambda-Fleming-Viot models with fluctuating environments, extending previous neutral case results.

Findings

Convergence of mutation dynamics to Feller diffusion in random environment

Extension to spatial populations leading to SuperBrownian motion in a random environment

First application of lookdown approach in this context

Abstract

We investigate the behaviour of an establishing mutation which is subject to rapidly fluctuating selection under the Lambda-Fleming-Viot model and show that under a suitable scaling it converges to the Feller diffusion in a random environment. We then extend to a population that is distributed across a spatial continuum. In this setting the scaling limit is the SuperBrownian motion in a random environment.The scaling results for the behaviour of the rare allele are achieved via particle representations which belong to the family of `lookdown constructions'. This generalises the results obtained for the neutral version of the model by Chetwynd-Diggle and Etheridge (2018), which was proved using a duality argument. To our knowledge this is the first instance of the application of the lookdown approach in which other techniques seem unavailable.