The spatial $Λ$-Fleming-Viot process in a random environment

Aleksander Klimek, Tommaso Cornelis Rosati

Published 2020-04-13Version 1

We study large scale behaviour of a population consisting of two types which evolve in dimension $d=1,2$ according to a spatial Lambda-Fleming-Viot model subject to random time-independent selection. If one of the two types is rare compared to the other, we prove that its evolution can be approximated by a superBrownian motion in a random time-independent environment. Without the sparsity assumption, a diffusion approximation leads to a Fisher-KPP equation in a random potential. We discuss the longtime behaviour of the limiting processes addressing Wright's claim that the variation in spatial conditions contributes positively to genetic variety in the populations. The crucial technical components of the proofs are two-scale Schauder estimates for semidiscrete approximations of the Laplacian and of the Anderson Hamiltonian.