Stochastic partial differential equations describing neutral genetic diversity under short range and long range dispersal

TL;DR

This paper develops stochastic PDE models for neutral genetic diversity considering both short and long range dispersal, deriving formulas for genetic identity probabilities and extending classical models to include long range dispersal effects.

Contribution

It introduces a unified framework for modeling genetic diversity with short and long range dispersal using the spatial Lambda-Fleming-Viot process, deriving new asymptotic formulas.

Findings

Recovered the classical Wright-Malécot formula for short range dispersal.

Derived a new formula for long range dispersal, enhancing inference methods.

Proved functional central limit theorems for the process under large population limits.

Abstract

In this paper, we consider a mathematical model for the evolution of neutral genetic diversity in a spatial continuum including mutations, genetic drift and either short range or long range dispersal. The model we consider is the spatial $ \Lambda $-Fleming-Viot process introduced by Barton, Etheridge and V\'eber, which describes the state of the population at any time by a measure on $ \R^d \times [0,1] $, where $ \R^d $ is the geographical space and $ [0,1] $ is the space of genetic types. In both cases (short range and long range dispersal), we prove a functional central limit theorem for the process as the population density becomes large and under some space-time rescaling. We then deduce from these two central limit theorems a formula for the asymptotic probability of identity of two individuals picked at random from two given spatial locations. In the case of short range dispersal, we recover the classical Wright-Mal\'ecot formula, which is widely used in demographic inference for spatially structured populations. In the case of long range dispersal we obtain a new formula which could open the way for a better appraisal of long range dispersal in inference methods.