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

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 Veber, which describes the state of the population at any time by a measure on R-d x [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-Malecot 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.

Automated summary

This paper considers a mathematical model for the evolution of neutral genetic diversity in a spatial continuum. The model incorporates mutations, genetic drift, and either short range or long range dispersal. The authors prove functional central limit theorems for the model in both cases, which allow for the calculation of the asymptotic probability of identity of individuals at different spatial locations. The results provide new formulas applicable to demographic inference and can potentially improve methods for studying long range dispersal.