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 Λ-Fleming-Viot process introduced by Barton, Etheridge and Véber, which describes the state of the population at any time by a measure on , where is the geographical space and 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-Malécot 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.
Acknowledgments
The author would like to thank Alison Etheridge and Amandine Véber for their valuable inputs at various stages of this work, including the suggestion to treat the case of long range dispersal with this approach. R.F. is supported in part by the Chaire Modélisation Mathématique et Biodiversité of Veolia Environnement-École Polytechnique-Museum National d’Histoire Naturelle-Fondation X. The author is also thankful to the associate editor and anonymous referee whose suggestions helped to greatly improve the presentation of this paper.
Citation
Raphaël Forien. "Stochastic partial differential equations describing neutral genetic diversity under short range and long range dispersal." Electron. J. Probab. 27 1 - 41, 2022. https://doi.org/10.1214/22-EJP827
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