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Differential Equations and Dynamical Systems

, Volume 27, Issue 4, pp 467–492 | Cite as

A General Solution of the Wright–Fisher Model of Random Genetic Drift

  • Tat Dat Tran
  • Julian Hofrichter
  • Jürgen JostEmail author
Original Research

Abstract

We introduce a general solution concept for the Fokker–Planck (Kolmogorov) equation representing the diffusion limit of the Wright–Fisher model of random genetic drift for an arbitrary number of alleles at a single locus. This solution will continue beyond the transitions from the loss of alleles, that is, it will naturally extend to the boundary strata of the probability simplex on which the diffusion is defined. This also takes care of the degeneracy of the diffusion operator at the boundary. We shall then show the existence and uniqueness of a solution. From this solution, we can readily deduce information about the evolution of a Wright–Fisher population.

Keywords

Random genetic drift Fokker–Planck equation Wright–Fisher model Several alleles 

Notes

Acknowledgments

We would like to thank the anonymous referee for helpful comments, in particular suggestions concerning SNPs and mitochondrial DNA, as mentioned in Sect. 4. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement no. 267087.

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Copyright information

© Foundation for Scientific Research and Technological Innovation 2016

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.Department of MathematicsLeipzig UniversityLeipzigGermany
  3. 3.Santa Fe Institute for the Sciences of ComplexitySanta FeUSA

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