Communications in Mathematical Sciences

Volume 14 (2016)

Number 4

A hierarchical extension scheme for solutions of the Wright–Fisher model

Pages: 1093 – 1110

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n4.a11

Authors

Julian Hofrichter (MPI for Mathematics in the Sciences, Leipzig, Germany)

Tat Dat Tran (MPI for Mathematics in the Sciences, Leipzig, Germany)

Jürgen Jost (MPI for Mathematics in the Sciences, Leipzig, Germany)

Abstract

We develop a global and hierarchical scheme for the forward Kolmogorov (Fokker–Planck) equation of the diffusion approximation of the Wright–Fisher model of population genetics. That model describes the random genetic drift of several alleles at the same locus in a population. The key of our scheme is to connect the solutions before and after the loss of an allele. Whereas in an approach via stochastic processes or partial differential equations, such a loss of an allele leads to a boundary singularity, from a biological or geometric perspective, this is a natural process that can be analyzed in detail. Our method depends on evolution equations for the moments of the process and a careful analysis of the boundary flux.

Keywords

Wright–Fisher model, forward Kolmogorov equation, random genetic drift, hierarchical solution

2010 Mathematics Subject Classification

35K65, 60J70

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Published 6 April 2016