A hierarchical extension scheme for solutions of the Wright–Fisher model
Julian Hofrichter, Tat Dat Tran,and Jürgen Jost
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Submission date: 19. Jun. 2014
published in: Communications in mathematical sciences, 14 (2016) 4, p. 1093-1110
DOI number (of the published article): 10.4310/CMS.2016.v14.n4.a11
Keywords and phrases: Wright-Fisher model, forward Kolmogorov equation, random genetic drift, hierarchical solution
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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.