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

TL;DR

This paper introduces a hierarchical scheme for solving the Wright-Fisher model's diffusion equation, connecting solutions across allele loss events by analyzing boundary fluxes and moments, offering a new perspective on genetic drift modeling.

Contribution

It presents a novel hierarchical extension scheme that handles boundary singularities in the Wright-Fisher model by linking solutions before and after allele loss.

Findings

Developed a global hierarchical solution scheme

Analyzed boundary fluxes at allele loss points

Connected solutions across different allele states

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.