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Numerical methods on random genetic drift

Abstract

Random genetic drift occurs at a single unlinked locus with two or more alleles. The probability density of alleles is governed by a degenerated Fokker-Planck equation. Due to the degeneration and convection, Dirac singularities will always be developed at boundary as time evolves, which is just the so-called fifixation phenomenon. In order to fifind a complete solution which should keep the conservation of positivity, total probability and expectation, difffferent schemes of FDM, FVM and FEM are tested to solve the equation numerically. We observed that the methods have totally difffferent behaviors. Some of them are stable and keep the conservation of positivity and probability, but fail to keep the expectation. Some of them fails to keep the positivity. Careful analysis is presented to show the reason why one central scheme does work and the others fail. Our study shows that the numerical methods should be carefully chosen and any method with intrinsic numerical viscosity and anti-viscosity must be avoided. Numerical methods for multi-alleles are also discussed. This is a joint work with Xinfu Chen, Chun Liu, David Waxman and Shixin Xu.