A standard forward model in population genetics is the Wright-Fisher model, which assumes random mating in a constant population. It models genetic drift, selection and mutation. An important infinite population limit is the Kimura diffusion equation. For a single gene with one 2 alleles the diffusion takes place on the unit interval, and with N alleles in an N-1 simplex. The 1-dimensional case is also important in the study of the allele frequency spectrum. The Kimura operator is degenerate at the boundary of the simplex, so its analysis falls outside of the standard elliptic/parabolic theory. I will discuss some of the analytic properties of the diffusion defined by this operator, joint with Rafe Mazzeo, and recent numerical work in the 1-dimensional case, joint with Jon Wilkening, which produces highly accurate solutions, even in very challenging situations.
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