JOINT MATHEMATICS COLLOQUIUMUNIVERSITY OF IDAHOWASHINGTON STATE UNIVERSITY |
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I present fundamental mathematical results on
the distribution of admixture fractions in an admixed population
undergoing continuous admixture events, founded by two source
populations. I show that the admixture fractions at one locus follow a
Hardy-Weinberg law. This is surprising due to gene flow between the
hybrid population and the two source populations, since Hardy-Weinberg
laws assume a closed population. For more than one locus, I show that a
stationary distribution of the admixture fractions exists: I solve for
this distribution analytically for two loci, and calculate it by
deterministic simulations when the number of loci is greater than two.
I provide a Bayesian method to estimate the introgression (admixture)
parameters and discuss the implications of reaching stationarity on estimability of parameters when there are more than one admixture processes. G.H. Hardy was an influential mathematician of the 20th century. He discovered the Hardy-Weinberg law, a fundamental mathematical rule stating that gene and genotype frequencies in a population will remain constant in the absence of external forces. Hardy has written a book ``A Mathematician's Apology" which has inspired me enormously.
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