Some Mathematical Models from Population Genetics: École D'Été de Probabilités de Saint-Flour XXXIX-2009, Issue 2012
This work reflects sixteen hours of lectures delivered by the author at the 2009 St Flour summer school in probability. It provides a rapid introduction to a range of mathematical models that have their origins in theoretical population genetics. The models fall into two classes: forwards in time models for the evolution of frequencies of different genetic types in a population; and backwards in time (coalescent) models that trace out the genealogical relationships between individuals in a sample from the population. Some, like the classical Wright-Fisher model, date right back to the origins of the subject. Others, like the multiple merger coalescents or the spatial Lambda-Fleming-Viot process are much more recent. All share a rich mathematical structure. Biological terms are explained, the models are carefully motivated and tools for their study are presented systematically.
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a-alleles allele frequencies alleles model allelic types ancestral lineages ancestral selection graph approximation assume Brownian motion Cetraro chromosome coalescent tree convergence corresponding deﬁned Deﬁnition deme denote density differential equation diploid diploid population Dirichlet distribution Editors effective population effective population size Etheridge evolution example F Univ ﬁnd ﬁnite ﬁrst ﬁtness ﬁxed gene tree genealogy genetic drift haploid independent inﬁnitely many alleles instantaneous rate Kingman coalescent labelled Lemma limit lookdown process Markov chain Markov process Martina Franca Mathematical Models migration model with selection Models from Population Moran model MRCA mutation rates neutral locus offspring one-dimensional diffusion pair of lineages panmictic parameter Poisson process population genetics probability random variables random walk reﬂects Remark rescaled Saint-Flour sample satisﬁed selective sweeps sequence spatial speed measure stationary distribution stepping stone model stochastic subdivided suppose Theorem Theory timescale trace backwards type a individuals variance vector Wright–Fisher diffusion Wright–Fisher model