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On the past history of an allele now known to have frequency p

1977; Cambridge University Press; Volume: 14; Issue: 3 Linguagem: Inglês

10.2307/3213447

1475-6072

Stanley Sawyer,

Diffusion and Search Dynamics

Consider a diploid population of N individuals undergoing random mating and mutation as in the infinite-allele Wright model. Choose a particular gene in generation Q for large Q , and let f t (0 ≦ t ≦ Q ) be the frequency of the allelic type of the predecessor of that particular gene (or the gene itself) in generation t . By considering a ‘diffusion approximation' { x t } of { f t }, we find the distribution of the age of an allelic type now known to have frequency p , and of its distribution of frequencies since the allele came into existence. A novelty here is that the process { x t } is not a diffusion, but a process with jumps; it has x = 0 as an inaccessible entrance boundary but periodically jumps to it from the interior of [0, 1]. The formulas obtained are the same as those derived by Maruyama and Kimura, who used a totally different approach.