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Journal Article

A Pain in the Torus: Some Difficulties with Models of Isolation by Distance

Joseph Felsenstein
The American Naturalist
Vol. 109, No. 967 (May - Jun., 1975), pp. 359-368
Stable URL: http://www.jstor.org/stable/2459700
Page Count: 10
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A Pain in the Torus: Some Difficulties with Models of Isolation by Distance
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Abstract

Malecot's models of isolation by distance seem to assume random distribution in space, Poisson distribution of offspring number, and independent migration of individuals. These assumptions are inconsistent. By calculating the probability of joint occupancy of two locations at a distance x, it is shown that a haploid model embodying the latter two assumptions violates the first, forming larger and larger clumps of individuals separated by greater and greater distances. Thus the model is biologically irrelevant, even though it is possible to use it to calculate probabilities of genetic identity at different distances. The formation of these clumps is verified by computer simulation. The same phenomenon occurs in two- and three-dimensional models. In models of finite regions (a circle or a torus), the formation of clumps takes the form of the certainty of ultimate local extinction of the organism in each region. This does not occur if we hold the number of individuals on the circle or torus constant, but considerable departure from random distribution still results. The stepping-stone models appear for the present to be the only well-defined models of geographically structured populations with finite population density.

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