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November 2002 The stepping stone model: New formulas expose old myths
J. Theodore Cox, Richard Durrett
Ann. Appl. Probab. 12(4): 1348-1377 (November 2002). DOI: 10.1214/aoap/1037125866


We study the stepping stone model on the two-dimensional torus. We prove several new hitting time results for random walks from which we derive some simple approximation formulas for the homozygosity in the stepping stone model as a function of the separation of the colonies and for Wright's genetic distance $F_{\mathit{ST}}$. These results confirm a result of Crow and Aoki (1984) found by simulation: in the usual biological range of parameters $F_{\mathit{ST}}$ grows like the $\log$ of the number of colonies. In the other direction, our formulas show that there is significant spatial structure in parts of parameter space where Maruyama and Nei (1971) and Slatkin and Barton (1989) have called the stepping model "effectively panmictic."


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J. Theodore Cox. Richard Durrett. "The stepping stone model: New formulas expose old myths." Ann. Appl. Probab. 12 (4) 1348 - 1377, November 2002.


Published: November 2002
First available in Project Euclid: 12 November 2002

zbMATH: 1016.60089
MathSciNet: MR1936596
Digital Object Identifier: 10.1214/aoap/1037125866

Primary: 60K35
Secondary: 92D10

Keywords: Coalescent , fixation indices , heterozygosity , identity by descent , Stepping stone model , voter model

Rights: Copyright © 2002 Institute of Mathematical Statistics


Vol.12 • No. 4 • November 2002
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