The Annals of Probability

Results for the Stepping Stone Model for Migration in Population Genetics

Stanley Sawyer

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Abstract

The stepping stone model describes a situation in which beasts alternately migrate among an infinite array of colonies, undergo random mating within each colony, and are subject to selectively neutral mutation at the rate $u$. Assume the beasts follow a random walk $\{X_n\}$. If $u = 0$, we show that two randomly chosen beasts in the $n$th generation in any bounded set are genetically identical at a given locus with probability converging to one iff the symmetrization of $\{X_n\}$ is recurrent. In general, if either $u = 0$ or $u$ is of order $1/n$, this probability converges to its limit at the rate $C/n^{\frac{1}{2}}$ for finite variance walks in one dimension and $C/(\log n)^a$ in two, with other rates for other classes of $\{X_n\}$. More complicated rates ensure for $u \neq O(1/n)$.

Article information

Source
Ann. Probab. Volume 4, Number 5 (1976), 699-728.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1176995980

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aop/1176995980

Mathematical Reviews number (MathSciNet)
MR682605

Zentralblatt MATH identifier
0341.92009

Subjects
Primary: 92A10
Secondary: 92A15 60J15 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40] 60K99: None of the above, but in this section

Keywords
Stepping stone model random walks genetics population genetics diploid migration mutation random mating rate of convergence

Citation

Sawyer, Stanley. Results for the Stepping Stone Model for Migration in Population Genetics. The Annals of Probability 4 (1976), no. 5, 699--728. doi:10.1214/aop/1176995980. http://projecteuclid.org/euclid.aop/1176995980.


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