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 $u \neq O(1/n)$.
Publié le : 1976-10-14
Classification:
Stepping stone model,
random walks,
genetics,
population genetics,
diploid,
migration,
mutation,
random mating,
rate of convergence,
92A10,
92A15,
60J15,
60J20,
60K99
@article{1176995980,
author = {Sawyer, Stanley},
title = {Results for the Stepping Stone Model for Migration in Population Genetics},
journal = {Ann. Probab.},
volume = {4},
number = {6},
year = {1976},
pages = { 699-728},
language = {en},
url = {http://dml.mathdoc.fr/item/1176995980}
}
Sawyer, Stanley. Results for the Stepping Stone Model for Migration in Population Genetics. Ann. Probab., Tome 4 (1976) no. 6, pp. 699-728. http://gdmltest.u-ga.fr/item/1176995980/