The stepping stone model: New formulas expose old myths

Abstract

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."