The Stepping Stone Model: New Formulas Expose Old Myths

Ted Cox and Rick Durrett

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 FST. These results confirm a result of Crow and Aoki (1984) found by simulation: in the usual biological range of parameters FST grows like the logarithm 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".

This paper has appeared in Annals of Applied Probability 12 (2002), 1348-1377
Reprint as a PDF file

The Stepping Stone Model, II: Genealogies and the Infinite Sites Model

Iljana Zahle, Ted Cox, and Rick Durrett

Abstract. This paper extends earlier work by Cox and Durrett who studied the coalescence times for two lineages in the stepping stone model on the two-dimensional torus. We show that the genealogy of a sample of size n is given by a time change of Kingman's coalescent. With DNA sequence data in mind, we investigate mutation patterns under the infinite sites model, which assumes that each mutation occurs at a new site. Our results suggest that the spatial structure of the human population contributes to the haplotype structure and slower than expected decay of genetic correlation with distance revealed by recent studies of the human genome.

This paper has appeared in Annals of Applied Probability 15 (2005), 671-699
Reprint as PDF file

Back to Durrett's home page