The stepping stone model: New formulas expose old myths

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 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.” 1. Introduction. The phrase “isolation by distance” was introduced by Wright (1943) to describe the accumulation of local genetic differences under geographically restricted dispersal. The effects of population subdivision have primarily been studied using two models. The first, introduced by Wright (1943), is the island model, in which a population consists of s colonies of equal size, and migration probabilities are equal for each pair of colonies. The great amount of symmetry of the island model makes it easy to solve the model exactly. An account of the theory can be found in Section 7 of Hudson (1990) or Nei and Takahata (1993). The second approach, which will be the focus of this investigation, is the stepping stone model of Kimura (1953). This process was studied extensively in the genetics literature for at least two decades before being rediscovered by probabilists Clifford and Sudbury (1973) and Holley and Liggett (1975) under the name the voter model. At that time Kimura and Weiss (1964), Weiss and Kimura (1965), Malecot (1967, 1969 1975), Maruyama (1970, 1971, 1972), Nagylaki (1974) and others had built a detailed theory that could be used to answer many questions of interest to geneticists. Work on the stepping stone model has, of course, continued during the last 25 years. See Crow and Aoki (1984), Strobeck (1987), Slatkin and Barton (1989), Slatkin (1991, 1993), Hey (1991) and Wilkinson-Herbots (1998). In a parallel endeavor, the voter model has been studied extensively by probabilists. Our investigations here have their roots in the work of Sawyer (1976, 1979) and Cox and Griffeath (1986) who examined the spatial structure of the