The random walk model of human migration

Abstract

It has been known that the distribution of matrimonial distances, the distance between birthplaces of mates, is highly leptokurtic and has been fitted empirically to a gamma function (including an exponential function as a special case) (Cavalli-Sforza, 1963), a beta function (Yasuda, 1966), a sum of normal probabilities (Cavalli-Sforza et al., 1966) and so forth. Also, theoretical models have been built to justify the skewness of the distribution. In this connection, the distribution has been used for the prediction of probabilities of consanguineous marriage (Cavalli-Sforza et al., 1966) as well as for the study of the probability of gene identity in terms of migration and mutation (Malecot, 1967). In these studies, it is assumed that the second moment of the distribution is finite and the convolution of the distribution is simple. The purpose of the present paper is to demonstrate that such a skewed distribution can be derived from a model of the behavior of man in terms of a random walk of the “Brownian motion type.” The resulting distribution is a K-distribution of the Bessel function. This not only satisfies the two theoretical requirements above, but also contains merely two parameters to be calculated. Some attempts to fit the K-distribution have been made in a helminthe (Broadbent and Kendall, 1953) and in a codling moth (Williams, 1961). In these cases, the stopping time of the movement of larvae is exponentially distributed.