Genetic correlation in the stepping stone model with non-symmetrical migration rates

Abstract

Kimura (1953) proposed a “stepping stone” model for the study of the genetic structure of a subdivided population. In this model, it was assumed that a population consists of infinitely many colonies located at grid points of ann-dimensional lattice and that each colony exchanges individuals with neighboring colonies in each generation, and also receives immigrants as random samples from the whole population. The former type of migration has been called “short range migration” and the latter type “long range migration”. Later, Kimura and Weiss (1964), and Weiss and Kimura (1965) have obtained formulas for the genetic correlation and variance between colonies for general cases of the model, assuming that the short range migration is symmetrical in each fixed direction. The purpose of the present report is to extend the results of Kimura and Weiss to cover situations where the restriction of symmetry of short range migration is removed, so that migration rates in opposite directions need not be equal. I believe that in nature there are cases which require the model presented in this report. For example, consider a plant population distributed along a river, or on a plain where the wind at the time when the seeds are scattered is stronger in one direction than in others. For animals and plants it is often true that the centre of habitat is more densely populated than the marginal regions where the environment is less suitable for the species. In such a case the migration rate toward the outside from the centre is larger than that in the opposite direction. As in other theories of population genetics, we will assume that the size of each colony is determined by the carrying capacity of the environment and is not affected by the migration rate. Thus we assume that in each generation each colony produces many more gametes than those which contribute to the next generation, and, among those many gametes, a certain number, say 2N,are chosen from various colonies to form the individuals of a particular colony.