A stability test for migration matrix models of genetic differentiation

Models of genetic differentiation which incorporate migration matrices generally assume that the recorded rates of migration between subdivisions of an solate have persisted for a very long time. A technique derived from the stability theory of linear differential equations is presented which can be used to test this assumption. The technique involves a simple transformation of the original migration matrix. If the transformed matrix possesses at least one eigenvalue with a positive real part, no set of equilibrium subdivision population sizes can be associated with the recorded migration rates, in which case it can be concluded that the rates have not existed for an indefinite period of time. This technique is applied to migration data from Papua New Guinea that have already been used in an analysis of genetic differentiation. The transformed data yield an eigenvalue of 0.016 which, though positive real, does not appear to be significantly different from zero. Therefore, it can be tentatively concluded that these data were appropriate for the migration matrix analysis that was carried out on them.