On the eigenvalue effective size of structured populations

A general theory is developed for the eigenvalue effective size ([Formula: see text] ) of structured populations in which a gene with two alleles segregates in discrete time. Generalizing results of Ewens (Theor Popul Biol 21:373–378, 1982), we characterize [Formula: see text] in terms of the largest non-unit eigenvalue of the transition matrix of a Markov chain of allele frequencies. We use Perron–Frobenius Theorem to prove that the same eigenvalue appears in a linear recursion of predicted gene diversities between all pairs of subpopulations. Coalescence theory is employed in order to characterize this recursion, so that explicit novel expressions for [Formula: see text] can be derived. We then study [Formula: see text] asymptotically, when either the inverse size and/or the overall migration rate between subpopulations tend to zero. It is demonstrated that several previously known results can be deduced as special cases. In particular when the coalescence effective size [Formula: see text] exists, it is an asymptotic version of [Formula: see text] in the limit of large populations.