On the eigenvalue effective size of structured populations

A general theory is developed for the eigenvalue effective size (N(e)E) 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 N(e)E 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 N(e)E can be derived. We then study N(e)E 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 N(e)C exists, it is an asymptotic version of N(e)E in the limit of large populations.