On the eigenvalue effective size of structured populations
A general theory is developed for the eigenvalue effective size ($$N_{eE}$$NeE) 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 $$N_{eE}$$NeE 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 $$N_{eE}$$NeE can be derived. We then study $$N_{eE}$$NeE 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 $$N_{eC}$$NeC exists, it is an asymptotic version of $$N_{eE}$$NeE in the limit of large populations.