Eigenvalue effective population numbers for populations that vary cyclically in size

Four types of effective population numbers have been discussed in the literature on population genetics. It was shown by Wang and Pollak [Math. Biosci. 166 (2000) 1] that if a large random mating dioecious population varies cyclically in size and the population is followed over a whole cycle, the variance, inbreeding and mutation effective numbers are equal. It was also shown, in a special case, that the variance effective number, N(eV), is equal to the eigenvalue effective number, N(eE). The former is related to the long-term rate at which the variance between lines increases and the latter is computed from the largest non-unit eigenvalue of the matrix of transition probabilities governing the change with time of frequencies of an allele among males and females. It is shown in this paper that N(eE) is quite generally at least approximately the same as N(eV) if the numbers of males and females are always large. Both autosomal and sex-linked loci are considered and the theory allows for reproduction partially by a regular system of inbreeding.