Estimating Effective Number of Breeders Under the Heterozygote-Excess Equilibrium

Estimating the effective number of breeders per reproductive cycle or cohort ( $N_b$ ) can contribute to understanding the viability of wild populations. However, previous $N_b$ estimators based on heterozygote-excess assume that parental genotypes are in accordance with Hardy-Weinberg equilibrium (HWE), and can only be applied to diploids. When these criteria are not fulfilled, estimates of expected heterozygosity may thus be biased. We extended a previous $N_b$ estimator to autopolyploids and account for parental genotypes either in HWE or heterozygote-excess equilibrium (HEE). Results include the following: the distributions of genotypes converge to HEE within nine generations (with a relative error of ${10}^{-3}$ ), the corrected estimators can asymptotically estimate $N_b$ without bias, and estimations for autopolyploids are slightly better than those for diploids. The relationships between ${\hat{N}}_b$ and $N_b$ and the influence of variable effective breeding numbers across generations on ${\hat{N}}_b$ were clarified. We find ${\hat{N}}_b$ mainly reflects $N_b$ in the previous generation in diploids but is a weighted harmonic mean of $N_b$ in previous generations in polyploids. Our methods are implemented in a software package, polygene v1.7, which is freely available at https://github.com/huangkang1987/polygene.