Multiple paternity in social Hymenoptera: estimating the effective mate number in single-double mating populations

Several hypotheses have been advanced to explain the widespread occurrence of multiple mating by queens in eusocial bees, wasps and ants. In order to test competing hypotheses, it is essential to obtain an accurate description of the distribution of mate number and the mates' respective contributions to the brood in populations under study. In this study we examine current methods for estimating the effective mate number and other paternity frequency parameters in populations dominated by single and double mating of queens, based on genetical data from parent-offspring combinations. Estimation errors are due to limited sample size of offspring (nonsampling error) and limited variation of genetic markers (nondetection error). We demonstrate that limited sample size of offspring leads to an underestimation of the paternity contribution from the majority male of double-mated queens (the paternity skew or sperm bias), thus overestimating the effectiveness of double matings and underestimating the population frequency of double-mated queens. We propose a calculation procedure which avoids this bias. Furthermore, we propose a general method to correct simultaneously for errors from nondetection and nonsampling when estimating the population frequency of double-mated queens. The calculation procedures presented are designed for male-haplodiploid social Hymenoptera but some can be applied directly to studies of diplo-diploid mating systems where single and double mating prevail. This has particular relevance to vertebrate studies where the number of offspring studied per mother is invariably low. The remaining calculation procedures are conceptually widely applicable, but details would have to be modified for other than haplodiploid mating systems.