Revisiting the Variance-based Selection Model of Diploid Drone Production for Multiple Mating in Honey Bees

The high number of mates of honeybee queens has lead to the proposal of several adaptive explanations. The competing hypotheses to explain multiple mating in honeybees and some other social insects have been mostly evaluated empirically with comprehensive theoretical analysis lacking behind. We report on the mathematical analysis of the diploid drone hypothesis for multiple mating, which suggests that multiple mating evolved as a safeguard against the production of infertile male offspring. In accordance with earlier models, our analysis shows that multiple mating does not reduce the average value of diploid drone production but reduces its variance. We combine this observation with a colony growth model to assess the impact of this reduction in variance to the colony fitness. Considering a plausible parameter space for the honeybee, we conclude that the reduction in variance of diploid drone production can be a significant selective force for multiple mating. We have also described rules of a game for which a problem of finding the best strategy is equivalent to the above biological problem of bee mating. We made a significant progress in the general solution of this game and conjectured that the best strategy is strongly related to the geometry of rational numbers.