We re-investigated the simulation results of [14] and compared them with the theoretical relations derived in the previous section. The simulations had been carried out with the program BeeSim [28]. They featured breeding populations of Nb=500, Nb=1000, or Nb=2000 colonies and passive populations of Np=500, Np=1000, or Np=2000 colonies per year. Although [14] also comprises simulations for Np=0 and Np=, in these cases no passive queens were modeled explicitly, and thus we found these scenarios to be unfit for our analysis. The breeding population was selected for a single trait with truncation selection based on a honeybee-specific best linear unbiased prediction (BLUP) procedure [25, 29]. The selection trait had a direct (worker) and a maternal (queen) component, and two different heritabilities were modelled. In the first set-up, the maternal heritability was hm2=0.53, the direct heritability was hd2=0.34, and the correlation between the effects was rmd=-0.53. The corresponding values in the second set-up were hm2=0.72, hd2=0.46, and rmd=-0.88. Heritabilities were calculated for the selection scheme with controlled mating as described in [26, 30]. Either controlled mating took place on 5, 10, or 20 isolated mating stations, each consisting of a sister group of eight drone producing queens (DPQ), or uncontrolled mating took place. In the case of uncontrolled mating, the probability of an involved drone to have a dam from the breeding population was p=NbNb+Np. The relative proportion of passive queens with dams from the breeding population was q. The simulations covered the values of q=0, q=0.25, q=0.5, q=0.75, and q=1. The generation interval between a breeding queen and her dam was always two years, while the generation interval between a passive queen and her dam varied between one and three years (average two years), as did the average generation interval between drones in uncontrolled matings and their dams. Controlled mating was modelled with a three year age difference between the DPQ and their dam. No age difference between DPQ and their drones was assumed (Fig. 3). Thus, the population structure assumptions imposed in the theoretical derivations were largely met in the simulations, except that the simulated age structure was slightly less rigid. The simulations covered a 20-year period. See [14] for a more detailed description of the simulations.

Illustration of the generation intervals in the simulations with uncontrolled or controlled mating of queens

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