Oxley and Oldroyd write

> Having established a closed breeding population, there are several broad methods available to minimize inbreeding, particularly at the sex locus.  The most straightforward method is to maintain a sufficiently large closed population that can maintain enough csd  alleles to ensure high brood viability.  Page and Laidlaw (1982a)  have shown that an effective population size of 107 has a greater than 90% chance of maintaining brood viability greater than 85% over 40 generations. Because selection tends to favour more closely related individuals, selection within the population will require a substantially larger population size to offset the effects of selection on inbreeding (Omholt and Adnoy, 1994 ). This means that large population size alone is unlikely to be sufficient to maintain high brood viability in a selection programme.

> few bee breeding programmes have been successful in the long term, constrained by limited progress in trait improvement, the detrimental effects of inbreeding and poor returns on investment. ... the social and genetic architecture of colonies significantly interferes with the process of artificial selection, slowing its progress and potentially limiting its long term viability. ... multiple mating results in a colony phenotype that contains multiple paternal contributions, and hampers identification of the singular paternal genetic contribution to any one queen offspring.

The Genetic Architecture of Honeybee Breeding
Peter R. Oxley and Benjamin P. Oldroyd
Behaviour and Genetics of Social Insects Laboratory, School of Biological
Sciences, The University of Sydney, New South Wales, Australia

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