A theoretical derivation of response to selection with and without controlled mating in honeybees

Background: In recent years, the breeding of honeybees has gained significant scientific interest, and numerous theoretical and practical improvements have been made regarding the collection and processing of their performance data. It is now known that the selection of high-quality drone material is crucial for mid to long-term breeding success. However, there has been no conclusive mathematical theory to explain these findings. Methods: We derived mathematical formulas to describe the response to selection of a breeding population and an unselected passive population of honeybees that benefits indirectly from genetic improvement in the breeding population via migration. This was done under the assumption of either controlled or uncontrolled mating of queens in the breeding population. Results: Our model equations confirm what has been observed in simulation studies. In particular, we have proven that the breeding population and the passive population will show parallel genetic gain after some years and we were able to assess the responses to selection for different breeding strategies. Thus, we confirmed the crucial importance of controlled mating for successful honeybee breeding. When compared with data from simulation studies, the derived formulas showed high coefficients of determination $$> 0.95$$ > 0.95 in cases where many passive queens had dams from the breeding population. For self-sufficient passive populations, the coefficients of determination were lower ( $$\sim 0.8$$ ∼ 0.8 ) if the breeding population was under controlled mating. This can be explained by the limited simulated time-frame and lower convergence rates. Conclusion: The presented theoretical derivations allow extrapolation of honeybee-specific simulation results for breeding programs to a wide range of population parameters. Furthermore, they provide general insights into the genetic dynamics of interdependent populations, not only for honeybees but also in a broader context.