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. 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. 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 $$> 0.95$$ in cases where many passive queens had dams from the breeding population. For self-sufficient passive populations, the coefficients of determination were lower ( $$\sim 0.8$$ ) if the breeding population was under controlled mating. This can be explained by the limited simulated time-frame and lower convergence rates. 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.

中文翻译：

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蜜蜂对受控交配和无受控交配选择反应的理论推导


近年来，蜜蜂育种引起了极大的科学兴趣，并且在其性能数据的收集和处理方面取得了许多理论和实践改进。现在众所周知，优质无人机材料的选择对于中长期育种成功至关重要。然而，还没有确凿的数学理论来解释这些发现。我们推导出数学公式来描述对繁殖种群和未选择的被动蜜蜂种群选择的反应，这些被动蜜蜂种群通过迁移间接受益于繁殖种群的遗传改良。这是在繁殖群体中蜂王受控或不受控交配的假设下完成的。我们的模型方程证实了模拟研究中观察到的情况。特别是，我们已经证明，几年后育种群体和被动群体将表现出平行的遗传增益，并且我们能够评估对不同育种策略的选择的反应。因此，我们证实了控制交配对于成功蜜蜂繁殖的至关重要性。与模拟研究的数据相比，在许多被动蜂王拥有来自繁殖群体的母亲的情况下，导出的公式显示出较高的决定系数$$> 0.95$$。对于自给自足的被动种群，如果繁殖种群处于受控交配状态，则决定系数较低（ $$\sim 0.8$$ ）。这可以通过有限的模拟时间范围和较低的收敛速度来解释。所提出的理论推导允许将育种计划的蜜蜂特定模拟结果外推到广泛的种群参数。 此外，它们提供了对相互依存种群的遗传动态的一般见解，不仅适用于蜜蜂，而且适用于更广泛的背景。