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Recursive equations with uncontrolled mating
This protocol is extracted from research article:
A theoretical derivation of response to selection with and without controlled mating in honeybees
Genet Sel Evol, Feb 17, 2021;

Procedure

First, we consider breeding colonies with uncontrolled mating (Fig. 1). A worker group W in year t receives half of its genetic material from its queen Q, which in turn is an offspring from a selected colony in year $t-2$, and thus on average, has a breeding value equal to $Bt-2+S1,t-2$. The other half of its genetic material comes from the drones which Q mated with. The drone producing queens are unselected and from year $t-2$. With probability $pt$, they are breeding queens of that year and thus have an average breeding value equal to $Bt-4+S1,t-4$. With probability $1-pt$, they are passive queens of year $t-2$. In this case, a further distinction is necessary, since the colony from which a passive queen originates may be an unselected breeding colony (probability $qt-2$) with an average breeding value equal to $Bt-4$ or a passive colony (probability $1-qt-2$) with an average breeding value equal to $Pt-4$. Combining all paths of inheritance, we arrive at the following recursive equation for the average true breeding values in the breeding population:

If we assume that $pt=p$, $qt=q$, and $S1,t=S1$ are constant over years, grouping terms for breeding and passive populations yields:

In the passive population, a worker group W in year t receives half of its genetic material from its queen Q, which in turn comes either from an unselected breeding colony (probability $qt$) with average true breeding value $Bt-2$, or from a passive colony (probability $1-qt$) with average breeding value $Pt-2$ (Fig. 1). Therefore, Q has an average breeding value that is equal to $qtBt-2+(1-qt)Pt-2$. For the paternally inherited genetic material, the same considerations as with uncontrolled mating for the breeding population apply. This yields:

which in analogy to Eq. 2, again under the assumption of constant $pt=p$, $qt=q$, and $S1,t=S1$, can be rearranged to

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