Recursive equations with uncontrolled mating

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