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Solving the 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

The recursive equations given by Eq. 2 and Eq. 4 are linked and thus cannot be solved independently. Therefore, first we calculate the genetic lag $Dt=Bt-Pt$. Subtracting Eq. 4 from Eq. 2 and simplifying terms results in:

For large t, this recursive equation converges, and its asymptotic value can be obtained by equating $Dt-2=Dt$ in Eq. 6:

This means that the genetic lag between the breeding population and the passive population is given by the genetic selection differential $S1$ if the passive population is self-sufficient in terms of queen replacement ($q=0$). By rearing passive queens from breeding colonies ($q>0$), this genetic lag can be reduced by up to 50% (for $q=1$).

Next, we examine the development of the breeding population. From Eq. 7, we know that for sufficiently large t, we can replace $Pt-4$ in Eq. 2 by $Bt-4-S11+q$, which results in:

From Eq. 8, we derive:

The stable value of $ΔBt$ for sufficiently large values of t can be obtained from equating $ΔBt=ΔBt-1=ΔBt-2=ΔBt-3$ in Eq. 9, resulting in:

Since the genetic lag $Dt$ is constant for large t, the annual rate of genetic improvement in the passive population has to be equal to that of the breeding population:

The annual rate of genetic progress in the breeding and passive populations can thus range from 0 ($p=q=0$) to $S13$ ($p=1$). For a fixed probability $p<1$ of queens to mate with drones from breeding colonies, the rate of genetic progress increases slightly with increasing q, i. e. when more passive queens originate from breeding colonies.

From the annual genetic gain $ΔBt=ΔPt$ and the genetic lag $Dt$, one can calculate the time lag between the breeding and the passive populations, i.e. how many years the genetic level of the passive population lags behind the genetic level of the breeding population. This value amounts to:

Thus, the time lag is at least 1.5 years but may become arbitrarily long if the probability for drones from breeding colonies to reproduce is low and few passive queens originate from breeding colonies.

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