Offspring total additive genetic values u are the average of their
parents u plus a Mendelian sampling deviation drawn from a normal
distribution with mean of 0 and variance equal to \(0.5V_{A} (1 -
f_{sd})\) where \(V_A\) is VAf and
\(f_{sd}\) is the average of the parents' coefficient of inbreeding
f (p. 447 Verrier et al. 1993). Each ‘immigrant’ (individual
with unknown parents in generations >1) is given a total additive genetic
effect that is drawn from a normal distribution with mean of mui and
variance equal to VAi. Residual deviations are sampled for
‘focal’ and ‘immigrant’ populations separately, using normal
distributions with means of murf and muri, respectively, and
variances of VRf and VRi, respectively. Phenotypes are the sum
of total additive genetic effects and residual deviations plus an overall
mean mup.
Trends in total additive genetic effects and/or residual deviations can be
specified for both the focal and immigrant populations. Trends in total
additive genetic effects occurring in the immigrants, in the residual
deviations occurring in the focal population, and in the residual deviations
occurring in the immigrants are produced by altering the mean each
generation for the separate distribution from which these effects are each
drawn. The change in mean over a generation is specified in units of
standard deviations of the respective distributions (e.g., square roots of
VAi, VRf, and VRi) and is set with d_bvi,
d_rf, or d_ri, respectively.
Trends in total additive genetic effects for the focal population are
produced by selecting individuals to be parents of the next generation
according to their predicted total additive genetic effects.
Individuals are assigned probabilities of being selected as a parent of the
next generation depending on how closely their predicted total additive
genetic effect matches an optimum value. Probabilities are assigned:
$$exp((\frac{-1}{2\sigma_{x}}) (x - \theta)^{2})$$ where x is the vector of predicted total additive
genetic effects (u), \(\sigma_{x}\) is the standard
deviation of x, and \(\theta\) is the optimum value.
Sampling is conducted with replacement based on these probabilities.
The parameter d_bvf specifies how much the optimal total additive
genetic effect changes per generation. The optimal total additive genetic
effect in a given generation is calculated as: muf + d_bvf
*sqrt(VAf) * (i-2). Individuals with predicted total additive
genetic effects closest to this optimum have a higher probability of being
randomly sampled to be parents of the next generation. This represents
selection directly on predicted total additive genetic effects.
Total additive genetic effects are predicted for the first generation of
focal individuals and all immigrants using equation 1.3 in Mrode (2005,
p.3): \(h^{2} * (phenotype_{i} - mean population phenotype)\). The
heritability is either VAf / (VAf + VRf) or VAi /
(VAi + VRi). Total additive genetic effects are predicted for all
other individuals using equation 1.9 in Mrode (2005, p. 10) - or as the
average of each individual's parents' predicted total additive genetic
effects.