where r is the rate of recombination times the birth rate of the double
heterozygote genotype, 
the linkage disequilibrium, 
and 
the mean malthusian
fitness.
Using these equations an explicit equation for the rate of
change of the first character can be derived:
Because of the additivity of the allelic effects the system is independent of linkage-disequilibrium. We obtain:
which can be rewritten into equation (7). However, in contrast to the general equation in the last section, we have obtained an explicit expression for the covariance term.
Depending on the frequencies of the gametes 
and 
, the covariance
term can be positive or negative.
Figure 1: Mean
value of character 1 over the dynamic of a 2-locus/2-allele model, starting
near fixed point 
; 
, 
, 
and d=1; black:
with pleiotropic effects, grey: without pleiotropic effects. Pleiotropy
slows down the evolution of the character significantly.
Consequently the rate of evolution can be accelerated or inhibited by selection on hidden pleiotropic effects.
Figure 2: Genetic variance of
character 2, grey, and covariance 
, black, over
the dynamic with pleiotropic effects in figure 2.
In Fig. 2 the
time development of 
in the dynamic of Fig. 1
is plotted in comparison to the genetic variance of the second character.
At low frequencies of the `1' alleles, the covariance term is positive and
is inhibiting the evolution of 
At higher frequencies of these
alleles, i.e. at higher values of the 
character, the covariance term
is negative and accelerates the rate of evolution along the corridor.
Inhibitory effects of the covariance term are associated with increasing
genetic variance of the character 
, which is under stabilizing
selection. Accelerating effects of the covariance term are associated with
decreasing genetic variance of the second character. These dynamics lead
to a biological interpretation of the covariance term. Because, in our
model, each gene that increases the value of 
also has pleiotropic
effects on the second character, the change of the mean value of
is associated with a change in the genetic variance of
the second character, 
. This variance is highest at intermediate
frequencies of the alleles. Consequently, at low frequencies of the `1'
alleles 
increases with increasing 
and at
higher frequencies 
decreases. In the initial phase the
expansion of 
is opposed by the stabilizing selection on the
second character and at higher frequencies of the `1' alleles the decrease
of 
is enhanced by the stabilizing selection. This is the
mechanistic link by which directional selection on 
and stabilizing
selection on 
interact in determining the rate of evolution of
An analogous argument holds for models of non-overlapping
generations.
Provided that the stabilizing selection on the pleiotropic effects can
inhibit the evolution of the first character, it is conceivable that these
effects completely prevent the evolution along the corridor, i.e., with
weak directional selection and strong stabilizing selection it is
conceivable that the rate of evolution is negative,
even if there are genotypes with higher fitness
`uphill' of the corridor. These genotypes may not be reached under these
conditions. To explore this possibility more completely, we perform a
qualitative analysis of the dynamical system underlying this genetic model.