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population. Suppose that genotype is equal to phenotype. Quality function is used
to assign a value to each phenotype:
q
q
i,j
where
q
i,j
can be explained as the difference between birthrate and death rate.
Suppose that
(
a
i
,a
j
)=
´
i,j
is the frequency of the next generation's phenotype. Then,
Darwin's selection adjusts the distribution of phenotype according to selection
equations:
p
q
i, j
p
′
= p
′
(13.5)
i, j
i, j
Q
=
Ã
Q
q
p
(13.6)
i j
,
i j
,
i j
,
Q
is the average fitness of population. Assume that
p
i
is the frequency of
alleles of population. If
where
p
i
p
j
then a selection equation in GS is obtained as follows:
p
i,j
=
Q
p'
i
=
p
i
Q
i
/
(13.7)
=
Ã
Q
q
p
(13.8)
i
i j
,
j
j
The above discrete selection equation can be approximated with the following
continuous equation:
dp
dt
i
Q
Q
=
p
i
(Q
i
-
)/
(13.9)
If
q
i,j
=
q
j,i
, then
dp
dt
i
Q
=
p
i
(Q
i
-
)
(13.10)
It is easy to verify that:
d Q
dt
= 2(E(Q
2
) -
Q
2
) = 2Var(Q)
²
0
(13.11)
This result is called Fisher's fundamental theorem, which shows that average
fitness is proportional to the difference of fitness. In fact, only some of all
possible genotypes are realized. This is the task of genetic operations to search
genotype space with small number of individuals. These genetic operations are
the main origin of the genetic variation of population. And the most important
genetic operations are mutation and recombination.
