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genotypic space does not change the results in the limit !0. Moreover, the
threshold between one and multiple quasi-species is dened as the value of parame-
ters for which the satellite quasi-species vanish. In this case the multiplicity factor
of satellite quasi-species does not inuence the competition, and thus we believe
that the threshold G
c
of Eq. (15.28) still holds in the genotypic hyper-cubic space.
For a variable population, the theory still works substituting G with
G
a
= NG = N
J=R
b=r
;
(15.29)
(for a detailed analysis see Ref. [3]).
This result is in a good agreement with numerical simulations, as shown in the
following Section.
15.5.2. Speciation and mutational meltdown in the hyper-cubic
genotypic space
Let us now study the consequences of evolution in presence of competition in the
more complex genotypic space. We were not able to obtain analytical results, so
we resort to numerical simulations. Some details about the computer code we used
can be found in Appendix B.
In the following we refer always to rule (a), that allows us to study both spe-
ciation and mutational meltdown. Rule (b) has a similar behavior for speciation
transition, while, of course, it does not present any mutational meltdown transition.
We considered the same static tness landscape of Eq. (15.27), (non-epistatic
interactions among genes).
We observe, in good agreement with the analytical approximation Eq. (15.28),
that if G
a
(Eq. (15.29)) is larger than the threshold G
c
(Eq. (15.28)), several
quasi-species coexist, otherwise only the master sequence quasi-species survives.
In Fig. 15.12 a distribution with multiple quasi-species is shown.
We can characterize the speciation transition by means of the entropy S of the
asymptotic phenotypic distribution p(u) as function of G
a
,
X
S =
p(u) ln p(u)
u
which increases in correspondence of the appearance of multiple quasi-species (see
Fig. 15.13).
We locate the transition at a value G
a
'2:25, while analytical approximation
predicts G
c
(0:1)'2:116. The entropy, however, is quite sensible to uctuations
of the quasi-species centered at the master sequence (which embraces the largest
part of distribution), and it was necessary to average over several runs in order to
obtain a clear curve; for larger values of it was impossible to characterize this
transition. A quantity which is much less sensitive of uctuations is the average
