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(N
(2)
= n
(2)
) and
8
<
= N
(3)
(1
s
)A
0
A
A
0
A
n
(3)
0
;
s
1
s
A
0
(qA
0
A
)
(A
0
A
)
2
n
(3)
1
= N
(3)
;
n
(3)
=
s
1
s
A
0
A
(A
0
A
)
2
;
:
n
(3)
= N
(3)
N
(3)
= 1
1
A
0
(1
s
)
:
The xed point n
(1)
corresponds to extinction of the whole population, i.e. to
mutational meltdown (MM). It is trivially stable if A
0
< 1, but it can become stable
also if A
0
> 1, A
< 1 and
s
> 1
1
A
0
:
(15.12)
The xed point n
(2)
corresponds to a distribution in which the master sequence
has disappeared even if it has larger tness than other phenotypes. This eect is
usually called Muller's ratchet (MR). The point P
2
is stable for A
0
> 1, A
> 1 and
s
>
A
0
=A
1
A
0
=A
:
(15.13)
The xed point n
(3)
corresponds to a coexistence of all phenotypes. It is stable
in the rest of cases, with A
0
> 1. The asymptotic distribution, however, can
assume two very dierent shapes. In the quasi-species (QS) distribution, the master
sequence phenotype is more abundant than other phenotypes; after increasing the
mutation rate, however, the numeric predominance of the master sequence is lost,
an eect that can be denoted error threshold (ET). The transition between these
two regimes is given by n
0
= n
1
, i.e.,
A
0
=A
1
2A
0
=A
1
:
s
=
(15.14)
Our denition of the error threshold transition needs some remarks: in Eigen's
original work [36, 37] the error threshold is located at the maximum mean Hamming
distance, which corresponds to the maximum spread of population. In the limit of
very large genomes these two denitions agree, since the transition becomes very
sharp [16]. See also Refs. [38, 39].
In Fig. 15.3 we reported the phase diagram of model (15.11) for A
> 1 (the
population always survives). There are three regions: for a low mutation probabil-
ity
s
and high selective advantage A
0
=A
of the master sequence, the distribution
has the quasi-species form (QS); increasing
s
the distribution undergoes the error
threshold (ET) eect; nally, for very high mutation probabilities, the master se-
quence disappears and we enter the Muller's ratchet (MR) region [32, 40]. The error
threshold phase transition is not present for smooth landscapes (for an example of
a study of evolution on a smooth landscape, see Ref. [41]).
