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We are interested in deriving the exact form of the asymptotic distribution near
the maximum. We take a static tness A(u) with a smooth, isolated maximum for
u = 0 (smooth maximum approximation). Let us assume that
A(u)'A
0
(1au
2
) ;
(15.19)
where A
0
= A(0). Substituting exp(w) = Ap in Eq. (15.17) we have (neglecting to
indicate the phenotype u, and using primes to denote dierentiation with respect
to it):
hAi
A
= 1 + (w
0
2
+ w
00
) ;
and approximating A
1
= A
1
0
(1 + au
2
), we have
hAi
A
0
(1 + au
2
) = 1 + (w
0
2
+ w
00
) :
(15.20)
A possible solution is
w(u) =
u
2
2
2
:
Substituting into Eq. (15.20) we nally get
p
4a + a
2
2
2
hAi
A
0
=
2 + a
:
(15.21)
SincehAi=A
0
is less than one we have chosen the minus sign. In the limit a!0
(small mutation rate and smooth maximum), we have
hAi
A
0
'1
p
a
(15.22)
and
r
a
:
2
'
(15.23)
The asymptotic solution is
1 + au
2
u
2
2
2
p
p(u) =
2(1 + a
2
)
exp
;
(15.24)
R
so that
p(u)du = 1. The solution is then a bell-shaped curve, its width being
determined by the combined eects of the curvature a of maximum and the mutation
rate .
For completeness, we study here also the case of a sharp maximum, for which
A(u) varies considerably with u. In this case the growth rate of less t strains has
a large contribution from the mutations of ttest strains, while the reverse ow is
negligible, thus
p(u1)A(u1)p(u)A(u)p(u + 1)A(u + 1)
