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distribution formed by more than one quasi-species, i.e. the speciation phenomenon.
Before doing that we need to know the shape of a quasi-species given a static tness
landscape. Some analytical results can be obtained by considering the dynamics
only in the phenotypic space [41].
We assume that the phenotypic index u ranges between1and1in unit steps
(the tness landscape provides that only a nite range of the phenotypic space is
viable), and that mutations connect phenotypes at unit distance; the probability of
observing a mutation per unit of time is . The mutational matrix M(u; v) has the
form:
8
<
:
ifju; vj= 1 ;
M(u; v) =
12
if u = v ;
0
otherwise :
Let us consider as before the evolution of phenotypic distribution p(u), that gives
the probability of observing the phenotype u. As before the whole distribution is
denoted by p.
Considering a phenotypic linear space and non-overlapping generations, we get
from Eq. (15.5)
p
0
(u) =
(12)A(u; p)p(u) + (A(u + 1; p)p(u + 1) + A(u1; p)p(u1)
hAi
:
In the limit of continuous phenotypic space, u becomes a real number and
A(u; p)p(u) +
@
2
A(u; p)p(u)
@u
2
1
hAi
p
0
(u) =
;
(15.17)
with
Z
Z
1
1
p(u)du = 1 ;
A(u; p)p(u)du =hAi:
(15.18)
1
1
Equation (15.17) has the typical form of a nonlinear reaction-diusion equation.
The numerical solution of this equation shows that a stable asymptotic distribution
exists for almost all initial conditions.
The tness A(u; p) = exp(H(u; p)) can be written as before, with
Z
1
H(u; p) = V (u) +
J(u; v)p(v)dv :
1
Before studying the eect of competition and the speciation transition let us
derive the exact form of p(u) in case of a smooth and sharp static tness landscape.
15.3.4.1. Evolution near a smooth and sharp maximum
In the presence of a single maximum the asymptotic distribution is given by one
quasi-species centers around the global maximum of the static landscape. The eect
of a nite mutation rate is simply that of broadening the distribution from a delta
peak to a bell-shaped curve.
