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in the limit of vanishing mutation probability (per generation) and weak selection, a
continuous approach (overlapping generations) is preferred. The mutation-selection
equation becomes
p = (AhAi)M p'(AhAi)p + p ;
where = M1, and 1 is the identity. Considering only symmetric point mu-
tations, is the L-dimensional diusion operator. One can recognize here the
structure of a reaction-diusion equation: evolution can be considered as a reaction-
diusion process in sequence space. The typical patterns are called species.
This simple model allows to recover some known results.
15.3. Evolution on a Fitness Landscape
The analysis of the replication equation, Eq. (15.5) is much simpler if the tness
A does not depend on the population structure, i.e., A = A(x). The lineage of an
individual corresponds to a walk on a static landscape, called the tness landscape.
A generic tness function may be considered a landscape for short times, in which
the species, other than the ones under investigations, can be considered constant.
In this case the evolution really corresponds to an optimization process: in the
case of innite population, and for mutations that are able to \connect" any two
genomes (in an arbitrary number of passes { M is an irreducible matrix), the system
may reach an equilibrium state (in the limit of innite time). The mutation-selection
equation may be linearized by considering that the relative probability q(x) of a
path x (composed by a time sequence of genotypes x = x
(1)
; x
(2)
; : : : ; x
(t)
) is
Q
T +1
t=1
M(g
(t+1)
jx
(t)
)A(x
(t)
)
Q
q(x) =
;
(15.6)
T +1
t=1
hAi(t)
where the denominator is common to all paths. Thus, the unnormalized probabili-
ties of genotypes at time t, z(x)z(x; t), evolve as
X
z
0
(x) =
z
0
= MAz :
M(xjy)A(y)z(y)
or
(15.7)
y
Notice that z(x; t) can be considered as a restricted partition function, see Sec-
tion 15.3.2.
The structure of probabilities in Eq. (15.6) suggests to use an exponential form
for A:
A(x; p) = exp(H(x; p)) ;
that makes evident the analogy with statistical mechanics: the tness landscape
behaves as a sort of energy, while mutations act like temperature [14{16]. See also
Section 15.3.2.
There are a few landscapes that have been studied in details [17]. The simplest
one is the at landscape, where selection plays no role. It is connected to the concept
