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0.2
0.2
p(u)
A(u)
p(u)
A(u)
m: 0.12
t: 5000
m: 0.14
t: 5000
0.15
0.15
0.1
0.1
0.05
0.05
0
0
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
u
u
Fig. 15.7. Mutation-induced speciation. A two peaks static tness landscape, increasing the
mutation rate we pass from a single quasi-species population (left, = 0:12) to the coexistence of
two quasi-species (right, = 0:14).
15.3.5. Coexistence on a static tness landscape
We investigate here the conditions for which more than one quasi-species can coexist
on a static tness landscape without competition.
Let us assume that the tness landscape has several distinct peaks, and that
any peak can be approximated by a quadratic function near its maximum. For
small but nite mutation rates, as shown by Eq. (15.24), the distribution around
an isolated maxima is a bell curve, whose width is given by Eq. (15.23) and average
tness by Eq. (15.22). Let us call thus distribution a quasi-species, and the peak a
niche.
If the niches are separated by a distance greater than , a superposition of quasi-
species (15.24) is a solution of Eq. (15.5). Let us number the quasi-species with the
index k:
X
p(u) =
p
k
(u) ;
k
each p
k
(u) is centered around u
k
and has average tnesshAi
k
. The condition for
the coexistence of two quasi-species h and k ishAi
h
=hAi
k
(this condition can be
extended to any number of quasi-species). In other terms one can say that in a stable
environment the tness of all co-existing individuals is the same, independently on
the species.
Since the average tness (15.22) of a quasi-species depends on the height A
0
and
the curvature a of the niche, one can have coexistence of a sharper niche with larger
tness together with a broader niche with lower tness, as shown in Fig. 15.7. This
coexistence depends crucially on the mutation rate . If is too small, the quasi-
species occupying the broader niche disappears; if the mutation rate is too high the
reverse happens. In this case, the dierence of tness establishes the time scale,
which can be quite long. In presence of a uctuating environment, these time scales
can be long enough that the extinction due to global competition is not relevant.
A transient coexistence is illustrated in Fig. 15.8. One can design a special form
of the landscape that allows the coexistence for a nite interval of values of , but
