Information Technology Reference
In-Depth Information
0
0.4
0.35
-0.1
V
0.3
-0.2
0.25
-0.3
H
p
0.2
-0.4
0.15
H
-0.5
p
0.1
-0.6
0.05
-0.7
0
-50
-30
-10
10
30
50
u
Fig. 15.10. Static tness V , eective tness H, and asymptotic distribution p numerically com-
puted for the following values of parameters: = 2, = 0:01, V
0
= 1:0, b = 0:04, J = 0:6, R = 10,
r = 3 and N = 100.
For illustration, we report in Fig. 15.10 the numerical solution of Eq. (15.5),
showing a possible evolutionary scenario that leads to the coexistence of three quasi-
species. We have chosen the smooth static tness V (u) of Eq. (15.27) and a Gaussian
( = 2) competition kernel. One can realize that the eective tness H is almost
degenerate (here > 0 and the competition eect extends on the neighborhood of
the maxima), i.e. that the average tness of all coexisting quasi-species is the same.
We now derive the conditions for the coexistence of multiple species. We are
interested in its asymptotic behavior in the limit !0, which is the case for actual
organisms. Actually, the mutation mechanism is needed only to dene the genotypic
distance and to populate all available niches. Let us assume that the asymptotic
distribution is formed byLquasi-species. Since !0 they are approximated by
delta functions p
k
(u) =
k
u;u
k
, k = 0; : : : ;L1, centered at u
k
. The weight of
each quasi species is
k
, i.e.
Z
L1
X
p
k
(u)du =
k
;
k
= 1 :
k=0
The quasi-species are ordered such as
0
1
; : : : ;
L1
.
The evolution equations for the p
k
are
p
0
k
(u) =
A(u
k
)
hAi
p
k
(u) ;
where A(u) = exp(H(u)) and
L1
X
uu
j
R
H(u) = V (u)J
K
j
:
j=0
