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The stability condition of the asymptotic distribution is (A(u
k
)hAi)p
k
(u) = 0,
i.e. either A(y
k
) =hAi= const (degeneracy of maxima) or p
k
(u) = 0 (all other
points). This supports our assumption of delta functions for the p
k
.
The position u
k
and the weight
k
of the quasi-species are given by A(u
k
) =
hAi= const and @A(u)=@uj
u
k
= 0, or, in terms of the tness H, by
L1
X
u
k
u
j
R
V (u
k
)J
K
j
= const ;
j=0
L1
X
V
0
(u
k
)
J
R
u
k
u
j
R
K
0
j
= 0 ;
j=0
where the prime in the last equation denotes dierentiation with respect to u.
Let us compute the phase boundary for coexistence of three species for two kinds
of kernels: the exponential ( = 1) and the Gaussian ( = 2) one. The diusion
kernel can be derived by a simple reaction-diusion model, see Ref. [63].
We assume that the static tness V (u) of Eq. (15.27). Due to the symmetries
of the problem, we have the master quasi-species at u
0
= 0 and, symmetrically,
two satellite quasi-species at u =u
1
. Neglecting the mutual inuence of the two
marginal quasi-species, and considering that V
0
(u
0
) = K
0
(u
0
=R) = 0, K
0
(u
1
=R) =
K
0
(u
1
=R), K(0) = J and that the three-species threshold is given by
0
= 1
and
1
= 0, we have
1
u
1
r
b
K(u
1
) =1 ;
b
r
+ K
0
(u
1
) = 0 ;
where u = u=R, r = r=R and b = b=J. We introduce the parameter G = r=b =
(J=R)=(b=r), that is the ratio of two quantities, the rst one related to the strength
of inter-species interactions (J=R) and the second to intra-species ones (b=r).
In the following we drop the tildes for convenience. Thus
z
rzG exp
=G ;
z
Gz
1
exp
= 1 :
For = 1 we have the coexistence condition
ln(G) = r1 + G :
The only parameters that satisfy these equations are G = 1 and r = 0, i.e. a at
landscape (b = 0) with innite range interaction (R =1). Since the coexistence
region reduces to a single point, it is suggested that = 1 is a marginal case. Thus
