Chemistry Reference
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and
U
D
.
C
S
/
v
;
(4.65)
where U
D
P
kD1
u
k
as before. Substituting (4.65) into (4.64) we obtain
a
k
r
k
.
C
S
C
ˇ
S
I.//
C
a
k
.1
C
r
k
/
u
k
D
v
;
where we can assume again that
¤
a
k
.1
C
r
k
/. By summing the last equation
over all k and using (4.65) we get for
v
the single equation
N
.
C
S
/
!
v
D
0:
X
a
k
r
k
.
C
S
C
ˇ
S
I./
C
a
k
.1
C
r
k
/
kD1
Noticing again that
v
¤
0, since otherwise the eigenvector would become zero, we
obtain the eigenvalue equation
X
N
C
S
C
S
C
ˇ
S
I./
:
a
k
r
k
C
a
k
.1
C
r
k
/
D
(4.66)
kD1
We have already seen in Theorem 4.2 that in the case T
D
0 the equilibrium is
locally asymptotically stable. The case T>0can be examined in a similar fashion to
the cases discussed earlier in this topic. In the general case computational methods
are used to locate the eigenvalues and check stability. In the symmetric case however
analytical results can be obtained.
Consider therefore the symmetric case of a
1
D D
a
N
D
a and r
1
D D
r
N
D
r. Then (4.66) has the form
C
S
C
S
C
ˇ
S
1
C
Nar
C
a.1
C
r/
D
p
.mC1/
;
T
that is,
.
C
S
/
1
C
p
mC1
T
Nar
C
a.1
C
r/
D
;
.
C
S
/
1
C
p
mC1
T
C
ˇ
S
or
1
C
mC1
T
p
.
C
S
/.
C
a.1
C
r.1
N///
Narˇ
S
D
0:
(4.67)
The roots of this equation can be examined in a similar way to the case of (2.58),
which was given in detail earlier in Sect. 2.6. The details are left as an exercise for
the interested reader.
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