Chemistry Reference
In-Depth Information
0
1
a
1
a
1
r
1
a
1
r
1
a
1
r
1
ˇ
S
@
A
a
2
r
2
a
2
a
2
r
2
a
2
r
2
ˇ
S
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
;
a
N
r
N
a
N
r
N
a
N
a
N
r
N
ˇ
S
1
1
1
S
where we use the notation of the previous section but for the sake of simplicity ˇ
S
is
now denoted by ˇ
S
, which is not necessarily the same as ˇ
S
in the case of discrete
time models. The eigenvalue equation of this matrix has the form
a
k
u
k
C
X
l
a
k
r
k
u
l
C
a
k
r
k
ˇ
S
v
D
u
k
;
.1
k
N/;
(4.54)
¤
k
X
N
u
k
S
v
D
v
;
(4.55)
kD1
where is an eigenvalue and .
u
1
;:::;
u
N
;
v
/ is an associated eigenvector. By letting
U
D
P
kD1
u
k
, these equations imply that
a
k
r
k
U
D
a
k
r
k
ˇ
S
v
C
.
C
a
k
C
a
k
r
k
/
u
k
and
U
D
.
C
S
/
v
:
(4.56)
Substituting this expression into the previous equation we obtain
a
k
r
k
.
C
S
C
ˇ
S
/
C
a
k
.1
C
r
k
/
u
k
D
v
:
(4.57)
Here we assume that
¤
a
k
.1
C
r
k
/, since in both the concave and isoelastic
cases r
k
>
1,so
a
k
.1
C
r
k
/<0and negative eigenvalues cannot destroy local
asymptotic stability. By summing (4.57) over all k and using (4.56) we get for
v
the
single equation
N
C
a
k
.1
C
r
k
/
.
C
S
/
!
v
D
0:
X
a
k
r
k
.
C
S
C
ˇ
S
/
k
D
1
If
v
D
0, then from (4.57),
u
k
D
0 for all k, so the eigenvector becomes zero, which
is impossible. Therefore
v
¤
0, and the eigenvalue equation becomes
X
N
a
k
r
k
C
a
k
.1
C
r
k
/
D
C
S
C
S
C
ˇ
S
:
(4.58)
k
D
1
The following theorem provides results on the local stability of the equilibrium.
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