Chemistry Reference
In-Depth Information
Hence we need to consider the four different Jacobian matrices given by
0
1
1
a
1
!
a
1
B
2.B
1
a
1
a
1
B
2.B
C
e
1
/
@
A
I
J
.1/
C
e
1
/
J
.2/
D
J
.6/
D
I
D
a
2
B
2.B
1
a
2
0
1
a
2
C
e
2
/
1
a
1
!
0
J
.4/
D
J
.8/
D
I
a
2
B
2.B
1
a
2
C
e
2
/
0
1
1
a
1
0
J
.3/
D
J
.5/
D
J
.7/
D
J
.9/
@
A
:
D
01
a
2
x0
01
with x>0, then the row norms of these
Select a diagonal matrix
P
D
Jacobians generated by the matrix
P
are bounded by the row norm of the matrix
x0
!
1
a
1
!
x
0
01
!
1
a
1
!
;
a
1
B
2.B
a
1
Bx
2.B
C
e
1
/
C
e
1
/
D
(1.39)
a
2
B
a
2
B
2.BCe
2
/
1
a
2
2.BCe
2
/x
1
a
2
01
which is below one if and only if
a
1
Bx
2.B
C
e
1
/
<1;
1
a
1
C
and
a
2
B
2.B
C
e
2
/x
<1:
1
a
2
C
Since we assume that 0<a
k
1.k
D
1;2/, these relations can be rewritten as
2.B
C
e
2
/
<x<
2.B
C
e
1
/
B
;
B
and a feasible x exists if and only if B
2
<4.B
C
e
1
/.B
C
e
2
/.
Hence under this condition the equilibrium is unique and is globally asymptoti-
cally stable regardless of whether it is interior or not. (See Appendix B, Theorem B.3
for the relevant theoretical background.)
Next we will examine the local asymptotic stability of an interior steady state E.
Let us consider the Jacobian matrix evaluated at the steady state,
1
a
1
!
:
B
2.B
a
1
C
e
1
/
J
D
B
2.B
a
2
1
a
2
C
e
2
/
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