Environmental Engineering Reference
In-Depth Information
where
i
denotes the row index, and
j
the column index. The Jacobi matrix changes
with the variables
c
i
and thus with time
t
, as the
c'
s are functions of
t
. The Jacobi
matrix depends on the position in the phase space where it is evaluated. Df is
a generalization of the derivative for multi-valued functions, depending on several
variables.
For the competing species system (
19.3
), the function f is given by:
f
1
ðc
1
; c
2
Þ
f
2
ðc
1
; c
2
Þ
c
1
r
1
a
1
Dh
ð
Þ
f
ð
c
Þ¼
f
ðc
1
; c
2
Þ¼
¼
(19.7)
c
2
r
2
a
2
Dh
ð
Þ
All four expressions in (
19.7
) are different writings of the same thing.
MATLAB
users, familiar with vector notation, probably prefer the most compact
writing on the left. As the reader may easily verify, the Jacobi matrix at location
(
c
1
,
c
2
) is given by:
®
r
1
a
1
2
h
1
c
1
þ h
2
c
2
ð
Þ
a
1
h
2
c
1
Df
ðc
1
; c
2
Þ¼
(19.8)
a
2
h
1
c
2
r
2
a
2
2
h
2
c
2
þ h
1
c
1
ð
Þ
0 the Jacobi matrix is diagonal with
r
1
in the upper-left position
and
r
2
in the lower right. The eigenvalues of the diagonal matrix are given by
two reaction rates
r
1
and
r
2
(compare Chap. 18). These are always positive; the
eigenvalues are positive, indicating that the equilibrium at zero concentrations is
unstable.
At the other two equilibrium positions the Jacobi matrices look as follows:
For
c
1
¼ c
2
¼
Df
Þ¼
r
1
h
2
r
1
=h
1
Df
ðr
1
=a
1
h
1
;
0
ð
0
; r
2
=a
2
h
2
Þ
0
r
2
a
2
r
1
=h
1
r
1
a
1
r
2
=a
2
0
¼
(19.9)
h
1
r
2
=h
1
r
2
The eigenvalues for these two triangular matrices can be read directly from the
diagonal:
r
1
at (
r
1
=a
1
h
1
;
Þ
r
1
a
1
r
2
=a
2
at (0,
r
2
=a
2
h
2
Þ
r
2
a
2
r
1
=a
1
at (
r
1
=a
1
h
1
;
Þ
0
0
l
1
¼
l
2
¼
(19.10)
r
2
at (0,
r
2
=a
2
h
2
Þ
The eigenvalues are real in all cases, because all parameters are real numbers.
The stability depends on the sign of both eigenvalues. The equilibria are stable if
both eigenvalues are negative. As the rates
r
1
and
r
2
are positive, there remains only
one condition. Stability is equivalent to the conditions:
r
1
a
1
<
r
2
a
2
r
1
a
1
>
r
2
a
2
at (
r
1
=a
1
h
1
,0)
and
at (0,
r
2
=a
2
h
2
Þ
(19.11)
