398
Chapter 9
same method used in Eq. (9.102). The analysis turns out to be remarkably simple since the only
nontrivial steady state with both species coexisting is that for which
A _ / ymax,2 A _ ^
A + Oj
A +
a 2
U
(9.114)
P
Fm
ax,la 2
/^mrn.2^1
_
n _
/^max,ia 2
^max,2^1
*^0-------------------------- >
U ~
_______
/^max,2
~ №m*r.,\
a
2 ~
X,0 + X2(j -
1
-
Slh
whereas the distribution between
X,0
and
X20
is unknown. The Jacobian of the
system consisting of Eqs. (9.113 a,b,c) is
A A
A +
al
J=
0
V
A A
A +
o
A A
A +
a 2
~
A A
a\
A A o
(A
+ai f
ai
A 'A o
( A + a 2)2
alA At)
+ ^AAo
^ (A +<*l)
(*^0 + a 2 )
+ 1
))
(9.115)
where
p ,
=
,
/
D
and
fa
= / w
.2
/
From Eq. (9.114), one obtains
= 1
R
A
_ -
A
A „
- A
A
a i
A + a 2
Consequently, the eigenvalues of Eq. (9.115) are zeros of
F(X)=X\X2
+
Ai
Pi
)
A
2 +
a iAo
a2X 2(j
I 1
(9.116)
(9.117)
or
X -i
0
-1
a
I At) + fl2 A d ^ I
A
Pi
) s l
(9.118)
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