404
Chapter 9
Model 3:
qn
= ^ nm,iA~,
' y
^ y
(13)
In all three cases, analytical integration is possible. The authors believe that the kinetics of Eq. (13) is the
most reasonable, but their data does not allow them to discriminate between the metamorphosis reaction
models of Eqs. (12) and (13).__________________________________________________________ ______
Example 9.16 Competition between a microbial prey and a predator
In the typical prey—predator situation, the prey grows on the substrate fed to the reactor while the predator
or parasite grows on the prey organism and thereby diminishes its net rate of growth. Equations (1) - (3)
form the simplest possible chemostat model for prey—predator interaction without substrate limitation for
growth of the prey *i (//, - ^ mai !) , while the growth rate of the predator is proportional to both the
concentration of prey and the concentration of the predator
xy.
^
= (ßl - \ ) X , - ß 1x,<
IX lX 2
au
(
1
)
dX±
dB
= ß 2Ysxsf X lX 2- X 2
(2)
dS_
= - ß lX l
+ 1
- S
(
3
)
The dimensionless variables are defined as follows:
6 = Dt
;
X,
=
sf Y*
X
, =-
X]0YX
iX2
(
4
)
Tl|l2 is the yield of predator (kilogram of predator per kilogram of prey consumed) and
xl0
is the initial
concentration of the prey organism,
fa
and
fa
are dimensionless rate constants,
fii
=
/
D
and
fa = ki2
/
D.
Comparison with the previous example shows that the present model is a variant of Eqs. 1 to 3 in
Example 9.15 with
p2
= 0, a yield coefficient of the metamorphosis reaction different from 1, and kinetics of
the metamorphosis reaction given by Eq. (12E9.15) or (13E9.15). In the present example,
X}
+
X2
is not
assumed to be constant.
There are 3 possible steady states:
Xj
and
x2
coexist, i.e., both
X/
and
X2
have nonzero steady-state values.
X/
is present in nonzero steady-state concentration, but
x2
is washed out.
both
Xj
and
x2
are washed out (i.e.,
S =
1 and
Xt~X2 =
0).
Each of these steady states will be considered and its stability will be analyzed based on an analysis of the
eigenvalues of the Jacobian:
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