310
Chapter 7
1
[X0\
l + ^W +AT.Jf.W I^rl
Problem 7.4 Oscillating yeast
We now want to analyze the simple morphologically structured model for oscillating yeast described by
Cazzador (1991). The metamorphosis reactions are given in Eqs. (2)-(3) of Example 7.9, and the growth of
each morphological form is given by Eq. (4) of the example. The rate constants of the metamorphosis
reactions (<fcb
and
kv)
and the specific growth rates of the two morphological forms
and
are all
assumed to be functions of the limiting substrate concentration only.
a.
For a chemostat with dilution rate
D,
write the steady-state balances for the two morphological
forms and for the limiting substrate. The concentration of the limiting substrate in the feed is / , and
the biomass concentration in the chemostat is
x.
b.
By combining the steady-state balances for the two morphological forms, it is possible to derive an
equation that relates
D,
IV
k^.
Specify this relation. If it is assumed that for any
s
there is one
and only one admissible dilution rate which satisfies this relation, show that this leads to the
constraint
^ + ^ > i
M b
M b
0)
c.
To analyze the stability of a given steady state (Z
6
,Z U
ts
, x ) (corresponding to the rates
(u b
>
Uu
>
Mb
>
Mu
»
D
)], you have to linearize the model, i.e.,
( z \
^b
ZH
^
5
)
(
2
)
where J is the Jacobi matrix. Show that the Jacobi matrix is given by
K
rb
- b 2
ru + \$
Mua ux
- 6 3 ,
(
3
)
and specify
b
,,
b2,
63
, rb, ru and 4* as functions of the model parameters and variables in the steady
state considered (see Example 7.9). What can be inferred about the signs of
bu b2, bh rb
and ru?
Show that the eigenvalues of the Jacobi matrix are given as zeros of the polynomial in Eq. (4)
=
A?
+
"h 03
(
4
)