Modeling of Growth Kinetics
311
and specify the coefficients in
F(k)
as functions of the model parameters and values of the variables
(or partial derivatives of these) taken at the considered steady state.
d.
Show by using the steady-state balances for the two morphological forms that
rb -<j> = 'fi'Zb + Z 'b(b] + ku)
(5)
rb + tp= Jl,Z u - Z 'b(b2 + kb)
(6)
where the prime is used to indicate the derivative with respect to the substrate concentration. By
combining eqs. (5) and (6), it is possible to derive an expression for
M'
M '~rb +ru + Z b(p b - p u)
(7)
Use eqs. (5)-{7) to eliminate 4* from the expressions for the coefficients in the polynomial in eq. (4).
e.
It can be shown that the requirements for stability (i.e., only eigenvalues with a negative real part)
are
a, > 0 ;
a2 >
0 ;
a 3 > 0 ; a 4 =
ala2
- a 3 > 0
(8)
If you are interested in mathematics, you may derive the four inequalities. There is a bifurcation,
i.e., change from a stable system to a non-stable system, when the parameters
a\
to a4 are all
positive except one, which changes sign. If
a A
becomes negative, the bifurcation is dynamic
(so-called Hopf bifurcation) and sustained oscillations are obtained. This corresponds to the
situation where a pair of complex conjugate eigenvalues cross the imaginary axis while the third
eigenvalue is real and negative. What does
au a2,
<a3
and o4< 0 imply concerning the parameters in
the morphologically structured model when
Z \>
0 ?
Analyze the following three cases:
*
rib=Mu and a b = a
*
and
*
Hh*Mu and < \= a u
For the last case a requirement for instability leads to
D'
<0. Is this physiologically reasonable?
Final note: with second case of above, Cazzador derived analytical expressions for the rates of the
metamorphosis reactions (see Problem 8.5) and showed that sustained oscillations can be simulated.
REFERENCES
Agger, T., Spohr, A. B., Carlsen, M. and Nielsen, J. (1998) Growth and product formation o f
A sp erg illu s o ryza e
during
submerged cultivations: V erification o f a m orphologically structured model using fluorescent probes.
B io tech n o l. B ioeng.
57,321-329
Bailey, J. E. and Ollis, D. F, (1986).
B io ch em ica l E n g in eerin g F undam entab,
2d. ed., M cGraw-Hill, New York.
Baltzis, B. C„ Fredrickson, A. G. (1988). Limitation o f growth by two complementary nutrients: Som e elementary, but neglected
considerations,
B io tec h n o l B ioeng.
31,75-86.