Sustainability and Stability - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 843

The maximum value of substrate concentration in the chemostat culture is the feed concen-
tration (prey washed out). Therefore, all washout condition is given by
D m b max S 0
K b þS 0

(16.94)

Fig. 16.22 shows the steady-state operating regions of the prey e predator system. Avariety
of types of dynamic coexistence behavior have been revealed by stability analysis for this
prey e predator interaction. When both prey and predator coexist in the reactor, there is no
stable solution as all the steady states are unstable. The trajectories to steady state are gov-
erned by Eqns (16.87) through (16.89) . The Eigen values of the Jacobian determine the nature
of the trajectories ( Fig. 16.11 ); there are three Eigen values when both predator and prey are
present, or when

k dp þ m p max YF b = S S 0
K p þ YF b=S S 0

D < (16.95)

The real Eigen value is positive. Therefore, there are no stable steady solutions. However, the
solutions are sustainable as there is a bound or limit on the variations.

16.6.5. LokkaeVolterra ModeldA Simplified PredatorePrey
Interaction Model
A classical model that describes oscillations in a prey e predator system is the
Lotka e Volterra model, which assumes K b S (substrate is unlimited) and K p [ X b (prey

population is very small). Eqns (16.87) through (16.89) are reduced to:

d t ¼ m 0 b X b m 0 p X b X p

d X b

(16.96)

YF p=b

d X p

d t ¼k 0 dp X p þm 0 p X b X p

(16.97)

0.30

FIGURE 16.22 Steady-
state operating regions of
prey e predator model with
Monod growth. The kinetic
parameters are given by:
k dp ¼ 0, m pmax ¼ 0.24/h, K p ¼
4 10 11 E. coli/L, m bmax ¼
0.25/h, K b ¼ 5 10 4 g/L,
YF p/b ¼ 7.14 10 4 D.
discoideum/E. coli,YF b/S ¼ 3
10 12 E. coli/g.

Total washout

0.25

0.20

Predator washout

0.15

0.10

0.05

0.00

0.00 0.02 0.04 0.06 0.08 0.10 1 10

S 0 , g/L