Sustainability and Stability - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 841

Noting that

r p ¼ðm p k dp ÞX p ¼ m p max X b

X p k dp X p (16.84)

K p þX b

YF p=b m p max X p

r b ¼ m b X b ¼ðm bG m b k d ÞX b ¼ m b max S

1

K b þS X b

X b (16.85)

K p þX b

1
YF b=S m bG X b ¼ YF b=S m bmax S

1

r S ¼

K b þS X b

(16.86)

where YF b/S and YF p/b are the yield coefficients for the growth of prey on substrate
and the growth of predator on prey, respectively. Eqns (16.81) through (16.83) can be
reduced to

d X p

d t ¼ðD k dp ÞX p þ m p max X b

X p (16.87)

K p þX b

m p max X p
K p þX b

d X b

d t ¼DX b þ m bmax S

1
YF p=b K b þS X b

X b (16.88)

d S
d t ¼ DðS 0 SÞ YF b=S m b max S

1

K b þS X b

(16.89)

Eqns (16.87) through (16.89) govern the substrate e prey e predator concentration change
with time in a continuous culture. This model was used to describe the behavior of Dic-
tyostelium discoideum and E. coli in a chemostat culture and was found to predict experi-
mental results quite well ( Fig. 16.21 ). Fig. 16.21 shows that the prey and predator
system is not stable, but marginally sustainable. Substrate (glucose), prey, and predator
concentrations are oscillatory with a clear phase shift as expected. When prey concentra-
tion (population) is high, nearly all the substrates are consumed. High prey concentration
is then followed by the increase in predator concentration (population), in turn causing
the prey concentration to drop. When the prey concentration is low, predator population
decrease is followed with a short delay while the substrate concentration gains. This trend
is shown in Fig. 16.21 from both the predictions based Eqns (16.87) though (16.89) and the
experimental measurements.
The washing out conditions can be obtained by examining Eqns (16.87) and (16.88) .

d X p
d t < 0 ,

Eqn (16.87) indicates that if predator is starting to wash out

Dk dp > m p max X b (16.90)

K p þX b

if X p > 0. The maximum value of X b is at which X p ¼ 0 (predator already washed out):

X b YF b=S ðS 0 SÞ < YF b=S S 0 (16.91)