Sustainability and Stability - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 835

A common example of parasitism is the destruction of microorganisms by microphages.
Although the physical mechanisms in predation and parasitism differ, these two phenomena
have many common features in their conceptual and mathematical descriptions. In an open
system, such as a chemostat where predator e prey interactions take place, the populations of
predator and prey do not necessarily reach steady state but can oscillate at certain dilution
rates. At the beginning of the operation prey concentration is high, but predator concentra-
tion is low. As the predators consume prey, the number of predators increases and the prey
concentration decreases. After a while, a small prey population cannot support the large
predator population, and the predator population decreases while prey population increases.
Depending on the dilution rate and feed substrate concentration, these oscillations may be
sustained or damped or may not exist.
Finally, note that these interactions can, and often do, exist in combination. For example,
A and B may compete for glucose as a nutrient, but A requires a growth factor from B to grow.
In such a case, both competition and commensalism would be present.

16.6.2. Interactions of Two Species Fed on the Same Limiting Substrate
Competition of two species for the same growth-rate-limiting substrate is common.
The two organisms may stably coexist in a chemostat if both A and B follow Monod
kinetics.
Fig. 16.18 shows a schematic of a chemostat with two competing species. Mass balances of
species A and B, as well as the common growth-limiting substrate:

d ð X j V Þ
d t

Qð0X j Þþr j V ¼ (16.61)

d ð SV Þ
d t

QðS 0 SÞþr S V ¼ (16.62)

where j ¼ Aorj ¼ B. Noting that the growth rate follows Monod equation, i.e.

m j ¼ m jmax S

(16.63)

K j þS

and

r j ¼ðm j k dj ÞX j (16.64)

r S ¼ m A X A

YF A=S þ m B X B

(16.65)

YF B=S

At steady state, there is no time variation of any quantity. Eqns (16.61) and (16.62) are
reduced to

Q
V X j þðm j k dj ÞX j ¼ 0

(16.66)