Sustainability and Stability - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 844

The first term in Eqn (16.96) describes the growth of prey on substrate and the second the
consumption of prey by predators. The first and second terms in Eqn (16.97) describe the death
of predator in the absence of prey and the growth of predator on prey, respectively. YF p/b is the
yield of predators on prey (g/g),

m 0 b ¼ m b D is the net specific growth rate of prey on

m 0 p ¼ m p is the specific growth rate of predator on prey, 1/(g-h),
and k 0 dp ¼ k dp þD is the apparent specific removal rate of the predator, per hour.
Eqns (16.96) and (16.97) allow a steady-state solution for either batch growth (D ¼ 0) or

a soluble substrate, per hour,

d X p
d t ¼ 0 . Under these conditions

d X b

continuous culture (D > 0), where

d t ¼ 0 and

k 0 dp
m 0 p ¼

k dp þ D
m p

X bF ¼ (16.98)

X pF ¼ m 0 b YF p = b

m 0 p ¼ m b D

YF p=b (16.99)

m p

Let

X b
X bF

X b ¼ (16.100)

X p
X pF

X p ¼ (16.101)

Eqns (16.96) and (16.97) can be rearranged to give

d X b

d t ¼ m 0 b ð1X p ÞX b

(16.102)

d X p

d t ¼k 0 dp ð1X b ÞX p

(16.103)

Eqns (16.102) and (16.103) can be solved with the initial conditions of X b (t ¼ 0) ¼ X b0 and
X p (t ¼ 0) ¼ X p0 . In this case, we would like to examine the stability behavior of the prey e
predator model. Dividing Eqn (16.102) by Eqn (16.103) yields

d X p ¼ m 0 b ð 1 X p Þ X b

d X b

(16.104)

k 0 dp ð1X b ÞX p

Integration of Eqn (16.104) leads to

k 0 dp k 0 dp

Þ m 0 b Þ m 0 b ðX p e X p ðX b e X b ¼ðX p 0 e X p 0 ðX b 0 e X b 0

Þ Þ (16.105)

which governs the trajectory of how the concentrations of prey and predator approaches the
steady-state sol ut ion. One ca n infer from Eqn (16.105) that the steady-state solution is not
achievable as X p ¼ 1 and X b ¼ 1 do not satisf y Eqn (16.105) simultaneously for most
m 0 b and k 0 dp values. However, the values of X p and X b are bound and not zero for any given
set of non trivial and bound initial data. Therefore, the system is sustainable although not
stable. This is a case similar to Fig. 16.11 h).