Sustainability and Stability - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 845

The phase-plane analysis of the system can be made using Eqn (16.105) . Fig. 16.23 a
describes the limit cycles (oscillatory trajectories) of prey e predator populations for different
initial population levels. The Lotka e Volterra model considers the exponential growth of prey
species in the absence of predator and neglects the utilization of substrate by prey species
according to Monod form. The Lotka e Volterra oscillations depend on initial conditions
and change their amplitude and frequency in the presence of an external disturbance. These
types of oscillation are called soft oscillations.
Eqns (16.87) e (16.89) explain the more stable and sustained oscillations observed in nature,
which are independent of initial conditions (that is, hard oscillations). A differential equation
integrator (such as Odexlims used in this text) can be employed to obtain the solutions from
the set of Eqns (16.87) e (16.89) , and the phase e plane data are shown in Fig. 16.23 b. Fig. 16.23 b
shows that the trajectory depends on the initial conditions, whereas the limit cycle (oscilla-
tion) is not dependent on the initial conditions. The dynamic solution does not approach
the steady-state value; however, the populations are bound and nonzero. Even when the
initial populations correspond to those of the steady-state solution (point F in Fig. 16.23 b),
minor fluctuations cause the dynamic populations to shift away and eventually confirm to
the limit cycle (solid line). A 3D plot of the trajectory toward sustainable state and corre-
sponding phase-plane trajectories on substrate versus prey population are shown in
Fig. 16.24 . Therefore, the system is sustainable, although not stable as the single-point/value
steady-state solution cannot be maintained.

16.6.6. Industrial Applications of Mixed Cultures
Growth behavior differences and cell e cell interactions are the main causes of instabilities
in mixed cultures as we have learned. Industrial applications are designed such instabilities
are minimized to maintain process integrity and/or product quality.

(a) (b)

10 1

4

A

10 0

F

E

B

10 -1 D

3

10 -2

10 -3

2

10 -4 C

C

10 -5

1

10 -6

B

A

0 10 -7

0 1 2 3 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1

X b / X bF

X b / X bF

FIGURE 16.23 Phase-plane trajectories for prey e predator interactions. (a) Limit cycles predicted by Lotka e

k 0 dp ¼ 1

2 m 0 b . (b) Trajectories when the substrate limitation and Monod predator growth

Volterra, Eqn (16.105) , with

m bmax , k dp ¼ 0, K b ¼ 5 10 4 S 0 ,YF p/b ¼ 7.14 10 4 ,YF b/S S 0 ¼ 3 X b0 , K p ¼
0.4 X b0 . The predicted steady-state point is defined by X pF and X bF . A, B, and C (and D, E, and F) represent different
initial conditions.

rate are used: D ¼ 0.6 m bmax , m pmax ¼ 0.96