
156 Feedback Control in Systems Biology
0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2
2.5
3
3.5
4
x
FIGURE 5.4: Phase plane of the simple chemical oscillator (5.7). The thick
curve denotes the limit cycle.
Example 5.1
On the basis of the results above, among the possible candidates for chemical
systems which exhibit limit cy c les, the s implest such reaction mechanism can
be shown to be, [4]:
X
k
1
−−⇀
↽−−
k
1i
A, B
k
2
−→ Y, 2X + Y
k
3
−→ 3X. (5.6)
Applying the law of mass ac tion, the reaction kinetics are describ ed by
˙x = k
3
x
2
y + k
1i
a − k
1
x (5.7)
˙y = k
2
b − k
3
x
2
y (5.8)
The system exhibits a limit cycle for certain choices of the parameters, as
shown by the phase plane in Fig. 5.4 ...