Systems Biology (Methods in Molecular Biology)

J¼

∂F 1

∂X 1



∂F 1

∂Xn

⋮⋱⋮

∂Fn

∂X 1



∂Fn

∂Xn

2

6

6

6

4

3

7

7

7

5

Solving the determinant of this Jacobian will provide the linear

stability of a nonlinear system near an equilibrium point.

Consider the Brusselator again. For simplicity, all rate constants

are set to 1. Eqs.16 and 17 become

dX 1

dt

¼Aþ½ŠX 12 ½ŠX 2 ðÞ 1 þB½ŠðX 1 31 Þ

dX 2

dt

¼BX½Š 1 ½ŠX 12 ½ŠðX 2 32 Þ

The Brusselator has an equilibrium point atX 1 ¼ Aand

X 2 ¼B/Awhen Eqs.31 and 32 are set to zero and solved. Its

Jacobian at this point is therefore

J¼ B^1 A

2

B A^2



ð 33 Þ

Solving the determinant of Eq.33 reveals

λ¼

ðÞBA 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðÞBA 12  4 A

q

2

ð 34 Þ

BecauseAis always positive, it can be shown that the Brusse-

lator is stable whenB<A^2 + 1 and outside this regime, instability

or Hopf bifurcation can be achieved.

The phase-space plots are a simple and powerful way to observe

stability of a nonlinear system. They show all possible states of a

system and by observing the focus, the stability of the system can be

observed. To illustrate, by tracking the time trajectories ofX 1 and

X 2 , we can deduce the type of stability for a range of parameter

values. For example, by choosing different values ofAandB,we

can achieve steady state or lose stability leading to oscillatory pat-

terns (Fig.11).

In a similar fashion, we can also determine the stability focus of

the Goodwin model at the equilibrium point. Figure12 shows the

phase-space plots of the Goodwin model for a range of parameter

values. From the Brusselator and Goodwin examples, understand-

ing nonlinear dynamics requires precisely tuned parameter values,

as any small variations can lead to drastic changes in dynamics or

stability.

192 Kumar Selvarajoo