Systems Biology (Methods in Molecular Biology)

Subheading3.2). The linearization at an equilibrium point

(u,w) gives a system for the first order perturbation (ξ,η) that is

dξ

dt

¼

1

τ

ðÞξþη,

dη

dt

¼λ

dg

du

ðÞuξη,

or, in vectorial form,

d

dt

ξ

η



¼

ξ

η



where the matrix

¼



1

τ

1

τ

λ

dg

du

ðÞ u 1

0

B

B

@

1

C

C

A

is known as thejacobian matrix. Spectral analysis is based on the

computation of the eigenvalues (and, specifically, on their sign) of,

which are the roots of thecharacteristic polynomialgiven by

pðμÞ:¼detðμIÞ¼ 

1

τ

μ



ð 1 μÞ

λ

τ

dg

du

ðuÞ

¼μ^2 þ 1 þ

1

τ



μþ

1

τ

1 λ

dg

du

ðuÞ



:

ð 17 Þ

Denoting byμ 1 andμ 2 the zeros of the above polynomial, the

following representation holds

pðμÞ¼ðμμ 1 Þðμμ 2 Þ¼μ^2 ðμ 1 þμ 2 Þμþμ 1 μ 2 ,

and therefore, comparing with Eq.17, we deduce that

μ 1 þμ 2 ¼ 1 þ

1

τ



, μ 1 μ 2 ¼

1

τ

1 λ

dg

du

ðÞu



:

Recalling thatdudhðÞ¼u;λ 1 λdgduðÞu from Eq.10,ifdudhðÞu;λ is

positive, the productμ 1 μ 2 of the two roots is positive—indicating

that they have the same sign—and their sumμ 1 +μ 2 is negative—

indicating that they are both negative—so that the equilibrium

state (u,w) is stable. Complementarily, ifdhduðÞu;λ is negative, one

root is positive and the other is negative, consistently with the

appearance of a saddle point, or, in other words, a metastable

equilibrium. The above fact is a special form of the more general

Routh–Hurwitz criterion[57].

120 Chiara Simeoni et al.