Systems Biology (Methods in Molecular Biology)

then it is called stable equilibrium point. As in scalar equations,

by expanding the Taylor’s series forf(x,y) andg(x,y);

u_¼x_¼fðx,yÞ

¼fðxþu,yþvÞ

¼fðx,yÞþfxðx,yÞuþfyðx,yÞvþhigher order terms...

¼fxðx,yÞuþfyðx,yÞv þhigher order terms...:

ð 31 Þ

Similarly,

v_¼y_¼gðx,yÞ

¼gðxþu,yþvÞ

¼gðx,yÞþgxðx,yÞuþgyðx,yÞv þhigher order terms...

¼gxðx,yÞuþgyðx,yÞvþhigher order terms...:

ð 32 Þ

Sinceuandvare assumed to be small, the higher order terms

are extremely small, we can neglect the higher order terms and

obtain the following linear system of equations governing the

evolution of the perturbationsuandv,

u_

v_



¼

fxðx,yÞ fyðx,yÞ

gxðx,yÞ gyðx,yÞ



u

v



where the matrix

fxðx,yÞ fyðx,yÞ

gxðx,yÞ gyðx,yÞ



is called Jacobian matrixJof the nonlinear system, where the

raws of the Jacobian are the derivatives computed in the steady

state. The above linear system foruandvhas the trivial steady

state (u,v)¼(0, 0), and the stability of this trivial steady state

is determined by the eigenvalues of the Jacobian matrix at the

equilibrium point (0, 0) whereJ(0, 0) give the eigenvalues by

solving the characteristic equation det(JλI)¼0, whereIis

the identity matrix andλare the eigenvalues.

As a summary,

l Asymptotically stable. A critical point is asymptotically stable

if all eigenvalues of the jacobian matrixJare negative, or

have negative real parts.

l Unstable. A critical point is unstable if at least one eigen-

value of the jacobian matrixJis positive, or has positive

real part.

Inverse Problems in Systems Biology 87