This means that the state is stable, at least for infinitely small perturbations of the state variables.
• If just one eigenvalue has a positive real part then the steady state is unstable.
If the imaginary part
, of all eigenvalues
a, ± i
, is zero then the deviation variable
increases exponentially away from
(a, > 0) or decreases exponentially to 0
0). If any
then the movement away from
or towards 0 is oscillatory.
By these rules all steady states are asymptotically stable for Monod kinetics while steady states
s > (K, Ks)%
are unstable for the substrate inhibition kinetics. Oscillations will not occur,
neither for Monod kinetics nor for substrate inhibition kinetics.
The asymptotic stability analysis gives no clue as to the final goal of the path away from an
asymptotically unstable steady state. It does not either tell what happens if an asymptotically
stable steady state is submitted to a perturbation of finite magnitude.
Figure 9,9. Time profiles of the response of a continuous stirred tank reactor which is disturbed by
applying a pulse
to the unstable steady state. Substrate inhibition kinetics with
fx = Q.5 S / (S2
+ S +
S = s/sf =
0.42162 at the steady state
= 0.00338 (a), 0.00038 (b), and - 0.00062 (c).