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Fig. 4.8 Zero isoclines of the Belousov-Zhabotinsky reaction for the vibrational (a) and the
excitable (b) modes: the time variations of the activator (
v
) and inhibitor (
u
) concentrations are
shown below
Based on the kinetic equations and equating their right sides to zero, it is easy to
obtain two equations for zero isoclines:
uu
ð
1
Þ
ð
u
þ ʼ
Þ
q
,
v
u
ʻ
:
v
¼
¼
ð
u
ʼ
Þ
q
The first of these equations is an S-shaped curve, while the second one is a linear
dependence of the activator of the reaction on the inhibitor concentration (Fig.
4.8
).
A theoretical analysis shows that the points of intersection of zero isoclines can
correspond to either stable or unstable states. In the first case, the derivative at the
crossing point must be negative; in the second case it is positive.
Consider the case of the stable point of intersection (Fig.
4.8c
). The figure shows
three possible options that correspond to the gradual approaching of the medium to
a steady state, based on the arbitrary concentrations of the molecular components of
the medium. After that the medium remains in this steady state until a perturbation
occurs. A detailed examination shows that when the diffusion of medium compo-
nents is taken into account, this variant of the intersection of isoclines, called
excitable regime, corresponds to concentration pulses propagating in the medium.
Moving concentration pulses in the reaction-diffusion medium are autowave struc-
tures, whose properties differ from those typical for conventional physical wave
phenomena. They are not reflected, but rather fade at an impermeable boundary.
When concentration pulses encounter each other, they annihilate. Figure
4.9
shows
how concentration pulses bend around an obstacle and pass through small holes.
