
Consider the following chemical reaction
where molecule of
combine reversibly to form
and,
are the reaction rates.
If are the concentrations of
respectively, then according to the Law of Mass Action, the reaction is governed by
Without solving this initial-value problem quantitatively, the future state of system can be predicted through qualitatively analyzing how the value of changes over the course of time.
To this end, we solve (0-1) for first:
Substitute it in (0-2),
It simplifies to
Let
we have
Substituting for
respectively in (1-1) gives
It means or
Integrate it with respect to ,
Let
,
we have
is a monotonically decreasing function.
In addition, Descartes’ rule of signs reveals that
has exactly one real positive root.
By definition, this root is the in an equilibrium point
.

Fig. 1
Hence,
As time advances, if
. Otherwise
,
Dividing (0-2) by (0-1) yields
That is,
By (0-3),
.
And so,
Since is a line with a negative slope,
is a monotonically decreasing function of
Moreover, from (0-1) and (0-2), we see that
is an equilibrium point.
i.e.,
All points on the curve in the first quadrant of x-y plane are equilibriums
.
Based on (1-1), (1-2) and (1-3), for a initial state ,
.
Similary,
.

Fig. 2
A phase portrait of the system is shown in Fig. 3.

Fig. 3
It shows that on the trajectory approaches the equilibrium point
over the course of time. Namely, the system is asymptotically stable.
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