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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