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
Substituting for respectively in (1-1) gives
It means or
Integrate it with respect to ,
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 .
As time advances, if . Otherwise ,
Dividing (0-2) by (0-1) yields
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.
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 ,
A phase portrait of the system is shown in Fig. 3.
It shows that on the trajectory approaches the equilibrium point over the course of time. Namely, the system is asymptotically stable.