Science:Math Exam Resources/Courses/MATH102/December 2014/Question C 02 (b)/Solution 1

To determine the stability of each steady state, we just need to find the increasing and decreasing regions. As $c$ represents the concentration of a substance, we only need to consider the positive half of the real axis. The three steady states divide it into three intervals

${\begin{aligned}c\in (0,5),\quad P(c)-D(c)<0\quad {\text{decreasing}},\\c\in (5,20),\quad P(c)-D(c)>0\quad {\text{increasing}},\\c\in (20,\infty ),\quad P(c)-D(c)<0\quad {\text{decreasing}},\end{aligned}}$

If we choose a initial condition in the interval $c_{0}\in (0,5)$, as $P(c)-D(c)$ is negative, $c$ would decrease and approach the steady state $c=0$ as time goes on. So $c=0$ is stable. Similarly, we find $c=5$ is unstable and $c=20$ is stable.