Science:Math Exam Resources/Courses/MATH102/December 2015/Question 18 (c)/Solution 1

First, draw the graph of the function $f(I)=\beta I\left(N-{\frac {\mu }{\beta }}-I\right)$. (In the figure below, the red line is the graph of $f$)

Note that in the part (a), we find two steady state solutions $I_{1}=0$ and $I_{2}=N-{\frac {\mu }{\beta }}$. (i.e., $f(I_{1})=f(I_{2})=0$)

Then, from the figure, we can easily see that on $(-\infty ,I_{1})$ and $(I_{2},\infty )$, $f(I)<0$, while on $(I_{1},I_{2})$, $f(I)>0$.

By the hint, this means that on $(-\infty ,I_{1})$ and $(I_{2},\infty )$, $I$ is decreasing, so that we have the left arrows in these intervals on the phase line.

On the other hand, on $(I_{1},I_{2})$, $I$ is increasing and hence we get the right arrow.

Once we draw the arrows, it is obvious that $I_{1}$ is unstable (make empty hole), while $I_{2}$ is stable (fill up the hole).

To summarize, we obtain the following phase line: