## Remarks about the solutions to the exercises

The solutions to ex 1,2 are similar to the solutions to the examples
discussed on p1, and their graphs (Figures 3,4) are all fairly similar
(one equilibrium value, and if the initial population is less than
that, then *P* grows to that value and levels off, whereas if the
initial population is more than that, then *P* decreases to that
value and levels off).
However, the solution to ex 3 is a bit different. In finding
when *P'* is 0, you have to solve a quadratic equation
in *P*, giving (generally) **two** solutions, two equilibrium
values. Depending on the actual values of the various constants, the
lower one could be quite close to 0, and barely shows on the graph
(Figure 5), but if it's higher (or if your scale is big enough!), then
it shows clearly, and you can see the general behaviour better (Figure
6). Which is: two equilibrium values, and if *P* starts lower
than the lower one, *P* will (quickly) decrease to 0; if *P*
starts between the lower one and the higher one, *P* will
increase to the higher equilibrium value; and if *P*
starts higher than the higher one, *P* will decrease to the
higher equilibrium value (Figure 6).