Any population, $P$, for which we can ignore immigration, satisfies $$\frac{dP}{dt} =\hbox{Birth rate} - \hbox{Death rate}.$$ For organisms which need a partner for reproduction but rely on a chance encounter for meeting a mate, the birth rate is proportional to the square of the population. Thus, the population of such a type of organism satisfies a differential equation of the form $$\frac{dP}{dt} = a P^2 - b P\quad\hbox{with } a,\,b>0.$$ This problem investigates the solutions to such an equation.

(a) Sketch a graph of $dP/dt$ against $P$. Note when $dP/dt$ is positive and negative.
$dP/dt < 0$ when $P$ is in
$dP/dt > 0$ when $P$ is in
(Your answers may involve a and b. Give your answers as an interval or list of intervals: thus, if dP/dt is less than zero for P between 1 and 3 and P greater than 4, enter (1,3),(4,infinity).)

(b) Use this graph to sketch the shape of solution curves with various initial values: use your answers in part (a), and where $dP/dt$ is increasing and decreasing to decide what the shape of the curves has to be. Based on your solution curves, why is $P=b/a$ called the threshold population?

If $P(0) > b/a$, what happens to $P$ in the long run?
$P\to$

If $P(0) = b/a$, what happens to $P$ in the long run?
$P\to$

If $P(0) < b/a$, what happens to $P$ in the long run?
$P\to$