Any population, p, for which we can ignore immigration, satisfies

dpdt=birth rate−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

dpdt=ap2−bpwith 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 : (0, b/a)

dp/dt> 0 when p is in : (b/a, infinity)
(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,,

(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→

if p(0)=b/a, what happens to p in the long run?
p→
if p(0) p→