WeBWorK using host: https://webwork-ptx.aimath.org, format: simple seed: 207

Any population, , for which we can ignore
immigration, satisfies
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
This problem investigates the solutions to such an equation.

(a)
Sketch a graph of against . Note when is
positive and negative.
when is in
when 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 is
increasing and decreasing to decide what the shape of the curves has
to be. Based on your solution curves, why is called the
threshold population?