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.
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?