Suppose the population model (4) is modifi­ed to be

dP/dt = P(bP - a).

(a) If a > 0, b > 0 show by means of a phase portrait (see page 40) that, depending on the initial condition P(0) = P₀, the mathematical model could include a doomsday scenario (P(t) → ∞) or an extinction scenario (P(t) → 0).

(b) Solve the initial-value problem

dP/dt = P(0.0005P - 0.1), P(0) = 300.

Show that this model predicts a doomsday for the population in a ­finite time T.

(c) Solve the differential equation in part (b) subject to the initial condition P(0) = 100. Show that this model predicts extinction for the population as t → ∞.