model (4) is modified to be (a) If show by means of a

QUESTION:

Doomsday or Extinction Suppose the population model (4) is modified to b

\(\frac{d P}{d t}=P(b P-a)\)

(a) If a > 0, b > 0 show by means of a phase portrait (see page 39) that, depending on the initial condition \(P(0)=P_{0}\) the mathematical model could include a doomsday scenario \((P(t) \longrightarrow \infty)\) or an extinction scenario \((P(t) \longrightarrow 0)\).

(b) Solve the initial-value problem Show that this model predicts a doomsday for the population in a finite time T.

\(\frac{d P}{d t}=P(0.0005 P-0.1), P(0)=300\)

(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 \(t \longrightarrow \infty\).

Text Transcription:

frac{d P}{d t}=P(b P-a)}

P0 = P_0

P(t) \longrightarrow \infty

P(t) \longrightarrow 0

frac{d P}{d t}=P(0.0005 P-0.1), P(0)=300

t \longrightarrow \infty