The differential equation in Example $3$ of Section $2\mathrm{.}1$ is a well$-$known population model. Suppose the DE is changed to

dPdt

$=$ P$($aP $$ b$),$

where a and b are positive constants. Discuss what happens to the population P as time t increases.

If

P$0>$ b$/$a$,$

then

P$($t$)?0$ a$/$b b$/$a

as t increases; if

$0<$ P$0<$ b$/$a$,$

then

P$($t$)?0$ a$/$b b$/$a

as t increases.