This (simplest!) model demonstrates logistic growth.The original differential equation looks like
y'(t) = b y(t) (1 - y(t)/K)
where K is the carrying capacity of the quantity y.
But if we divide each side of the equation by K, we obtain
d(y/K)/dt = b (y/K) (1-y/K)
Defining a new variable w, the population relative to its carrying capacity, we obtain
dw/dt = b w (1 - w)
Finally we divide both sides by b, to write
dw/d(bt) = w (1 - w)
So if we work in dimensionless time units of bt, we have
w' = w (1 - w)
where the derivative is with respect to the variable bt=τ .
This equation, as simple as possible, contains all the dynamics (all the ways
the population can behave), while masking the "trivialities"; but it kind of
hides the physical aspects of the problem. So it's easy to study, but harder to
interpret: alas, you can't have it all!:)
τ=1 when t=1b: so if b=.5/year, then τ=1 when t=2.
the larger b (the greater the birthrate), the shorter the real time t to give τ=1.