Shortly we will solve specific examples of the logistic d.e. (1), instead of obtaining the general
solution (2), since the latter is algebraically rather messy. (It is somewhat less complicated to verify
that y ′ in (1) can be obtained by taking the derivative of (2).)
Unrestricted Versus Restricted Growth
FIGURE N9–9aFIGURE N9–9b
BC ONLY
In Figures N9–9a and N9–9b we see the graphs of the growth functions of Cases I and III. The
growth function of Case I is known as the unrestricted (or uninhibited or unchecked) model. It is not
a very realistic one for most populations. It is clear, for example, that human populations cannot
continue endlessly to grow exponentially. Not only is Earth’s land area fixed, but also there are
limited supplies of food, energy, and other natural resources. The growth function in Case III allows
for such factors, which serve to check growth. It is therefore referred to as the restricted (or
inhibited) model.
The two graphs are quite similar close to 0. This similarity implies that logistic growth is
exponential at the start—a reasonable conclusion, since populations are small at the outset.
The S-shaped curve in Case III is often called a logistic curve. It shows that the rate of growth y ′:
(1) increases slowly for a while; i.e., y ′′ > 0;
(2) attains a maximum when y = A/2, at half the upper limit to growth;
(3) then decreases (y ′′ < 0), approaching 0 as y approaches its upper limit.
It is not difficult to verify these statements.
Applications of Logistic Growth