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RICE UNIVERSITY STUDIES
Since two constants determine a straight line, it follows that two
constants completely determine the equation of exponential growth.
Complex social or natural processes will usually require more than
two constants for their accurate representation, so that exponential
growth laws should not be able to represent such processes for
long periods except in special circumstances.
If the growth exponent is positive, then P(t) will increase in-
definitely as time passes. For populations on the earth, this can-
not happen, so it must be the case that a population which follows
an exponential growth process must ultimately change its rate of
growth. This change cannot be described without introducing
new assumptions; it is by no means clear what these assumptions
should be for human or book populations, although a number of
proposals have been made.
If P(t) does grow exponentially, then it doubles every log2∕α0
units of time; if time is measured in years (as will be assumed
from here on), then P(t) doubles every (0.69315∕α0) years.
The annual rate of increase is given by (ea°—1) ; if α0 is small,
this is approximately equal to (α0 ÷ αo2∕2). For instance, if α0 =
0.1, then the annual growth rate will be 0.1052. . . , just more than
10 percent. For growth rates, or growth exponents, less than 0.1,
the growth exponent is essentially the same as the annual growth
rate.
Recognition that most growth processes could not be described
by a two-parameter curve such as the exponential led many in-
vestigators to attempt generalizations having a greater number
of parameters that could be adjusted so as to fit the data. This is
a difficult problem. There are mathematical theorems which state
that any sufficiently smooth curve—and all of the growth curves
that we are considering satisfy this condition—can be represented
as closely as desired if enough parameters are used. The practical
problem is to provide a representation that uses as many param-
eters as are necessary, but no more, so that there is some hope
that the representation actually corresponds to the actual under-
lying physical or probabilistic processes in a natural manner. The
earliest well-reasoned generalization of the exponential to fit
growth data was made by Verhulst (Ref. 14) in the mid-nineteenth
century; his discovery of the logistic curve was independently re-
peated by Pearl and Reed (Ref. 11) in 1920. Verhulsfs idea was
to replace the differential equation (1) by the “simplest” generali-
zation. Upon dividing both sides of eq(l) by P(t), one finds