The name is absent

ON THE ANALYSIS OF LIBRARY GROWTH

dP / dt - _n
P ^o '

Verhulst assumed that the right-hand side should be replaced by
a more general function, say f(t,P(t)), which might depend on
the time t as well as on the population P(t). If the population
growth rate depends only on the population, and not on the time
(this is realistic; for instance, had the United States been dis-
covered 300 years earlier, its population growth is likely to have
proceeded in the same manner as actually occurred) then f depends
only on P(t). Verhulsfs next assumption was that this depend-
ence could be expressed by means of a power series, that is, in
the form

_g) f(P(t) ) = a_o + a₁P(t) + a₂P(t)² +......

(a_o >o)

further, he made the approximation that all of the terms on the
right except the first two in eq(3) could be neglected, and arrived
at the differential equation

The solution is the logistic function,

5)                   P(t) = ⅛^laJ- . .

» . ∙α t + c
1 + e о

where c is a constant which must be determined from the value of
P(t₀) at some time t₀∙ This equation involves three parameters
ɑɑ, aɪ, and c instead of the two appearing in the exponential. For
P(t) to be positive it is necessary that aɪ be negative; then the
term a₁P(t) in the differential eq (4) corresponds to an influence
retarding the growth of P with time, and in fact, as t becomes
indefinitely large, P(t) does not, but rather approaches the value
-α⁰∕α¹ as an absolute maximum. The positive number α₀ repre-
sents, as before, the growth exponent. Pearl (Ref. 10) provides
a useful example of a logistic curve, making use of Carlson’s
data on the growth of yeast cells; we have reproduced this curve
in Figure 6. It illustrates both the shape of the logistic and the
fact that logistic curves do sometimes provide accurate representa-
tions of growth processes. Unfortunately, the complex processes
of library imprint growth, human population growth, or other

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