By N. Keyfitz, Hal Caswell
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2). 1) is in terms of r¯, the arithmetic mean rate over the interval from zero to T : N (T ) = N0 er¯T , where r¯ = 1 T T r(t) dt. 0 In words, the numerical eﬀect on the population total of a varying rate of growth is the same as though the arithmetic average rate applied at each moment over the time in question. Special Cases of Changing Rates. We may try various special functions for r(t). 1) reduces to N (T ) = N0 erT . If r(t) is equal to k/t and we start at time 1, then T N (T ) = N1 exp 1 k dt t = N1 exp(k log T ) = N1 T k .
His demographic transition refers to the uniformity of change from high to low birth and death rates among the countries of Europe and those overseas that had developed industrially. They showed ﬁrst a decline in death rates, starting at the beginning of the nineteenth century or earlier, followed after a longer or shorter interval by a decline in birth rates. In France the fall in births was nearly simultaneous with that in deaths, whereas in England births did not begin to decline until about 1870, but all countries resembled one another to some degree (Flieger 1967).
But the 1940 Census was a disappointment—it fell far below the curve, and the logistic was dropped amid a barrage of criticism no less intense than the earlier enthusiasm. The fall in the birth rate over most of the last 20 years, however, makes the logistic look much better, and this, along with visible diﬃculties in the environment, has brought the logistic back into fashion in some circles. 41 million persons, a ceiling not sharply contradicted by current tendencies. As Pearl expected, the logistic is hardly invariant with respect to the period over which it is ﬁtted.