Abstract
The formula developed in the preceding chapter
is in a certain sense incomplete. It expresses a relation between the birth rate b and the constant rate of increase r without, however, definitively fixing their values. For this a second, independent relation is required. Evidently there exists an essential factor of some sort that up to now has not entered into our computations. That factor is the fertility of the population.
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The period of reproduction is more precisely defined for women than for men, and maternal parentage is almost always known, even in the case of illegitimate children.
Potential fertility, a quantity whose measurement escapes us, will not interest us at all in this discussion.
We note in passing that equation (134) still remains valid if p(a) and m(a) are functions of time t. However, in the present discussion we treat the case in which the life table and fertility by age are fixed.
R. Kuczynski, Fertility and Reproduction, 1932, pp. 62, 90.
Translators’ note: These comments reflect Lotka’s irritation with Kuczynski, who did not understand the fundamental nature of Lotka’s contributions to stable theory and assigned credit for key parts of it, at least in concept, to Bortkiewicz. Lotka’s correction is accurate, if also strident. The various threads are disentangled in P. Samuelson, “Resolving a historical confusion in population analysis,” Human Biology,1976, v. 48,pp. 559–580; also in D. P. Smith and N. Keyfitz, editors, Mathematical Demography: Selected Papers,pp. 109–129. Springer—Verlag, Berlin, 1976.
See A. J. Lotka, “The progeny of a population element,” American Journal of Hygiene, 1928, v. 8, p. 900.
We could equally write + 2’rtni upon making n span entirely negative values.
It must be noted that the curves in the upper part of Figure 14 were traced only to complete the symmetry of the design. The values of u and v corresponding to points of intersection in the two upper quadrants should be rejected, as not satisfying the condition mentioned earlier, by which there is but one real root p, and the real part of every complex root is less than p.
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Lotka, A.J. (1998). Relations Involving Fertility. In: Analytical Theory of Biological Populations. The Springer Series on Demographic Methods and Population Analysis. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-9176-1_7
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