Analytical Theory of Biological Populations
In the 50 years that have passed since Alfred Latka's death in 1949 his position as the father of mathematical demography has been secure. With his first demographic papers in 1907 and 1911 (the latter co authored with F. R. Sharpe) he laid the foundatio
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The Plenum Series on Demographic Methods and Population Analysis Series Editor: Kenneth C. Land, Duke University, Durham, North Carolina ADVANCED TECHNIQUES OF POPULATION ANALYSIS Shiva S. Halli and K. Vaninadha Rao ANALYTICAL THEORY OF BIOLOGICAL POPULATIONS Alfred J. Lotka CONTINUITIES IN SOCIOLOGICAL HUMAN ECOLOGY Edited by Michael Micklin and Dudley L. Poston, Jr. CURBING POPULATION GROWTH: An Insider's Perspective on the Population Movement Oscar Harkavy THE DEMOGRAPHY OF HEALTH AND HEALTH CARE Louis G. Pol and Richard K. Thomas FORMAL DEMOGRAPHY David P. Smith HOUSEHOLD COMPOSITION IN LATIN AMERICA Susan M. De Vos HOUSEHOLD DEMOGRAPHY AND HOUSEHOLD MODELING Edited by Evert van Imhoff, Anton Kuijsten, Pieter Hooimeijer, and Leo van Wissen MODELING MUL TIGROUP POPULATIONS Robert Schoen THE POPULATION OF MODERN CHINA Edited by Dudley L. Poston, Jr. and David Yaukey A PRIMER OF POPULATION DYNAMICS Krishnan Namboodiri
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Analytical Theory of Biological Populations Alfred J. Latka Translated and with an Introduction by
David P. Smith University of Texas Houston, Texas
and
Helene Rossert AIDES Federation Nationale Paris, France
Springer Science+Business Media, LLC
Library of Congress Cataloging-1n-Publicat1on Data
Lotka, Alfred J. . 1880-1949. [Theorle analyt1que des assoc1at1ons biolog1ques. Engl1shl Analyt1cal theory of blolog1cal populat1ons 1 Alfred J. Lotka translated and w1th •n Jntroductlon by David P. Sm1th and Helene Rossert. p. cm. -- ' l A.o •
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( Sb) O'r.
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so
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60
70
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80
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100
AGE
Figure 1. Age distribution [Malthusian and stationary populations].
curve tilts so to speak on a pivot, whose position we can find, namely the age a' satisfying the equation b1e
-r 1a'
p(a')=b2 e
-r.
2
a'
p(a')
(52)
thus
(53)
then, in view of(64), p. 71 ,
(54)
CHAPTER 2. RELATIONS INVOLVING MORTALITY AND BIRTHS
69
which gives the age a'
corresponding to the intersection of the age distribution curves b1 e -rta p(a) and b2 e-r;fJ p(a).
Instantaneous Center of Tilt. Rather than jumping abruptly from r 1 to r 2 , let us see what happens when r varies from r 1 to r 2 taking all intermediate values in succession. The momentary center of tilt then changes from moment to moment. We can find the position of the center a' corresponding to a given value of r on putting 8c(a) a (e - ra p(a)) =0 -=-b
ar
ar
(57)
or directly using formula (56). The instantaneous center of tilt is clearly the limiting value of a' when r 2 approaches infinitesimally near to r 1. This limiting value is thus obtained very simply by putting r 2 = r 1 in formula (56), which gives (58)
Consequently, the instantaneous center of tilt of the curve c(a) has as its abscissa the mean age Ar of the Malthusian population growing at the rate of incre
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