Resumen
Este artículo presenia una aproximación elemental al estudio de la matriz de proyeccian de poblaci'on, el reverso (o inversa) matriz de proyección, el proceso de desarrollo de la poblacian generado por la matriz de proyección, el “a priori” proceso de desarrollo de población asociado con la matriz de proyección inversa, la distribución de edades eetoble que pertenece a la matriz de proyección, la ecuación característica de la matriz de proyección y las raices laienies de la ecuación característica.
Nosotros introducimos algunos nuevos indices que miden el “valor reproductivo eventual” de los individuos en los varios intervalos de edades de una población, y mostramos porque los índices presentados aquí son preferibles a los indices de la misma naturaleza presentados anteriormente por R. A. Fisher y P. H. Leslie.
Para poder seguir la mayor parte de la presenie exposición, el lector precisa iener solamenie nociones elemeniales de algunos de los terminosusados en algebra de matrics. Esta aproximación elemental va a llevarnos a la introducción de algunas nuevas fórmulas las cuoles simplijican a la vez la comprensión y el cálculo de varias cantidades que son relevantes a la demografiá y a la teoría matemática del desarrollo de la población.
Summary
This article presents an elementary approach to the study of the population projection-matrix, the reverse (or inverse) projection-matrix, the process of population growth generated by the projection-matrix, the “prior” process of population growth associated with the reverse projection-matrix, the stable age-distribution that pertains to the projection-matrix, the characteristic equation of the projection-matrix, and the latent roots of the characteristic equation. We also introduce some new indices that measure the “eventual reproductive value” of the individuals in the various ageintervals in a population, and we show why the indices presented here. are preferable to a related index presented earlier by R. A. Fisher and P. H. Leslie.
In order to follow the major part of the present exposition, the reader will need only a beginner's understanding of a few of the terms used in matrix algebra. This elementary approach will lead to the introduction of some new formulas that will simplify both the understanding and the calculation of various quantities that are relevant to demography and to the mathematical theory of population growth.
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References
See, for example, H. Bernardelli, “Population Waves,”Journal of the Burma Research Society, XXXI (1941), 1–18; E. G. Lewis, “On the Generation and Growth of a Population,”Sankya, VI (1942),93-96; P. H. Leslie, “On the Use of Matrices in Population Mathematics,”Biometrika, XXXIII (1945), 183-212; N. Keyfitz, “The Population Projection as a Matrix Operator,”Demography, I (1964), 56-73; E. M. Murphy, “The Latent Roots of the Population Projection Matrix,”Demography, III (1966), 259–75.
See, for example, A. J. Lotka, “Théories Analytique des Associations Biologiques, Part II: Analyse Démographique avec Application Particuliere a l'Espèce Humaine,”Actualites Sci., DCCLXXX (Paris: Hermann, 1939), 1–149.
See W. Feller, “On the Integral Equation of Renewal Theory,”Annals of Mathematical Statistics, XII (1941), 243–67; T. E. Harris,The Theory of Branching Processes (New Jersey: Prentice-Hall, Inc., 1963), 161-63.
R. A. Fisher,The Genetical Theory of Natural Selection (Oxford: Clarendon Pres, 1930); and P. H. Leslie, “Some Further Notes on the Use of Matrices in Population Mathematics,”Biometrika, XXXV (1948), 213–45.
For a discussion of the problems that arise when both sexes are considered as sub-populations within a single system of population growth, see, for example, L. A. Goodman, “Population Growth of the Sexes,”Biometrics, IX (1953), 212–25; N. Keyfitz, “On the Interaction of Populations,”Demography, II (1965), 276–88; E. M. Murphy, “A Generalization of Stable Population Techniques” (Ph.D. thesis, University of Chicago, 1966); and L. A. Goodman, “On the Age-SexComposition of the Population That Would Result from Given Fertility and Mortality Conditions,”Demography, IV, 2 (1967), 423–41.
ee, for example, P. H. Leslie, “On the Use of Matrices in Certain Population Mathematics,”loc. cit. ; N. Keyfitz, “The Population Projection as a Matrix Operator,”loc. cit. Demography, I (1964), 56–73; N. Keyfitz and E. M. Murphy,Comparative Demographic Computations (Chicago: Population Research and Training Center, University of Chicago, 1964).
Note that an understanding of the preceding derivation of equations (6) and (8), and the corresponding explicit expression (10a) and (10b), does not require any familiarity with the concept of the inverse of a matrix as it is presented in the theory of matrix algebra. For a different derivation based on the calculation of the inverse of a matrix, see P. H. Leslie, “On the Use of Matrices in Certain Population Mathematics,”loc. cit.
N. Keyfitz, “The Population Projection as a Matrix Operator,”loc. cit., first calculated the adjoint matrix ofM and the determinant of M, and then divided each entry in the adjoint matrix by the determinant. E. M. Murphy, “The Latent Roots of the Population Projection Matrix,”loc. cit. Demography, III (1966), 259–75, first calculated the characteristic equation ofM and the adjoint of the characteristic equation, and then he divided the appropriate coefficient in the latter equation by the appropriate coefficient in the former equation.
E. M. Murphy, “The Latent Roots of the Population Projection Matrix,”loc. cit.,
J. H. Pollard, “On the Use of the Direct Matrix Product in Analyzing Certain Stochastic Population Models,”Biometrika, LIII (1966), 397–416.
N. Keyfitz, “The Population Projection as a Matrix Operator,”loc. cit., p. 62.
P. H. Leslie, “On the Use of Matrices in Population Mathematics,”loc. cit.
——.
N. Keyfitz, “The Intrinsic Rate of Natural Increase and the Dominant Root of the Projection Matrix,”Population Studies, XVIII (1965), 293–308.
P. H. Leslie, “On the Use of Matrices in Certain Population Mathematics,”loc. cit.
E. M. Murphy, “The Latent Roots of the Population Projection Matrix,”loc. cit.
See note 6.
P. H. Leslie, “On the Distribution in Time of the Births in Successive Generations,”Journal of the Royal Statistical Society, Series A (General), CXI (1948), 44–53; and by L. A. Goodman, “ the Reconciliation of Mathematical Theories of Population Growth,”Journal of the Royal Statistical Society, Series A (General), CXXX (1967), 541–53.
L. A. Goodman, “On the Reconciliation of Mathematical Theories of Population Growth,”loc. cit..
L. A. Goodman, “On the Reconciliation of Mathematical Theories of Population Growth,”loc. cit..
See, for example, P. H. Leslie, “On the Use of Matrices in Population Mathematics”loc. cit., and N. Keyfitz, ╒econciliation of Population Models: Matrix, Integral Equation, and Partial Fraction,”Joumal of the Royal Statistical Society, Series A (General), CXXX (1967), pp. 61–83; and “Estimating the Trajectory of a Population,”Fifth Berkeley Symposium on Mathematical Statistics and Probability (California: University of California Press, 1967).
See note 55 See, for example,.
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The research was supported in part by a grant from the National Science Foundation and in part by a research contract from the Army Research Office,the Office of Naval Research, and Air Force Office of Scientific Research.
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Goodman, L.A. An elementary approach to the population projection-matrix, to the population reproductive value, and to related topics in the mathematical theory of population growth. Demography 5, 382–409 (1968). https://doi.org/10.1007/BF03208583
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DOI: https://doi.org/10.1007/BF03208583