The Discrete Time Matrix Model

  • John Impagliazzo
Part of the Biomathematics book series (BIOMATHEMATICS, volume 13)


The discrete time recurrence model and the continuous time model which were discussed in Chapters Three and Four focused on the births that occurred at a particular time and their relationship to the births that occurred prior to this time. In each case the age distribution of the population was discussed almost as an adjunct to the model and not as an integral part of it. Perhaps this was unfortunate since it was the asymptotic age distribution of the model that gave meaning to the concept of a ‘stable’ population.


Transition Matrix Stable Theory Projection Matrix Transition Matrice Principal Eigenvalue 
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Notes for Chapter Five

  1. [1]
    Bernardelli, H. (1941), Population waves. Journal of the Burma Research Society, vol. 31, part 1, pp. 1–18.Google Scholar
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    Lewis, E.G. (1942), On the generation and growth of a population. Sank- hya, vol. 6, pp. 93–96.Google Scholar
  3. [3]
    Leslie, P.H. (1945), On the use of matrices in certain population mathematics. Biometrika, vol. 33, pp. 183–212.CrossRefMATHMathSciNetGoogle Scholar
  4. [4]
    Ibid., p. 187.Google Scholar
  5. [5]
    Ibid., p. 187.Google Scholar
  6. [6]
    Gantmacher, F.R. (1959), Applications of the theory of matrices, p. 61.MATHGoogle Scholar
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    Perron, O. (1907), Zur Theorie der Matrizen. Mathematische Annalen, vol. 64, p. 248 ff.CrossRefMATHMathSciNetGoogle Scholar
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    Frobenius, G. (1912), Über Matrizen aus nicht negativen Elementen. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, Berlin, p. 456ff.Google Scholar
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    Gantmacher, op. cit., p. 64 ff.Google Scholar
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    Meyer, W. (1982), Asymptotic birth trajectories in the discrete form of stable population theory. Theoretical Population Biology, vol. 21, no. 2, pp. 167–170.CrossRefMATHMathSciNetGoogle Scholar
  12. [12]
    The development of this section is primarily due to the invention of the author.Google Scholar
  13. [13]
    Moore, John T. (1968), Elements of linear algebra and matrix theory, p. 327 ff.Google Scholar
  14. [14]
    Noble, Ben (1969), Applied linear algebra, p. 361 ff.MATHGoogle Scholar
  15. [15]
    Pollard, J.H. (1973), Mathematical models for the growth of human populations, pp. 44–45.MATHGoogle Scholar
  16. [16]
    Noble, Ben, op. cit., pp. 363–366.Google Scholar
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    Lefkovitch, L.P. (1965), The study of population growth in organisms grouped by stages. Biometrics, vol. 21, pp. 1–18.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • John Impagliazzo
    • 1
  1. 1.Nassau Community CollegeState University of New YorkGarden CityUSA

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