Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T05:23:11.506Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic stable population growth

Published online by Cambridge University Press:  14 July 2016

Kenneth Lange  and
William Holmes
Kenneth Lange*
Affiliation:
University of California
William Holmes*
Affiliation:
University of California
*
∗Postal address: Department of Biomathematics, University of California, Los Angeles, CA 90024, U.S.A.
∗Postal address: Department of Biomathematics, University of California, Los Angeles, CA 90024, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

In classical demographic theory, the age structure of a population eventually stabilizes, and the population as a whole grows at a geometric rate. It is possible to prove stochastic analogues of these results if vital rates fluctuate according to a stationary stochastic process. The approach taken here is to study the action of random matrix products on random vectors. This permits the application of Hilbert's projective metric and leads to considerable simplification of the ergodic and central limit theory of population growth.

Keywords

DEMOGRAPHYHILBERT'S PROJECTIVE METRICERGODICCENTRAL LIMIT THEOREM
Type
Research Papers
Information
Journal of Applied Probability , Volume 18 , Issue 2 , June 1981 , pp. 325 - 334
Copyright
Copyright © Applied Probability Trust 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This research supported in part by the University of California, Los Angeles; the Mailman Research Center; NIH Research Career Development Award, K04 HD00307; and NRSA Training Grant GM 7191.

References

Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Bushell, P. J. (1973a) Hilbert's metric and positive contraction mappings in a Banach space. Arch. Rational Mech. Anal. 52, 330–338.Google Scholar
Bushell, P. J. (1973b) On the projective contraction ratio for positive linear mappings. J. London Math. Soc. (2) 6, 256–258.Google Scholar
Cohen, J. E. (1976) Ergodicity of age structure in populations with Markovian vital rates, I: Countable states. J. Amer. Statist. Assoc. 71, 335–339.Google Scholar
Cohen, J. E. (1977a) Ergodicity of age structure in populations with Markovian vital rates, II: General states. Adv. Appl. Prob. 9, 18–37.Google Scholar
Cohen, J. E. (1977b) Ergodicity of age structure in populations with Markovian vital rates, III: Finite-state moments and growth rate; an illustration. Adv. Appl. Prob. 9, 462–475.Google Scholar
Cohen, J. E. (1979a) Contractive inhomogeneous products of non-negative matrices. Math. Proc. Camb. Phil. Soc. 86, 351–364.Google Scholar
Cohen, J. E. (1979b) Ergodic theorems in demography. Bull. Amer. Math. Soc. 1, 275–295.Google Scholar
Furstenberg, H. and Kesten, H. (1960) Products of random matrices. Ann. Math. Statist. 31, 457–469.Google Scholar
Golubitsky, M., Keeler, E. B. and Rothschild, M. (1975) Convergence of the age structure: application of the projective metric. Theoret. Popn. Biol. 7, 84–93.Google Scholar
Ishitani, H. (1977) A central limit theorem for the subadditive process and its application to products of random matrices. Publ. RIMS, Kyoto Univ. 12, 565–575.Google Scholar
Kingman, J. F. C. (1968) The ergodic theory of subadditive stochastic processes. J. R. Statist. Soc. B 30, 499–510.Google Scholar
Kingman, J. F. C. (1973) Subadditive ergodic theory. Ann. Prob. 1, 883–909.Google Scholar
Kingman, J. F. C. (1976) Subadditive ergodic theory. In Ecole d'Eté de Probabilités de Saint-Flour V-1976, ed. Badrikian, A., Kingman, J. F. C. and Kuelbs, J. Springer-Verlag, New York.Google Scholar
Lange, K. (1978) The momentum of a population whose birth rates gradually change to replacement levels. Math. Biosciences 40, 225–231.Google Scholar
Lange, K. (1979) On Cohen's stochastic generalization of the strong ergodic theorem of demography. J. Appl. Prob. 16, 496–504.Google Scholar
Lopez, A. (1961) Some Problems in Stable Population Theory. Office of Population Research, Princeton University, Princeton, NJ.Google Scholar
Walter, P. (1975) Ergodic Theory — Introductory Lectures. Springer-Verlag, New York.Google Scholar
Ziebur, A. D. (1979) New directions in linear differential equations. SIAM Review 21, 57–70.Google Scholar