Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-26T01:21:01.747Z Has data issue: false hasContentIssue false
  • English
  • Français

Ergodicity of age structure in populations with Markovian vital rates, III: Finite-state moments and growth rate; an illustration

Published online by Cambridge University Press:  01 July 2016

Joel E. Cohen
Joel E. Cohen*
Affiliation:
The Rockefeller University, New York
Get access Rights & Permissions [Opens in a new window]

Abstract

Leslie (1945) models the evolution in discrete time of a closed, single-sex population with discrete age groups by multiplying a vector describing the age structure by a matrix containing the birth and death rates. We suppose that successive matrices are chosen according to a Markov chain from a finite set of matrices. We find exactly the long-run rate of growth and expected age structure. We give two approximations to the variance in age structure and total population size. A numerical example illustrates the ergodic features of the model using Monte Carlo simulation, finds the invariant distribution of age structure from a linear integral equation, and calculates the moments derived here.

Keywords

ERGODIC SETSLESLIE MATRICESNON-NEGATIVE MATRICESPOPULATION DYNAMICSPRODUCTS OF RANDOM MATRICESSTOCHASTIC POPULATION MODELS
Type
Research Article
Information
Advances in Applied Probability , Volume 9 , Issue 3 , September 1977 , pp. 462 - 475
Copyright
Copyright © Applied Probability Trust 1977 

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.)

References

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. (1977) Ergodicity of age structure in populations with Markovian vital rates, II: General states. Adv. Appl. Prob. 9, 18–37.*.Google Scholar
Furstenberg, H. and Kesten, H. (1960) Products of random matrices. Ann. Math. Statist. 31, 457–469.Google Scholar
Hajnal, J. (1976) On products of non-negative matrices. Math. Proc. Camb. Phil. Soc. 79, 521–530.Google Scholar
Karlin, S. (1968) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
Leslie, P. H. (1945) On the use of matrices in certain population mathematics. Biometrika 33, 183–212.Google Scholar
Parzen, E. (1962) Stochastic Processes. Holden-Day, San Francisco.Google Scholar
Sykes, Z. M. (1969) On discrete stable population theory. Biometrics 25, 285–293.Google Scholar
* Typographical errors on p. 22 of this paper should be corrected. In Theorem 3(iv), the region of integration should be Z, not C. In Theorem 3(v), II.3 and 4, x should be x' and y should be y'.Google Scholar