Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-06-27T13:07:40.706Z Has data issue: false hasContentIssue false
  • English
  • Français

The asymptotic behaviour due to a piecewise time dependent net maternity function

Published online by Cambridge University Press:  17 February 2009

P. Cerone  and
K. P. Tognetti
P. Cerone
Affiliation:
Department of Mathematics, The University of Wollongong, Box 1144, Wollongong, N.S.W. 2500
K. P. Tognetti
Affiliation:
Department of Mathematics, The University of Wollongong, Box 1144, Wollongong, N.S.W. 2500
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The long term asymptotic behaviour of a population is evaluated where the age specific fertility behaviour is allowed to change with time. Thus in this article the behaviour of a population is determined with a time dependent net maternity function. It is shown that methods used when the net maternity function was independent of time are still applicable if the change with time is explicit only for the initial population. Further, using the fact that for realistic situations the net maternity function is non-zero over a finite interval α < x < β, it is shown that traditional methods can again be used if the time dependence is associated with ages less than α, the minimum age of childbearing. Recent extensions of Cerone and Keane to include exponential time dependence are utilized and models are presented which are piecewise defined, allowing general and exponential time dependence for the parent and new-born populations respectively. The Sharpe-Lotka single sex determinstic population model is used as the basis for the analysis.

Type
Research Article
Information
The ANZIAM Journal , Volume 23 , Issue 4 , April 1982 , pp. 438 - 450
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Bellman, R. and Cooke, K. L., Differential difference equations, (Academic Press, New York, 1963).CrossRefGoogle Scholar
[2]Cerone, P. and Keane, A., “The momentum of population growth with time dependent net maternity function”, Demography 15(1978), 131–134.CrossRefGoogle ScholarPubMed
[3]Cerone, P. and Keane, A., “The stable births resulting from a time dependent change between two net maternity functions”, Demography 15 (1978), 135–137.CrossRefGoogle ScholarPubMed
[4]Frauenthal, J. C., “Birth trajectory under changing fertility conditions”, Demography 12 (1975), 447–454.CrossRefGoogle ScholarPubMed
[5]Frejka, T., The future of population growth: alternative paths to equilibrium (Wiley, New York, 1973).Google Scholar
[6]Keyfitz, N., Introduction to the mathematics of population, (Addison-Wesley, Reading, Mass., 1968).Google Scholar
[7]Keyfitz, N., “On the momentum of population growth”, Demography 8 (1971), 71–80.Google Scholar
[8]Keyfitz, N., “Reproductive value: with applications to migration, contraception and zero population growth”, in Quantitative sociology: international perspectives on mathematical and statistical modelling. Blalock, H. M. et al. (Eds.), (Academic Press, New York, 1975), 587–612.CrossRefGoogle Scholar
[9]Keyfitz, N., Applied mathematical demography, (Wiley, New York, 1977).Google Scholar
[10]Lopez, A., Problems in stable population theory (Office of Population Research, Princeton, 1961).Google Scholar
[11]Mitra, S., “Influence of instantaneous fertility decline to replacement level on population growth”, Demography 13(1976), 513–519.CrossRefGoogle ScholarPubMed
[12]Pollard, J. H., Mathematical models for the growth of human populations (Cambridge University Press, 1973).Google Scholar
[13]Ruzicka, L. T., Reflections on zero growth of the Australian population, National Population Inquiry–Research Report No. 7 (Canberra: Australian Government Publishing Service, 1977).Google Scholar
[14]Sharpe, F. R. and Lotka, A. J., “A problem in age distribution”, Philosophical Magazine 21 (1911), 435–438.Google Scholar
[15]Tognetti, K. P., “Some extensions of the Keyfitz momentum relationship”, Demography 13 (1976), 507–512.CrossRefGoogle ScholarPubMed