Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T10:42:50.030Z Has data issue: false hasContentIssue false
  • English
  • Français

The solution and the stability of a nonlinear age-structured population model

Published online by Cambridge University Press:  17 February 2009

Norhayati  and
G. C. Wake
Norhayati
Affiliation:
Department of Mathematics, University Brunei Darussalem, NegaraBrunei Darussalem; e-mail: [email protected].
G. C. Wake
Affiliation:
Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand; e-mail: [email protected].
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.

We consider an age-structured population model achieved by modifying the classical Sharpe-Lotka-McKendrick model, incorporating an overcrowding effect or competition for resources term. This term depends on the whole population rather than on any specific age group, in the case of overcrowding or limitation of resources. We investigate the solutions for arbitrary initial conditions. We consider the existence of a steady age distribution and its stability and are able to determine this for a simple illustrative case. If the non-trivial steady age distribution is unstable, there is a critical initial population size beyond which the population explodes. This watershed is independent of the shape of the initial age distribution.

Type
Research Article
Information
The ANZIAM Journal , Volume 45 , Issue 2 , October 2003 , pp. 153 - 165
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Gurtin, M. E. and MacCamy, R. C., “Non-linear age-dependent population dynamics”, Arch. Ration. Mech. Anal. 5 (1974) 281–300.Google Scholar
[2]Hoppensteadt, F. C., Mathematical theories of populations: demographics, genetics and epidemicsh (Soc. Industr. Appl. Math., Arrowsmith, England, 1975).CrossRefGoogle Scholar
[3]Leslie, P. H., “On the use of matrices in certain population mathematics”, Biometrika 33 (1945) 183–212.CrossRefGoogle ScholarPubMed
[4]Leslie, P. H., “Some further notes on the use of matrices in population mathematics”, Biometrika 35 (1949) 213–245.CrossRefGoogle Scholar
[5]Lotka, A. J., “The stability of the normal age distribution”, Proc. Natl. Acad. Sci. USA 8 (1922) 339–345.CrossRefGoogle ScholarPubMed
[6]Malthus, T. R., An essay on the principle of population (St. Paul's, London, 1798),Google Scholar
reprinted in: Malthus, T. R., An essay on the principle of population and a summary view of the principle of population (Penguin, Harmondsworth, England, 1970).Google Scholar
[7]McKendrick, A. G., “Applications of mathematics to medical problems”, Proc. Edinburgh Math. Soc. 44 (1926) 98–130.CrossRefGoogle Scholar
[8]Nisbet, R. M. and Gurney, W. S. C., Modelling fluctuating populations (Wiley, New York, 1982).Google Scholar
[9]Pollak, E., “The effective population size of some age-structured populations”, Math. Bioscience 168 (2000) 36–56.Google Scholar
[10]Sharpe, F. R. and Lotka, A. J., “A problem in age distributions”, Phil. Mag. 21 (1911) 435–438.CrossRefGoogle Scholar
[11]Wake, G. C., Louie, K. and Roberts, M. G., “The regulation of an age-structured population by a fatal disease with or without dispersion”, in Proceedings of June 1–4 of Claremont international conference dedicated to the memory of Stravos Busenberg, (World Scientific, Singapore, 1994) 553–563.Google Scholar