Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T06:22:56.086Z Has data issue: false hasContentIssue false
  • English
  • Français

A nonlinear model of age and size-structured populations with applications to cell cycles

Published online by Cambridge University Press:  17 February 2009

S. J. Chapman ,
M. J. Plank ,
A. James  and
B. Basse
S. J. Chapman
Affiliation:
Mathematical Institute, Oxford University Oxford UK email: [email protected].
M. J. Plank
Affiliation:
Biomathematics Research Centre University of Canterbury, Christchurch New Zealand email: [email protected], [email protected], [email protected]..
A. James
Affiliation:
Biomathematics Research Centre University of Canterbury, Christchurch New Zealand email: [email protected], [email protected], [email protected]..
B. Basse
Affiliation:
Biomathematics Research Centre University of Canterbury, Christchurch New Zealand email: [email protected], [email protected], [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.

The Sharpe-Lotka-McKendrick (or von Foerster) equations for an age-structured population, with a nonlinear term to represent overcrowding or competition for resources, are considered. The model is extended to include a growth term, allowing the population to be structured by size or weight rather than age, and a general solution is presented. Various examples are then considered, including the case of cell growth where cells divide at a given size.

Keywords

age-structuresize-structurepopulationscell cyclemathematical model.
Type
Articles
Information
The ANZIAM Journal , Volume 49 , Issue 2 , October 2007 , pp. 151 - 169
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Alberts, B., Bray, D., Lewis, J., Raff, M., Roberts, K. and Watson, J. D., The molecular biology of the cell, 3rd ed. (Garland, New York, 1994).Google Scholar
[2]Basse, B., Baguley, B., Marshall, E., Wake, G. and Wall, D., “Modelling the flow cytometric data obtained from unperturbed human tumour cell lines: Parameter fitting and comparison.”, Bull Math Biol. 67 (2005) 815–830.Google Scholar
[3]Basse, B., Baguley, B. C., Marshall, E. S., Joseph, W. R.Brunt, B. van, Wake, G. and Wall, D. J. N., “A mathematical model for analysis of the cell cycle in cell lines derived from human tumours”, J Math Biol. 47 (2003) 295–312.CrossRefGoogle Scholar
[4]Basse, B. and Ubezio, P., “A generalised age and phase structured model of human tumour cell populations both unperturbed and exposed to a range of cancer therapies”, Bull Math Biol. 69 (2007) 1673–1690.CrossRefGoogle ScholarPubMed
[5]Basse, B., Wake, G. C., Wall, D. J. N. and Brunt, B. van, “On a cell-growth model for plankton”, Math Med Biol. 21 (2004) 49–61.Google Scholar
[6]Botsford, L. W., Smith, B. D. and Quinn, J. F., “Bimodality in size distributions: The red sea urchin Strongylocentrotus franciscanus as an example”, Ecol Appl. 4 (1994) 42–50.Google Scholar
[7]Chen, P., Brenner, D. and Sachs, R., “Ionizing radiation damage to cells: effects of cell cycle redistribution”, Math Biosci. 126 (1994) 147–170.Google Scholar
[8]Coombes, D. A., Duncan, R. P., Allen, R. B. and Truscott, J., “Disturbances prevent stem size density distributions in natural forests from following scaling relationships”, Ecol Lett. 6 (2003) 980–989.Google Scholar
[9]Cushing, J. M., “Existence and stability of equilibria in age-structured population dynamics”, Math Biol. 20 (1984) 259–276.Google Scholar
[10]Folkman, J., “Tumour angiogenesis: therapeutic implications”, New Engl J Med. 285 (1971) 1182–1186.Google ScholarPubMed
[11]Gurtin, M. and MacCamy, R., “Some simple models for nonlinear age-dependent population dynamics”, Math Biosci. 43 (1978) 199–211.Google Scholar
[12]Hilborn, R. and Walters, C. J., Quantitative Fisheries Stock Assessment (Chapman and Hall, London, 1992).CrossRefGoogle Scholar
[13]Hoppensteadt, F., Mathematical theories of populations: demographics, genetics and epidemics (SIAM, Philadelphia, 1975).Google Scholar
[14]McKendrick, A., “Applications of mathematics to medical problems”, Proc Edinburgh Math Soc. 44 (1926) 98–130.CrossRefGoogle Scholar
[15]Murray, J. D., Mathematical biology (Springer-Verlag, Berlin, 1989).Google Scholar
[16]Norhayati, and Wake, G. C., “The solution and stability of a nonlinear age-structured population model”, ANZ1AMJ. 45 (2003) 153–165.Google Scholar
[17]Sharpe, F. and Lotka, A. J., “A problem in age distribution”, Phil Mag. 21 (1911) 435–438.CrossRefGoogle Scholar
[18]Takada, T. and Caswell, H., “Optimal size at maturity in size-structured populations”, J Theor Biol. 187 (1997) 81–93.Google Scholar
[19]Foerster, H. von, “Some remarks on changing populations”, in The kinetics of cell proliferation (ed. F., Stohlman), (Grune and Stratton, New York, 1959) 382–407.Google Scholar
[20]Webb, G., Theory of nonlinear age-dependent population dynamics (Dekker, New York, 1985).Google Scholar