Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-21T13:52:33.307Z Has data issue: false hasContentIssue false
  • English
  • Français

A CELL GROWTH MODEL ADAPTED FOR THE MINIMUM CELL SIZE DIVISION

Part of: Functional-differential and differential-difference equations

Published online by Cambridge University Press:  01 December 2015

B. VAN BRUNT ,
S. GUL  and
G. C. WAKE
B. VAN BRUNT*
Affiliation:
Institute of Fundamental Sciences, Massey University, Palmerston North 4442, New Zealand email b.vanbrunt@massey.ac.nz, S.Gul@massey.ac.nz
S. GUL
Affiliation:
Institute of Fundamental Sciences, Massey University, Palmerston North 4442, New Zealand email b.vanbrunt@massey.ac.nz, S.Gul@massey.ac.nz
G. C. WAKE
Affiliation:
Institute of Natural and Mathematical Sciences, Massey University at Albany, P.B. 102904, North Shore MC, Auckland, New Zealand email G.C.Wake@massey.ac.nz
*
✉b.vanbrunt@massey.ac.nz
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 study a cell growth model with a division function that models cells which divide only after they have reached a certain minimum size. In contrast to the cases studied in the literature, the determination of the steady size distribution entails an eigenvalue that is not known explicitly, but is defined through a continuity condition. We show that there is a steady size distribution solution to this problem.

Keywords

pantograph equationcell growth modelfunctional differential equation

MSC classification

Primary: 34K06: Linear functional-differential equations
Secondary: 34K10: Boundary value problems
Type
Research Article
Information
The ANZIAM Journal , Volume 57 , Issue 2 , October 2015 , pp. 138 - 149
Copyright
© 2015 Australian Mathematical Society 

References

Ambartsumyan, V. A., “On the fluctuation of the brightness of the Milky Way”, Dokl. Akad. Nauk SSSR 44 (1944) 223–226.Google Scholar
Andrews, G. E., The theory of partitions, Volume 2 of Encyclopedia of Mathematics and its Applications (Addison-Wesley, Reading, MA, 1976) 19–19; doi:10.1017/CBO9780511608650.Google Scholar
da Costa, F. P., Grinfeld, M. and McLeod, J. B., “Unimodality of steady size distributions of growing cell populations”, J. Evol. Equ. 1 (2001) 405–409; doi:PL00001379.CrossRefGoogle Scholar
Diekmann, O., Heijmans, H. and Thieme, H., “On the stability of the cell size distribution”, J. Math. Biol. 19 (1984) 227–248; doi:BF00277748.CrossRefGoogle Scholar
Gaver, D. P., “An absorption probability problem”, J. Math. Anal. Appl. 9 (1964) 384–393; doi:10.1016/0022-247X(64)90024-1.CrossRefGoogle Scholar
Hall, A. J. and Wake, G. C., “A functional differential equation arising in the modelling of cell-growth”, J. Austral. Math. Soc. (Series B) 30 (1989) 424–435; doi:10.1017/S0334270000006366.CrossRefGoogle Scholar
Hall, A. J. and Wake, G. C., “A functional differential equation determining steady size distributions for populations of cells growing exponentially”, J. Austral. Math. Soc. (Series B) 31 (1990) 344–353; doi:10.1017/S0334270000006779.CrossRefGoogle Scholar
Hall, A. J., Wake, G. C. and Gandar, P. W., “Steady size distributions for cells in one dimensional plant tissues”, J. Math. Biol. 30 (1991) 101–123; doi:10.1007/BF00160330.CrossRefGoogle Scholar
Iserles, A., “On the generalized pantograph functional differential equation”, European J. Appl. Math. 4 (1993) 1–38; doi:10.1017/S0956792500000966.CrossRefGoogle Scholar
Kato, T. and McLeod, J. B., “The functional–differential equation $y^{\prime }(x)=ay({\it\lambda}x)+by(x)$”, Bull. Amer. Math. Soc. 77 (1971) 891–937; doi:10.1090/S0002-9904-1971-12805-7.Google Scholar
Laurençot, P. and Perthame, B., “Exponential decay for the growth–fragmentation/cell-division equation”, Commun. Math. Sci. 7 (2009) 503–510; doi:10.4310/CMS.2009.v7.n2.a12.CrossRefGoogle Scholar
Michel, P., Mischler, S. and Perthame, B., “General relative entropy inequality: an illustration on growth models”, J. Math. Pures Appl. 84 (2005) 1235–1260; doi:j.matpur.2005.04.001.CrossRefGoogle Scholar
Ockendon, J. and Tayler, A., “The dynamics of a current collection system for an electric locomotive”, Proc. R. Soc. Lond. A 322 (1971) 447–468; doi:rspa.1971.0078.Google Scholar
Perthame, B. and Ryzhik, L., “Exponential decay for the fragmentation or cell-division equation”, J. Differential Equations 210 (2005) 155–177; doi:10.1016/j.jde.2004.10.018.CrossRefGoogle Scholar
Priestley, H. A., Introduction to complex analysis, revised edn (Oxford University Press, Oxford, 1990).Google Scholar
Sinko, J. W. and Streifer, W., “A new model for age–size structure of a population”, Ecology 48 (1967) 910–918; doi:10.2307/1934533.CrossRefGoogle Scholar
Sinko, J. W. and Streifer, W., “A model for populations reproducing by fission”, Ecology 52 (1971) 330–335; doi:10.2307/1934592.CrossRefGoogle Scholar
Suebcharoen, T., van-Brunt, B. and Wake, G. C., “Asymmetric cell division in a size-structured growth model”, Differential Integral Equations 24 (2011) 787–799; http://www.riddet.ac.nz/publicaton/aysmmetric-cell-division-in-a-size-structured-growth-model.CrossRefGoogle Scholar
van Brunt, B. and Vlieg-Hulstman, M., “An eigenvalue problem involving a functional differential equation arising in a cell growth model”, ANZIAM J. 51 (2010) 383–393; doi:10.1017/S1446181110000866.CrossRefGoogle Scholar
van Brunt, B. and Vlieg-Hulstman, M., “Eigenfunctions arising from a first order functional differential equation in a cell growth model”, ANZIAM J. 52 (2010) 46–58; doi:10.1017/S1446181111000575.CrossRefGoogle Scholar
van Brunt, B. and Wake, G. C., “A Mellin transform solution to a second-order pantograph equation with linear dispersion arising in a cell growth model”, European J. Appl. Math. 22 (2011) 151–168; doi:10.1017/S0956792510000367.CrossRefGoogle Scholar
van Brunt, B., Wake, G. C. and Kim, H. K., “On a singular Sturm–Liouville problem involving an advanced functional differential equation”, European J. Appl. Math. 12 (2001) 625–644; doi:10.1017/S0956792501004624.CrossRefGoogle Scholar
Wake, G. C., Cooper, S., Kim, H. K. and van Brunt, B., “Functional differential equations for cell-growth models with dispersion”, Commun. Appl. Anal. 4 (2000) 561–574; http://www.researchgate.net/publication/265494015_Functional_differential_equations_for_cell-growth_models_with_dispersion.Google Scholar