Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T09:28:14.345Z Has data issue: false hasContentIssue false
  • English
  • Français

Branching processes and functional differential equations determining steady-size distributions in cell populations

Part of: Mathematical biology in general Markov processes

Published online by Cambridge University Press:  14 July 2016

Ziad Taib
Ziad Taib*
Affiliation:
Chalmers University of Technology and the University of Göteborg
*
∗Postal address: Department of Mathematics, Chalmers University of Technology and the University of Göteborg, S-412 96 Göteborg, Sweden.
Get access Rights & Permissions [Opens in a new window]

Abstract

The functional differential equation y′(x) = ay(λx) + by(x) arises in many different situations. The purpose of this note is to show how it arises in some multitype branching process cell population models. We also show how its solution can be given an intuitive interpretation as the probability density function of an infinite sum of independent but not identically distributed random variables.

Keywords

MULTITYPE BRANCHING PROCESSPOPULATION MODELS

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes 92B05: General biology and biomathematics 92D25: Population dynamics (general)
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 1 , March 1995 , pp. 1 - 10
Copyright
Copyright © Applied Probability Trust 1995 

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.)

Footnotes

Supported by the Swedish Institute.

References

Fox, L., Mayers, D. F., Ockendon, J. R. and Tayler, A. B. (1971) On a functional differential equation. J. Inst. Math. Appl. 8, 271–307.Google Scholar
Hall, A. J. and Wake, G. C. (1990) Functional differential equations determining steady size distributions for populations of cells growing exponentially. J. Austral. Math. Soc., B31, 434–453.CrossRefGoogle Scholar
Jagers, P. (1989) General branching processes as Markov fields. Stoch. Proc. Appl. 32, 183–242.CrossRefGoogle Scholar
Jagers, P. and Nerman, O. (1992) The asymptotic composition of supercritical multitype branching populations. Department of Mathematics, Chalmers University of Technology and Gothenburg University. Preprint.Google Scholar
Kato, T. and Mcleod, J. B. (1971) The functional differential equation y'(x) = ay(?x) + by(x). Bull. Amer. Math. Soc. 77, 891–937.Google Scholar
Koch, A. L. and Schaechter, M. (1962) A model for statistics of the cell division process. J. Gen. Microbiol. 29, 435–454.Google Scholar
Mahler, K. (1940) On a special functional equation. J. London Math. Soc. 15, 115–123 (MR 2, 133).CrossRefGoogle Scholar
Shurenkov, V. M. (1992) Markov renewal theory and its applications to Markov ergodic processes. Department of Mathematics, Chalmers University of Technology and Gothenburg University. Preprint.Google Scholar
Taib, Z. (1993) A note on modelling the dynamics of budding yeast populations using branching processes. J. Math. Biol. 31, 805–815.CrossRefGoogle ScholarPubMed
Tyson, J. and Diekmann, O. (1986) Sloppy size control of the cell division cycle. J. Theoret. Biol. 118, 405–426.Google Scholar