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On the existence of the stable birth-type distribution in a general branching process cell cycle model with unequal cell division

Part of: Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Marina Alexandersson
Marina Alexandersson*
Affiliation:
University of California, Berkeley
*
∗ Postal address: University of California, Department of Statistics, 367 Evans Hall #3860, Berkeley, CA 94720-3860, USA. Email address: marina@stat.berkeley.edu
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Abstract

We use multi-type branching process theory to construct a cell population model, general enough to include a large class of such models, and we use an abstract version of the Perron-Frobenius theorem to prove the existence of the stable birth-type distribution. The generality of the model implies that a stable birth-size distribution exists in most size-structured cell cycle models. By adding the assumption of a critical size that each cell has to pass before division, called the nonoverlapping case, we get an explicit analytical expression for the stable birth-type distribution.

Keywords

Branching processcell populationunequal cell divisionPerron-FrobeniusCrump-Mode-Jagersstable type distribution

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) 92D25: Population dynamics (general)

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 3 , September 2001 , pp. 685 - 695
Copyright
Copyright © by the Applied Probability Trust 2001 

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Footnotes

This work is part of the Bank of Sweden TercentenaryF oundation project ‘Dependence and Interaction in Stochastic Population Dynamics’.

References

Alexandersson, M. (2000). An application of general branching processes to a cell cycle model with two uncoupled subcycles and unequal cell division. Int. J. Appl. Math Comput. Sci. 10, 131–145.Google Scholar
Bell, G. I., and Anderson, E. C. (1967). Cell growth and division I: a mathematical model with applications to cell volume distributions in mammalian suspension cultures. Biophys. J. 7, 329–351.CrossRefGoogle Scholar
Jagers, P. (1989). General branching processes as Markov fields. Stoch. Proc. Appl. 32, 183–242.CrossRefGoogle Scholar
Jagers, P. (1992). Stabilities and instabilities in population dynamics. J. Appl. Prob. 29, 770–780.CrossRefGoogle Scholar
Jagers, P. (2000). The deterministic evolution of general branching populations. In State of the Art in Statistics and Probability Theory. Festschrift for Willem R. van Zwet, eds de Gunst, M., Klaassen, C. and van der Vaart, A. Institute of Mathematical Statistics, Beachwood, OH.Google Scholar
Jagers, P., and Nerman, O. (1996). The asymptotic composition of multitype branching populations. In Séminaire de Probabilités (Lecture Notes Math. 1626), eds Azéma, J., Emery, M. and Yor, M. Springer, Berlin, pp. 40–54.CrossRefGoogle Scholar
Nerman, O. (1984). The growth and composition of supercritical branching populations on general type spaces. Preprint 1984-18, Department of Mathematics, Chalmers University of Technology and Göteborg University.Google Scholar
Shurenkov, V. (1992). Markov renewal theory and its applications to Markov ergodic processes. Preprint 1992-24, Department of Mathematics, Chalmers University of Technology and Göteborg University.Google Scholar
Shurenkov, V. (1992). On the existence of a Malthusian parameter. Preprint 1992-20. Department of Mathematics, Chalmers University of Technology and Göteborg UniversityGoogle Scholar
Taib, Z. (1999). A branching process version of the Bell–Anderson cell population model. Commun. Statist. Stoch. Models 15, 719–729.CrossRefGoogle Scholar