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Transient behaviour of a Galton–Watson process with a large number of types

Part of: Limit theorems Markov processes Nonstandard models

Published online by Cambridge University Press:  14 July 2016

Christine Reder
Christine Reder*
Affiliation:
Université de Bordeaux 1
*
∗Postal address: Institut de Mathématiques, Université de Bordeaux 1, 351 Cours de la Libération, F-33405 Talence Cedex, France. Email address: christine.reder@math.u-bordeaux.fr
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Abstract

Modelling the distribution of mutations of mitochondrial DNA in exponentially growing cell cultures leads to the study of a multitype Galton–Watson process during its transient phase. The number of types corresponds to the number of mtDNA per cell and may be considered as large. By taking advantage of this fact we prove that the stochastic process is deterministic-like on the set of nonextinction. On this set almost all trajectories are well approximated by the unique solution of a partial differential problem. This result allows also the comparison of trajectories corresponding to different modelling assumptions, for instance different values of the number of types.

Keywords

Mitochondrial geneticsmtDNA mutationmultitype Galton–Watson processbranching processnonstandard analysis

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes 60F99: None of the above, but in this section 92D10: Genetics 92D25: Population dynamics (general) 03H10: Other applications of nonstandard models (economics, physics, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 4 , September 2003 , pp. 1007 - 1030
Copyright
Copyright © Applied Probability Trust 2003 

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