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The multitype continuous-time Markov branching process in a periodic environment

Published online by Cambridge University Press:  01 July 2016

B. Klein  and
P. D. M. MacDonald
B. Klein*
Affiliation:
Institut de Cancérologie et d'Immunogénétique, Villejuif
P. D. M. MacDonald*
Affiliation:
McMaster University
*
∗Present address: Centre Paul Lamarque, Hôpital St. Eloi, 34 Montpellier, France.
∗∗Postal address: Department of Mathematical Sciences, McMaster University, Hamilton, Ont., Canada L8S 4K1. Research carried out while the author was visiting the Laboratoire de Biostatistique de l'Université Paris 7, Unité INSERM U.21, France.
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Abstract

The multitype continuous-time Markov branching process has many biological applications where the environmental factors vary in a periodic manner. Circadian or diurnal rhythms in cell kinetics are an important example. It is shown that in the supercritical positively regular case the proportions of individuals of various types converge in probability to a non-random periodic vector, independent of the initial conditions, while the absolute numbers of individuals of various types converge in probability to that vector multiplied by a random variable whose distribution depends on the initial conditions. It is noted that the proofs are straightforward extensions of the well-known results for a constant environment.

Keywords

BRANCHING PROCESSCELL KINETICSCHRONOBIOLOGYCIRCADIAN RHYTHMSCONVERGENCE IN PROBABILITYMARTINGALEMULTITYPE BRANCHING PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 1 , March 1980 , pp. 81 - 93
Copyright
Copyright © Applied Probability Trust 1980 

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Footnotes

Financial support from INSERM contract #77-5-049-2 and from the National Research Council of Canada is acknowledged.

References

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