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Exponential Growth of Bifurcating Processes with Ancestral Dependence

Part of: Markov processes

Published online by Cambridge University Press:  22 February 2016

Sana Louhichi  and
Bernard Ycart
Sana Louhichi*
Affiliation:
Université Grenoble Alpes
Bernard Ycart*
Affiliation:
Université Grenoble Alpes
*
∗ Postal address: Laboratoire Jean Kuntzmann, Université Grenoble Alpes, 51 rue des Mathématiques, 38041 Grenoble cedex 9, France.
∗ Postal address: Laboratoire Jean Kuntzmann, Université Grenoble Alpes, 51 rue des Mathématiques, 38041 Grenoble cedex 9, France.
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Abstract

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Branching processes are classical growth models in cell kinetics. In their construction, it is usually assumed that cell lifetimes are independent random variables, which has been proved false in experiments. Models of dependent lifetimes are considered here, in particular bifurcating Markov chains. Under the hypotheses of stationarity and multiplicative ergodicity, the corresponding branching process is proved to have the same type of asymptotics as its classic counterpart in the independent and identically distributed supercritical case: the cell population grows exponentially, the growth rate being related to the exponent of multiplicative ergodicity, in a similar way as to the Laplace transform of lifetimes in the i.i.d. case. An identifiable model for which the multiplicative ergodicity coefficients and the growth rate can be explicitly computed is proposed.

Keywords

Branching processbifurcating Markov chainmultiplicative ergodicitycell kinetics

MSC classification

Primary: 60J85: Applications of branching processes
Secondary: 92D25: Population dynamics (general)
Type
Research Article
Information
Advances in Applied Probability , Volume 47 , Issue 2 , June 2015 , pp. 545 - 564
Copyright
© Applied Probability Trust 

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