Positive Recurrence of Multi-Dimensional Population-Dependent Branching Processes

Abstract

Continuous time branching processes are appropriate stochastic models for the temporal description of cell populations as long as the cells existing simultaneously can be assumed to develop independently. A one-dimensional process is sufficient for the examination of the number of cells, but if one wants to consider internal states (for example cell cycle phases, see Baserga, 1976, for the biological aspects), one has to pass to the multi-dimensional processes. As up to now the information about the distributions of the phase durations is limited (see Prescott, 1976 especially Chapter 3), it is surely sufficient to confine oneself to Markov branching processes, which are easier to handle than the age-dependent processes. This restriction is not so severe as it seems because the sojourn times in the different biological phases can be represented as sums of independent exponentially distributed random variables without losing the Markov property of the whole process (see for example Kendall, 1948; Rittgen & Tautu, 1976; Rittgen, 1978).