The Bellman–Harris Process

Marek Kimmel, David E. Axelrod

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

The Bellman–Harris branching process is more general than the processes considered in the preceding chapters. Lifetimes of particles are nonnegative random variables with arbitrary distributions. It is described as follows. A single ancestor particle is born at t = 0. It lives for time τ which is a random variable with cumulative distribution function (Formula Presented). At the moment of death, the particle produces a random number of progeny according to a probability distribution with pgf f(s). Each of the first generation progeny behaves, independently of each other and the ancestor, as the ancestor particle did, i.e., it lives for a random time distributed according to (Formula Presented) and produces a random number of progeny according to f(s). If we denote Z(t) the particle count at time t, we obtain a stochastic process (Formula Presented). This so-called age-dependent process is generally non-Markov, but two of its special cases are Markov: the Galton–Watson process and the age-dependent branching process with exponential lifetimes. The Bellman–Harris process is more difficult to analyze, but it has many properties similar to these two processes.

Original languageEnglish (US)
Title of host publicationInterdisciplinary Applied Mathematics
PublisherSpringer Nature
Pages87-105
Number of pages19
DOIs
StatePublished - 2015

Publication series

NameInterdisciplinary Applied Mathematics
Volume19
ISSN (Print)0939-6047
ISSN (Electronic)2196-9973

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Management Science and Operations Research

Keywords

  • Cell Cycle Duration
  • Cell Cycle Phase
  • Cell Cycle Time
  • Renewal Equation
  • Watson Process

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