Acta Applicandae Mathematicae

, Volume 138, Issue 1, pp 279–312

A Two-Type Bellman–Harris Process Initiated by a Large Number of Particles

  • Vladimir Vatutin
  • Alexander Iksanov
  • Valentin Topchii
Article

DOI: 10.1007/s10440-014-9966-0

Cite this article as:
Vatutin, V., Iksanov, A. & Topchii, V. Acta Appl Math (2015) 138: 279. doi:10.1007/s10440-014-9966-0

Abstract

We investigate a two-type critical Bellman–Harris branching process with the following properties: the tail of the life-length distribution of the first type particles is of order o(t−2); the tail of the life-length distribution of the second type particles is regularly varying at infinity with index −β, β∈(0,1]; at time t=0 the process starts with a large number N of the second type particles and no particles of the first type. It is shown that the time axis 0≤t<∞ splits into several regions whose ranges depend on β and the ratio N/t within each of which the process at time t exhibits asymptotics (as N,t→∞) which is different from those in the other regions.

Keywords

Evolutionary stages Two-type critical Bellman–Harris process Regular variation Renewal theory 

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Vladimir Vatutin
    • 1
  • Alexander Iksanov
    • 2
  • Valentin Topchii
    • 3
  1. 1.Steklov Mathematical institute RASMoscowRussia
  2. 2.Faculty of CyberneticsNational Taras Shevchenko University of KyivKyivUkraine
  3. 3.Sobolev Institute of Mathematics SB RASNovosibirskRussia

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