Advances in Applied Probability

Limit theorems for supercritical age-dependent branching processes with neutral immigration

M. Richard

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate θ, new individuals singly enter the total population and start a new population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagers process: individuals have independent and identically distributed lifetime durations (nonnecessarily exponential) during which they give birth independently at a constant rate b. First, using spine decomposition, we relax previously known assumptions required for almost-sure convergence of the total population size. Then, we consider three models of structured populations: either all immigrants have a different type, or types are drawn in a discrete spectrum or in a continuous spectrum. In each model, the vector (P1, P2,...) of relative abundances of surviving families converges almost surely. In the first model, the limit is the GEM distribution with parameter θ / b.

Article information

Adv. in Appl. Probab. Volume 43, Number 1 (2011), 276-300.

First available in Project Euclid: 15 March 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G55: Point processes 92D25: Population dynamics (general) 60J85: Applications of branching processes [See also 92Dxx] 60F15: Strong theorems 92D40: Ecology

Splitting tree Crump-Mode-Jagers process spine decomposition immigration structured population GEM distribution biogeography almost-sure limit theorem


Richard, M. Limit theorems for supercritical age-dependent branching processes with neutral immigration. Adv. in Appl. Probab. 43 (2011), no. 1, 276--300. doi:10.1239/aap/1300198523.

Export citation


  • Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
  • Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121). Cambridge University Press.
  • Bertoin, J. (2006). Random Fragmentation and Coagulation Processes (Camb. Stud. Adv. Math. 102). Cambridge University Press.
  • Caswell, H. (1976). Community structure: a neutral model analysis. Ecological Monographs 46, 327–354.
  • Donnelly, P. and Tavaré, S. (1986). The ages of alleles and a coalescent. Adv. Appl. Prob. 18, 1–19. (Correction: 18 (1986), 1023.)
  • Durrett, R. (1999). Essentials of Stochastic Processes. Springer, New York.
  • Ethier, S. N. (1990). The distribution of the frequencies of age-ordered alleles in a diffusion model. Adv. Appl. Prob. 22, 519–532.
  • Fisher, R. A., Corbet, A. S. and Williams, C. B. (1943). The relation between the number of species and the number of individuals in a random sample of an animal population. J. Animal Ecology 12, 42–58.
  • Geiger, J. and Kersting, G. (1997). Depth-first search of random trees, and Poisson point processes. In Classical and Modern Branching Processes (Minneapolis, MN, 1994; IMA Vol. Math. Appl. 84), Springer, New York, pp. 111–126.
  • Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.
  • Hubbell, S. (2001). The Unified Neutral Theory of Biodiversity and Biogeography. Princeton University Press, NJ.
  • Jagers, P. (1975). Branching Processes with Biological Applications. Wiley-Interscience, London.
  • Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.
  • Karlin, S. and McGregor, J. (1967). The number of mutant forms maintained in a population. In Proc. 5th Berkeley Symp. Math. Statist. Prob., Vol. IV, University of California Press, Berkeley, CA, pp. 415–438.
  • Kendall, D. G. (1948). On some modes of population growth leading to R. A. Fisher's logarithmic series distribution. Biometrika 35, 6–15.
  • Lambert, A. (2008). Population dynamics and random genealogies. Stoch. Models 24, 45–163.
  • Lambert, A. (2010). Species abundance distributions in neutral models with immigration or mutation and general lifetimes. Preprint. Available at To appear in J. Math. Biol.
  • Lambert, A. (2010). The contour of splitting trees is a Lévy process. Ann. Prob. 38, 348–395.
  • MacArthur, R. H. and Wilson, E. O. (1967). The Theory of Island Biogeography. Princeton University Press, NJ.
  • Nerman, O. (1981). On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrscheinlichkeithsth. 57, 365–395.
  • Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.
  • Tavaré, S. (1987). The birth process with immigration, and the genealogical structure of large populations. J. Math. Biol. 25, 161–168.
  • Tavaré, S. (1989). The genealogy of the birth, death, and immigration process. In Mathematical Evolutionary Theory, Princeton University Press, Princeton, NJ, pp. 41–56.
  • Volkov, I., Banavar, J. R., Hubbell, S. P. and Maritan, A. (2003). Neutral theory and relative species abundance in ecology. Nature 424, 1035–1037.
  • Watterson, G. A. (1974). Models for the logarithmic species abundance distributions. Theoret. Pop. Biol. 6, 217–250.
  • Watterson, G. A. (1974). The sampling theory of selectively neutral alleles. Adv. Appl. Prob. 6, 463–488.