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

Part of: Markov processes Limit theorems Stochastic processes

Published online by Cambridge University Press:  01 July 2016

M. Richard
M. Richard*
Affiliation:
Université Paris VI
*
∗ Postal address: Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, Université Paris VI, Case courrier 188, 4 Place Jussieu, 75252 Paris Cedex 05, France. Email address: [email protected]
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Abstract

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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.

Keywords

Splitting treeCrump-Mode-Jagers processspine decompositionimmigrationstructured populationGEM distributionbiogeographyalmost-sure limit theorem

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G55: Point processes 92D25: Population dynamics (general) 60J85: Applications of branching processes 60F15: Strong theorems 92D40: Ecology
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 1 , March 2011 , pp. 276 - 300
Copyright
Copyright © Applied Probability Trust 2011 

References

Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.Google Scholar
Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121). Cambridge University Press.Google Scholar
Bertoin, J. (2006). Random Fragmentation and Coagulation Processes (Camb. Stud. Adv. Math. 102). Cambridge University Press.Google Scholar
Caswell, H. (1976). Community structure: a neutral model analysis. Ecological Monographs 46, 327–354.CrossRefGoogle Scholar
Donnelly, P. and Tavaré, S. (1986). The ages of alleles and a coalescent. Adv. Appl. Prob. 18, 1–19. (Correction: 18 (1986), 1023.)Google Scholar
Durrett, R. (1999). Essentials of Stochastic Processes. Springer, New York.Google Scholar
Ethier, S. N. (1990). The distribution of the frequencies of age-ordered alleles in a diffusion model. Adv. Appl. Prob. 22, 519–532.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.Google Scholar
Hubbell, S. (2001). The Unified Neutral Theory of Biodiversity and Biogeography. Princeton University Press, NJ.Google Scholar
Jagers, P. (1975). Branching Processes with Biological Applications. Wiley-Interscience, London.Google Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar
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.Google Scholar
Kendall, D. G. (1948). On some modes of population growth leading to R. A. Fisher's logarithmic series distribution. Biometrika 35, 6–15.Google Scholar
Lambert, A. (2008). Population dynamics and random genealogies. Stoch. Models 24, 45–163.CrossRefGoogle Scholar
Lambert, A. (2010). Species abundance distributions in neutral models with immigration or mutation and general lifetimes. Preprint. Available at http://www.proba.jussieu.fr/pageperso/amaury/index_fichiers/MigraMut.pdf. To appear in J. Math. Biol. Google Scholar
Lambert, A. (2010). The contour of splitting trees is a Lévy process. Ann. Prob. 38, 348–395.CrossRefGoogle Scholar
MacArthur, R. H. and Wilson, E. O. (1967). The Theory of Island Biogeography. Princeton University Press, NJ.Google Scholar
Nerman, O. (1981). On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrscheinlichkeithsth. 57, 365–395.Google Scholar
Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.Google Scholar
Tavaré, S. (1987). The birth process with immigration, and the genealogical structure of large populations. J. Math. Biol. 25, 161–168.Google Scholar
Tavaré, S. (1989). The genealogy of the birth, death, and immigration process. In Mathematical Evolutionary Theory, Princeton University Press, Princeton, NJ, pp. 41–56.Google Scholar
Volkov, I., Banavar, J. R., Hubbell, S. P. and Maritan, A. (2003). Neutral theory and relative species abundance in ecology. Nature 424, 1035–1037.Google Scholar
Watterson, G. A. (1974). Models for the logarithmic species abundance distributions. Theoret. Pop. Biol. 6, 217–250.Google Scholar
Watterson, G. A. (1974). The sampling theory of selectively neutral alleles. Adv. Appl. Prob. 6, 463–488.Google Scholar