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Approximations in population-dependent branching processes

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Workshop on Branching Processes and Their Applications

Part of the book series: Lecture Notes in Statistics ((LNSP,volume 197))

Abstract

We consider models of population size dependent branching processes with the feature that they are supercritical when population reaches some threshold K, near critical around that value, and subcritical below it. Although such population die out with probability one, their time to extinction is large. We show that this time is exponential in K. Approximations to the populations size in various domains are given, and a problem of small initial population size is discussed. From a technical point of view, analysis of such processes involves techniques of density dependent models as small random perturbations of dynamical systems, and size-dependent bounds.

Mathematics Subject Classifications (2000): 60J80, 60F10

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Acknowledgements

The author wishes to express his gratitude to the University of Extremadura probability group for being such perfect hosts, and for organising this workshop. Research supported by the Australian Research Council Grant DP08810011.

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Authors and Affiliations

  1. School of Mathematical Sciences, Monash University, Clayton, Victoria, 3800, Australia

    Fima C. Klebaner

Authors
  1. Fima C. Klebaner
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Correspondence to Fima C. Klebaner .

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Editors and Affiliations

  1. Fac. Ciencias, Universidad de Extremadura, Avenida Elvas s/n, Badajoz, 06071, Spain

    Miguel González Velasco

  2. Fac. Ciencias, Universidad de Extremadura, Avenida Elvas s/n, Badajoz, 06071, Spain

    Inés M. Puerto

  3. Universidad de Extremadura, Avda. Virgen del Puerto, Plasencia, 10600, Spain

    Rodrigo Martínez

  4. Fac. Ciencias, Universidad de Extremadura, Avenida Elvas s/n, Badajoz, 06071, Spain

    Manuel Molina

  5. Fac. Ciencias, Universidad de Extremadura, Avenida Elvas s/n, Badajoz, 06071, Spain

    Manuel Mota

  6. Facultad de Veterinaria, Universidad de Extradura, Avda Universidad s/n, Caceres, 10071, Spain

    Alfonso Ramos

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Klebaner, F.C. (2010). Approximations in population-dependent branching processes. In: González Velasco, M., Puerto, I., Martínez, R., Molina, M., Mota, M., Ramos, A. (eds) Workshop on Branching Processes and Their Applications. Lecture Notes in Statistics(), vol 197. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11156-3_5

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