Advertisement

Which Critically Branching Populations Persist?

  • J. Alfredo López-Mimbela
  • Anton Wakolbinger
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 84)

Abstract

The question how quick the individuals must spread out in order to overcome the clumping caused by a critical branching has been studied in [GRW] and [GW2] for a class of multitype branching populations on Rd. Here, an extension of the results to the genuine multitype branching case is presented, which relies on a more detailed analysis of the individual ancestral process [LWa]. Also, several advances and open problems around the question whether a recurrent migration together with a spatially homogeneous critical branching excludes persistence of the population are discussed.

Key words

Persistence local extinction Palm populations multitype branching process backward trees Key words and phrases 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [BCG]
    M. Bramson, J. T. Cox and A. Greven, Ergodicity of critical branching processes in low dimensions, Ann. Probab. 21 (1993) 1946–1957.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [CRW]
    B. Chauvin, A. Rouault and A. Wakolbinger, Growing conditioned trees, Stochastic Processes Appl. 39 (1991) 117–130.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [Daw]
    D. A. Dawson, The critical measure diffusion process, Z. Wahrsch. Verw. Geb., 40 (1977) 125–145.zbMATHCrossRefGoogle Scholar
  4. [DaF]
    D. A. Dawson and K. Fleischmann, Critical dimension for a model of branching in a random medium, Z. Wahrsch. Verw. Geb., 70 (1985) 315–334.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [DaH]
    D. A. Dawson and K. Hochberg, A multilevel branching model, Adv. Appl. Probab., 23 (1991) 701–715.MathSciNetzbMATHGoogle Scholar
  6. [Dyn]
    E. B. Dynkin, Branching particle systems and superprocesses, Ann. Probab. 19 (1991) 1157–1194.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [Eth]
    A. Etheridge, Asymptotic behavior of measure-valued critical branching processes, Proc. A.M.S., 118 (1993) 1251–1261.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [Fel]
    W. Feller, An introduction to probability theory and its applications,Vol. 2, Wiley, New York, (1966).zbMATHGoogle Scholar
  9. [GHW]
    L. Gorostiza, K. Hochberg and A. Wakolbinger, Persistence of a critical super-2 process, to appear in J. Appl. Probab., (1994).Google Scholar
  10. [GRW]
    L. Gorostiza, S. Roelly and A. Wakolbinger, Persistence of critical multitype particle and measure branching processes, Probab. Theory Relat. Fields 92 (1992) 313–335.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [GW1]
    L. Gorostiza, and A. Wakolbinger, Persistence criteria for a class of critical branching particle systems in continuous time, Ann. Probab. 19 (1991) 266–288.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [GW2]
    L. Gorostiza and A. Wakolbinger, Asymptotic behavior of a reaction-diffusion system. A probabilistic approach, Random and Computational Dynamics, 1(4) (1993) 445–463.MathSciNetzbMATHGoogle Scholar
  13. [HWa]
    K. Hochberg and A. Wakolbinger, Non-persistence of two-level branching particle systems in low dimensions, preprint (1994).Google Scholar
  14. [JaN]
    P. Jagers and O. Nerman, The growth and composition of branching populations, Ad. Appl. Probab., 16 (1984) 221–259.MathSciNetzbMATHGoogle Scholar
  15. [Kal]
    O. Kallenberg, Random Measures, Akademie-Verlag, Berlin, and Academic Press, (1983).zbMATHGoogle Scholar
  16. [Ka2]
    O. Kallenberg, Stability of critical cluster fields, Math. Nachr., 77 (1977) 7–43.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [LMW]
    A. Liemant, K. Matthes and A. Wakolbinger, Equilibrium Distributions of Branching Processes. Akademie-Verlag, Berlin, and Kluwer Academic Pub-lishers, Dordrecht, (1988).Google Scholar
  18. [L-M]
    J. A. López-Mimbela, Fluctuation limits of multitype branching random fields, J. of Multiv. Anal., 40 (1992) 56–83.zbMATHCrossRefGoogle Scholar
  19. [LWa]
    J. A. López-Mimbela and A. Wakolbinger, Clumping in multitype-branching trees, preprint (1994).Google Scholar
  20. [NJa]
    O. Nerman and P. Jagers, The stable doubly infinite pedigree process of supercritical branching processes, Z. Wahrsch. Verw. Geb., 65 (1984) 445–460.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [SWa]
    A. Stoeckl and A. Wakolbinger, On clan-recurrence and -transience in time stationary branching Brownian particle systems, in “Measure-Valued Processes, Stochastic Partial Differential Equations, and Interacting Systems,” D. A. Dawson (Editor), CRM Proc. and Lect. Notes, 5 (1994) 213–219.MathSciNetGoogle Scholar
  22. [Wu]
    Wu, Y. Asymptotic behavior of the two level measure branching process, Ann. Probab., 22 (1994) 854–874.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • J. Alfredo López-Mimbela
    • 1
  • Anton Wakolbinger
    • 2
  1. 1.Centro de Investigación en MatemáticasGuanajuato, Gto.México
  2. 2.Fachbereich MathematikJ.W. Goethe-UniversitätFrankfurt am MainGermany

Personalised recommendations