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Bisexual Galton-Watson branching process with population-size-dependent mating

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

M. Molina ,
M. Mota  and
A. Ramos
M. Molina*
Affiliation:
Universidad de Extremadura
M. Mota*
Affiliation:
Universidad de Extremadura
A. Ramos*
Affiliation:
Universidad de Extremadura
*
∗ Postal address: Departamento de Matemáticas, Facultad de Ciencias, 06071 Badajoz, Spain.
∗ Postal address: Departamento de Matemáticas, Facultad de Ciencias, 06071 Badajoz, Spain.
∗ Postal address: Departamento de Matemáticas, Facultad de Ciencias, 06071 Badajoz, Spain.
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Abstract

In this paper, we introduce a bisexual Galton-Watson branching process with mating function dependent on the population size in each generation. Necessary and sufficient conditions for the process to become extinct with probability 1 are investigated for two possible conditions on the sequence of mating functions. Some results for the probability generating functions associated with the process are also given.

Keywords

Bisexual Galton-Watson branching processespopulation-size-dependent branching processesextinction probability

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 3 , September 2002 , pp. 479 - 490
Copyright
Copyright © Applied Probability Trust 2002 

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Footnotes

Research supported by the Plan Nacional de Investigación Cientifíca, Desarrollo e Innovación Tecnológica, grant BFM2000-0356.

References

Alsmeyer, G. and Rösler, U. (1996). The bisexual Galton–Watson process with promiscuous mating: extinction probabilities in the supercritical case. Ann. Appl. Prob. 6, 922–939.Google Scholar
Bagley, J. (1986). On the asymptotic properties of a supercritical bisexual branching process. J. Appl. Prob. 23, 820–826.Google Scholar
Bruss, F. T. (1984). A note on extinction criteria for bisexual Galton–Watson branching processes. J. Appl. Prob. 21, 915–919.Google Scholar
Daley, D. J. (1968). Stochastically monotone Markov chains. Z. Wahrscheinlichkeitsth. 10, 305–317.Google Scholar
Daley, D. J., Hull, D. M., and Taylor, J. M. (1986). Bisexual Galton–Watson branching processes with superadditive mating functions. J. Appl. Prob. 23, 585–600.Google Scholar
González, M., and Molina, M. (1996). On the limit behaviour of a superadditive bisexual Galton–Watson branching process. J. Appl. Prob. 33, 960–967.Google Scholar
González, M., and Molina, M. (1997). On the L 2-convergence of a superadditive bisexual Galton–Watson branching process. J. Appl. Prob. 34, 575–582.Google Scholar
González, M., Molina, M., and Mota, M. (2001). Estimation of the offspring distribution and the mean vector for a bisexual Galton–Watson process. Commun. Statist. Theory Meth. 30, 497–516.CrossRefGoogle Scholar
González Fragoso, A. (1995). Ratio estimation for the offspring means of a bisexual Galton–Watson branching process. Estadística 47, 17–36.Google Scholar
Grimmett, G. R., and Stirzaker, D. R. (1992). Probability and Random Processes, 2nd edn. Oxford University Press.Google Scholar
Hull, D. M. (1982). A necessary condition for extinction in those bisexual Galton–Watson branching processes governed by superadditive mating functions. J. Appl. Prob. 19, 847–850.Google Scholar
Hull, D. M. (1984). Conditions for extinction in certain bisexual Galton–Watson branching processes. J. Appl. Prob. 21, 414–418.Google Scholar
Molina, M., González, M., and Mota, M. (1998). Bayesian inference for bisexual Galton–Watson processes. Commun. Statist. Theory Meth. 27, 1055–1070.Google Scholar