Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-24T19:02:29.747Z Has data issue: false hasContentIssue false
  • English
  • Français

Bisexual Galton–Watson branching processes with superadditive mating functions

Published online by Cambridge University Press:  14 July 2016

D. J. Daley ,
David M. Hull  and
James M. Taylor
D. J. Daley*
Affiliation:
The Australian National University
David M. Hull*
Affiliation:
Valparaiso University
James M. Taylor
Affiliation:
The Australian National University
*
∗Postal address: Statistics Department, (IAS), Australian National University, GPO Box 4, Canberra, ACT 2601, Australia.
∗∗Postal address: Department of Mathematics and Computer Science, Valparaiso University, Valparaiso, IN 46383, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

For a bisexual Galton–Watson branching process with superadditive mating function there is a simple criterion for determining whether or not the process becomes extinct with probability 1, namely, that the asymptotic growth rate r should not exceed 1. When extinction is not certain (equivalently, r > 1), simple upper and lower bounds are established for the extinction probabilities. An example suggests that in the critical case that r = 1, some condition like superadditivity is essential for ultimate extinction to be certain. Some illustrative numerical comparisons of particular mating functions are made using a Poisson offspring distribution.

Keywords

EXTINCTION PROBABILITYOFFSPRING DISTRIBUTION
Type
Research Paper
Information
Journal of Applied Probability , Volume 23 , Issue 3 , September 1986 , pp. 585 - 600
Copyright
Copyright © Applied Probability Trust 1986 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

∗∗∗

Present address: 73 Torrington Road, Maroubra, NSW 2035, Australia.

References

Asmussen, S. (1980) On some two-sex population models. Ann. Prob. 8, 727–744.CrossRefGoogle Scholar
Asmussen, S. and Hering, H. (1983) Branching Processes. Birkhauser, Boston.CrossRefGoogle Scholar
Bruss, F. T. (1984) A note on extinction criteria for bisexual Galton–Watson processes. J. Appl. Prob. 21, 915–919.CrossRefGoogle Scholar
Daley, D. J. (1968a) Extinction conditions for certain bisexual Galton–Watson branching processes. Z. Wahrscheinlichkeitsth. 9, 315–322.CrossRefGoogle Scholar
Daley, D. J. (1968b) Stochastically monotone Markov chains. Z. Wahrscheinlichkeitsth. 10, 305–317.CrossRefGoogle Scholar
Hille, E. and Phillips, R. (1957) Functional Analysis and Semi-Groups. American Mathematical Society, Providence, RI.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.CrossRefGoogle Scholar
Hull, D. M. (1984) Conditions for extinction in certain bisexual Galton–Watson branching processes. J. Appl. Prob. 21, 414–418.CrossRefGoogle Scholar
Kamae, T., Krengel, U. and O'brien, G. L. (1977) Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.CrossRefGoogle Scholar
Karlin, S. and Kaplan, N. (1973) Criteria for extinction of certain population growth processes with interacting types. Adv. Appl. Prob. 5, 183–199.CrossRefGoogle Scholar
Keilson, J. and Kester, A. (1977) Monotone matrices and monotone Markov processes. Stoch. Proc. Appl. 5, 231–241.CrossRefGoogle Scholar
Seneta, E. (1981) Non-negative Matrices and Markov Chains. Springer-Verlag, Heidelberg.CrossRefGoogle Scholar
Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester.Google Scholar
Tweedie, R. L. (1976) Criteria for classifying general Markov chains. Adv. Appl. Prob. 8, 737–771.CrossRefGoogle Scholar