Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-29T21:42:57.298Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniform bounding of probability generating functions and the evolution of reproduction rates in birds

Published online by Cambridge University Press:  14 July 2016

C. C. Heyde  and
H.-J. Schuh
C. C. Heyde
Affiliation:
CSIRO Division of Mathematics and Statistics, Canberra
H.-J. Schuh
Affiliation:
CSIRO Division of Mathematics and Statistics, Canberra
Get access Rights & Permissions [Opens in a new window]

Abstract

Many species of birds have a characteristic clutch size which is either fixed at k or is of the form k or k + 1 for some appropriate integer k. In this paper we show, using a multitype Galton–Watson process to model a bird population, that such behaviour can correspond to maximization of the probability of survival of the species to time t for each finite t. This is also a conclusion which might be drawn from the theory of natural selection and hence provides some mathematical evidence of the force of evolution. The results of the paper rest on a bounding of probability generating functions.

Keywords

BRANCHING PROCESSESMULTITYPE GALTON–WATSON PROCESSPROBABILITY GENERATING FUNCTIONEVOLUTIONCLUTCH SIZE FOR BIRDSREPRODUCTION RATES
Type
Research Papers
Information
Journal of Applied Probability , Volume 15 , Issue 2 , June 1978 , pp. 243 - 250
Copyright
Copyright © Applied Probability Trust 1978 

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.)

References

[1] Braun, H. (1975) Polynomial bounds for probability generating functions. J. Appl. Prob. 12, 507–514.Google Scholar
[2] Daley, D. J. (1969) Extinction probabilities in branching processes: a note on Holgate and Lakhani's paper. Bull. Math. Biophys. 31, 35–37.CrossRefGoogle Scholar
[3] Freedman, D. and Purves, R. (1967) Timid play is optimal II. Ann. Math. Statist. 38, 1284–1285.CrossRefGoogle Scholar
[4] Goodman, L. A. (1967) The probabilities of extinction for birth-and-death processes that are age-dependent or phase dependent. Biometrika 54, 579–596.CrossRefGoogle ScholarPubMed
[5] Goodman, L. A. (1968) How to minimize or maximize the probabilities of extinction in a Galton–Watson process and in some related multiplicative population processes. Ann. Math. Statist. 39, 1700–1710.Google Scholar
[6] Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
[7] Lack, D. (1954) The evolution of reproduction rates. In Evolution as a Process, ed. Huxley, J., Hardy, A. C. and Ford, E. B. Allen and Unwin, London.Google Scholar
[8] Lack, D. (1966) Population Studies of Birds. Clarendon Press, Oxford.Google Scholar
[9] Mountford, M. D. (1973) The significance of clutch size. In Mathematical Theory of the Dynamics of Biological Populations, ed. Bartlett, M. S. and Hiorns, R. W. Academic Press, London and New York, 315–323.Google Scholar