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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
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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 

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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