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The growth of general population-size-dependent branching processes year by year

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Peter Jagers  and
Serik Sagitov
Peter Jagers*
Affiliation:
Chalmers University/University of Göteborg
Serik Sagitov*
Affiliation:
Chalmers University/University of Göteborg
*
∗Postal address: School of Mathematical and Computing Sciences, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden
∗Postal address: School of Mathematical and Computing Sciences, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden
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Abstract

We study discrete-time population models where the nearest future of an individual may depend on the individual's life-stage (age and reproduction history) and the current population size. A criterion is given for whether there is a positive probability that the population survives forever. We identify the cases when population size grows exponentially and linearly and show that in the latter population size scaled by time is asymptotically Γ-distributed.

Keywords

Population-size dependencebranching processmultitype Galton-Watson processstochastic difference equationdemographic models

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F25: $L^p$-limit theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 1 , March 2000 , pp. 1 - 14
Copyright
Copyright © 2000 by The Applied Probability Trust 

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Footnotes

This work is part of the Bank of Sweden Tercentenary Foundation project Dependence and Interaction in Stochastic Population Dynamics

References

Charlesworth, B. (1980). Evolution in Age-Structured Populations. Cambridge University Press, Cambridge.Google Scholar
Höpfner, R. (1985). On some class of population size dependent Galton–Watson processes. J. Appl. Prob. 22, 25–36.Google Scholar
Ivanoff, B. G., and Seneta, E. (1985). The critical branching process with immigration stopped at zero. J. Appl. Prob. 22, 223–227.CrossRefGoogle Scholar
Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, Chichester.Google Scholar
Jagers, P. (1997). Coupling and population dependence in branching processes. Ann. Appl. Prob. 7, 281–298.CrossRefGoogle Scholar
Jagers, P. (1999). Branching processes with dependence but homogeneous growth. Ann. Appl. Prob. 9.Google Scholar
Jagers, P., and Klebaner, F. C. (1999). Population-size-dependent and age-dependent branching processes. To appear in Stoch. Proc. Appl. Preprint 1998:56, Dept. Mathematics, Chalmers and Gothenburg Universities.Google Scholar
Kersting, G. (1986). On recurrence and transience of growth models. J. Appl. Prob. 23, 614–625.Google Scholar
Kersting, G. (1992). Asymptotic Γ-distribution for stochastic difference equations. Stoch. Process. Appl. 40, 15–28.Google Scholar
Kesten, H., and Stigum, B. (1967). Limit theorems for decomposable multidimensional Galton–Watson processes. J. Math. Anal. Appl. 17, 309–338.CrossRefGoogle Scholar
Keyfitz, N. (1968). Introduction to the Mathematics of Population. Addison-Wesley, Reading, MA.Google Scholar
Klebaner, F. C. (1984). On population-size-dependent branching processes. Adv. Appl. Prob. 16, 30–55.Google Scholar
Klebaner, F. C. (1985). A limit theorem for population-size-dependent branching processes. J. Appl. Prob. 22, 48–57.Google Scholar
Klebaner, F. C. (1989a). Stochastic difference equations and generalised gamma distributions.CrossRefGoogle Scholar
Klebaner, F. C. (1989b). Linear growth in near-critical population-size-dependent multitype Galton–Watson processes. J. Appl. Prob. 26, 431–445.Google Scholar
Klebaner, F. C. (1989c). Geometric growth in near-supercritical population size dependent multi-type Galton–Watson processes. Ann. Prob. 17, 1466–1477.Google Scholar
Klebaner, F. C. (1991). Asymptotic behaviour of near-critical multi-type branching processes. J. Appl. Prob. 28, 512–519.Google Scholar
Klebaner, F. C. (1992). Correction. J. Appl. Prob. 29, 246.Google Scholar
Küster, P. (1985). Asymptotic growth of controlled Galton–Watson processes. Ann. Prob. 13, 1157–1178.Google Scholar
Mode, C. J. (1971). Multi-type Branching Processes. Elsevier, New York.Google Scholar
Tuljapurkar, S., and Caswell, H. (1997). Structured-Population Models in Marine, Terrestrial, and Freshwater Systems. Chapman and Hall, New York.CrossRefGoogle Scholar
Vatutin, V. A. (1977). A conditional limit theorem for a critical branching process with immigration (in Russian). Mat. Zametki 21, 727–736.Google Scholar
Zubkov, A. M. (1972). Life-periods of a branching process with immigration. Theory Prob. Appl. 17, 174–183.CrossRefGoogle Scholar