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On the emergence of random initial conditions in fluid limits

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  09 December 2016

A. D. Barbour ,
P. Chigansky  and
F. C. Klebaner
A. D. Barbour*
Affiliation:
Universität Zürich
P. Chigansky*
Affiliation:
The Hebrew University of Jerusalem
F. C. Klebaner*
Affiliation:
Monash University
*
* Postal address: Institut für Mathematik, Universität Zürich, Winterthurertrasse 190, CH-8057 Zürich, Switzerland. Email address: [email protected]
** Postal address: Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem, 91905, Israel. Email address: [email protected]
*** Postal address: School of Mathematical Sciences, Monash University, Monash, VIC 3800, Australia. Email address: [email protected]
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Abstract

In the paper we present a phenomenon occurring in population processes that start near 0 and have large carrying capacity. By the classical result of Kurtz (1970), such processes, normalized by the carrying capacity, converge on finite intervals to the solutions of ordinary differential equations, also known as the fluid limit. When the initial population is small relative to the carrying capacity, this limit is trivial. Here we show that, viewed at suitably chosen times increasing to ∞, the process converges to the fluid limit, governed by the same dynamics, but with a random initial condition. This random initial condition is related to the martingale limit of an associated linear birth-and-death process.

Keywords

Birth‒death processpopulation dynamics with carrying capacityfluid approximation

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 92D25: Population dynamics (general) 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 4 , December 2016 , pp. 1193 - 1205
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1] Barbour, A. D. (1976).Quasi-stationary distributions in Markov population processes.Adv. Appl. Prob. 8,296–314.CrossRefGoogle Scholar
[2] Barbour, A. D.,Hamza, K.,Kaspi, H. and Klebaner, F. C. (2015).Escape from the boundary in Markov population processes.Adv. Appl. Prob. 47,1190–1211.CrossRefGoogle Scholar
[3] Bartlett, M. S. (1949).Some evolutionary stochastic processes.J. R. Statist. Soc. Ber. B. 11,211–229.Google Scholar
[4] Bartlett, M. S. (1960).Stochastic Population Models in Ecology and Epidemiology.Methuen,London.Google Scholar
[5] Beverton, R. J. H. and Holt, S. J. (1957).On the Dynamics of Exploited Fish Populations.Her Majesty's Stationery Office,London.Google Scholar
[6] Bhaskaran, B. G. (1986).Almost sure comparison of birth and death processes with application to M/M/s queueing systems.Queueing Systems Theory Appl. 1,103–127.CrossRefGoogle Scholar
[7] Grimmett, G. R. and Stirzaker, D. R. (2001).Probability and Random Processes,3rd edn.Oxford University Press.CrossRefGoogle Scholar
[8] Hassell, M. P. (1975).Density-dependence in single-species populations.J. Animal Ecology 44,283–295.CrossRefGoogle Scholar
[9] Jagers, P. and Klebaner, F. C. (2011).Population-size-dependent, age-structured branching processes linger around their carrying capacity.In New Frontiers in Applied Probability: A Festschrift for Søren Asmussen,Applied Probabilty Trust,Sheffield,pp. 249–260.Google Scholar
[10] Kendall, D. G. (1956).Deterministic and stochastic epidemics in closed populations.In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954‒1955,Vol. IV,University of California Press,Berkeley, CA,pp. 149–165.Google Scholar
[11] Klebaner, F. C. (2012).Introduction to Stochastic Calculus with Applications,3rd edn.Imperial College Press,London.CrossRefGoogle Scholar
[12] Kurtz, T. G. (1970).Solutions of ordinary differential equations as limits of pure jump Markov processes.J. Appl. Prob. 7,49–58.CrossRefGoogle Scholar
[13] Maynard Smith, J. and Slatkin, M. (1973).The stability of predator‒prey systems.Ecology 54,384–391.CrossRefGoogle Scholar
[14] McKendrick, A. G. (1925).Applications of mathematics to medicalproblems.Proc. Edinburgh Math. Soc. 44,98–130.CrossRefGoogle Scholar
[15] Ricker, W. E. (1954).Stock and recruitment.J. Fisheries Res. Board Canada 11,559–623.CrossRefGoogle Scholar
[16] Thorisson, H. (2000).Coupling, Stationarity, and Regeneration.Springer,New York.CrossRefGoogle Scholar
[17] Whittle, P. (1955).The outcome of a stochastic epidemic–a note on Bailey's paper.Biometrika 42,116–122.Google Scholar