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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: a.d.barbour@math.uzh.ch
** Postal address: Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem, 91905, Israel. Email address: pchiga@mscc.huji.ac.il
*** Postal address: School of Mathematical Sciences, Monash University, Monash, VIC 3800, Australia. Email address: fima.klebaner@monash.edu
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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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