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Escape from the boundary in Markov population processes

Part of:Markov processesProbability theory on algebraic and topological structures

Published online by Cambridge University Press:  21 March 2016

A. D. Barbour,
K. Hamza,
Haya Kaspi and
F. C. Klebaner
A. D. Barbour*
Affiliation:
Universität Zürich
K. Hamza*
Affiliation:
Monash University
Haya Kaspi*
Affiliation:
Technion, Haifa
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: School of Mathematical Sciences, Monash University, Clayton, VIC 3800, Australia.
∗∗∗ Postal address: Faculty of Industrial Engineering and Management, Technion, Haifa, 32000, Israel.
∗∗ Postal address: School of Mathematical Sciences, Monash University, Clayton, VIC 3800, Australia.
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Abstract

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Density dependent Markov population processes in large populations of size N were shown by Kurtz (1970), (1971) to be well approximated over finite time intervals by the solution of the differential equations that describe their average drift, and to exhibit stochastic fluctuations about this deterministic solution on the scale √N that can be approximated by a diffusion process. Here, motivated by an example from evolutionary biology, we are concerned with describing how such a process leaves an absorbing boundary. Initially, one or more of the populations is of size much smaller than N, and the length of time taken until all populations have sizes comparable to N then becomes infinite as N → ∞. Under suitable assumptions, we show that in the early stages of development, up to the time when all populations have sizes at least N 1-α for 1/3 < α < 1, the process can be accurately approximated in total variation by a Markov branching process. Thereafter, it is well approximated by the deterministic solution starting from the original initial point, but with a random time delay. Analogous behaviour is also established for a Markov process approaching an equilibrium on a boundary, where one or more of the populations become extinct.

Keywords

Markov population processboundary behaviourbranching process

MSC classification

Primary:92D30: Epidemiology
Secondary:60J27: Continuous-time Markov processes on discrete state spaces60B12: Limit theorems for vector-valued random variables (infinite-dimensional case)

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 4 , December 2015, pp. 1190 - 1211
Copyright
Copyright © Applied Probability Trust 2015 

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