Header

UZH-Logo

Maintenance Infos

Escape from the boundary in Markov population processes


Barbour, A D; Hamza, K; Kaspi, Haya; Klebaner, F C (2015). Escape from the boundary in Markov population processes. Advances in Applied Probability, 47(4):1190-1211.

Abstract

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 $N^{1-\alpha}$ 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.

Abstract

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 $N^{1-\alpha}$ 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.

Statistics

Citations

Dimensions.ai Metrics
6 citations in Web of Science®
4 citations in Scopus®
8 citations in Microsoft Academic
Google Scholar™

Altmetrics

Downloads

1 download since deposited on 28 Dec 2017
1 download since 12 months
Detailed statistics

Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2015
Deposited On:28 Dec 2017 16:02
Last Modified:18 Apr 2018 11:49
Publisher:Applied Probability Trust
ISSN:0001-8678
Funders:Australian Research Council
OA Status:Closed
Publisher DOI:https://doi.org/10.1017/S0001867800049077

Download