Escape from the boundary in Markov population processes

Andrew Barbour, Kais Hamza, Haya Kaspi, Fima C Klebaner

Research output: Contribution to journalArticleResearchpeer-review

7 Citations (Scopus)

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 N1-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.
Original languageEnglish
Pages (from-to)1190-1211
Number of pages22
JournalAdvances in Applied Probability
Volume47
Issue number4
DOIs
Publication statusPublished - 2015

Keywords

  • Markov population process
  • boundary behaviour
  • branching process

Cite this