Linear birth and death processes under the influence of disasters with time-dependent killing probabilities

NanFu Peng, Dennis K. Pearl*, Wenyaw Chan, Robert Bartoszyński

*Corresponding author for this work

Research output: Contribution to journalArticle

6 Scopus citations

Abstract

Supercritical linear birth-and-death processes are considered under the influence of disasters that arrive as a renewal process independently of the population size. The novelty of this paper lies in assuming that the killing probability in a disaster is a function of the time that has elapsed since the last disaster. A necessary and sufficient condition for a.s. extinction is found. When catastrophes form a Poisson process, formulas for the Laplace transforms of the expectation and variance of the population size as a function of time as well as moments of the odds of extinction are derived (these odds are random since they depend on the intercatastrophe times). Finally, we study numerical techniques leading to plots of the density of the probability of extinction.

Original languageEnglish
Pages (from-to)243-258
Number of pages16
JournalStochastic Processes and their Applications
Volume45
Issue number2
DOIs
StatePublished - 1 Jan 1993

Keywords

  • catastrophes
  • delay differential equations
  • edgeworth expansion
  • extinction probability
  • linear birth-and-death process
  • time-dependent killing

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