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

NanFu Peng; Dennis K. Pearl; Wenyaw Chan; Robert Bartoszyński

doi:10.1016/0304-4149(93)90072-C

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

Institute of Statistics

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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 language	English
Pages (from-to)	243-258
Number of pages	16
Journal	Stochastic Processes and their Applications
Volume	45
Issue number	2
DOIs	https://doi.org/10.1016/0304-4149(93)90072-C
State	Published - 1 Jan 1993

Keywords

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

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10.1016/0304-4149(93)90072-C

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

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.",

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AU - Peng, NanFu

AU - Pearl, Dennis K.

AU - Chan, Wenyaw

AU - Bartoszyński, Robert

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N2 - 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.

AB - 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.

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KW - delay differential equations

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

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