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Extinction of population-size-dependent branching processes in random environments

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Han-xing Wang
Han-xing Wang*
Affiliation:
Shanghai University
*
∗Postal address: Department of Mathematics, Shanghai University, Shanghai, 201800, P.R. China. Email address: [email protected].
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Abstract

We generalize a population-size-dependent branching process to a more general branching model called the population-size-dependent branching process in random environments. For the model where {Zn}n≥0 is associated with the stationary environment ξ− = {ξn}n≥0, let B = {ω : Zn(ω) = for some n}, and q(ξ−) = P(B | ξ−, Z0 = 1). The result is that P(q(̅ξ) = 1) is either 1 or 0, and sufficient conditions for certain extinction (i.e. P(q(ξ−) = 1) = 1) and for non-certain extinction (i.e. P(q(ξ−) < 1) = 1) are obtained for the model.

Keywords

Stochastic population modelsMarkov chains in random environmentsextinction probabilities

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 1 , March 1999 , pp. 146 - 154
Copyright
Copyright © Applied Probability Trust 1999 

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References

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