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Population-size-dependent, age-structured branching processes linger around their carrying capacity

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Peter Jagers  and
Fima C. Klebaner
Peter Jagers
Affiliation:
Chalmers University of Technology and University of Gothenburg, Mathematical Sciences, Chalmers University of Technology, Chalmers, SE-412 96 Gothenburg, Sweden. Email address: jagers@chalmers.se
Fima C. Klebaner
Affiliation:
Monash University, School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia.
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Abstract

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Dependence of individual reproduction upon the size of the whole population is studied in a general branching process context. The particular feature under scrutiny is that of reproduction changing from supercritical in small populations to subcritical in large populations. The transition occurs when the population size passes a critical threshold, known in ecology as the carrying capacity. We show that populations either die out directly, never coming close to the carrying capacity, or grow quickly towards the carrying capacity, subsequently lingering around it for a time that is expected to be exponentially long in terms of a carrying capacity tending to infinity.

Keywords

Age structurebranching processcarrying capacity

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 92D25: Population dynamics (general)
Type
Part 5. Stochastic Growth and Branching
Information
Journal of Applied Probability , Volume 48 , Issue A: New Frontiers in Applied Probability (Journal of Applied Probability Special Volume 48A) , August 2011 , pp. 249 - 260
Copyright
Copyright © Applied Probability Trust 2011 

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