Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T06:52:48.606Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Fima C. Klebaner
Affiliation:
Monash University, School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

[1] Asmussen, S. and Hering, H., (1983). Branching Processes. Birkhäuser. Boston, MA.CrossRefGoogle Scholar
[2] Borde-Boussion, A.-M., (1990). Stochastic demographic models: age of a population. Stoch. Process. Appl. 35, 279–291.CrossRefGoogle Scholar
[3] Champagnat, N., Ferrière, R. and Méléard, S., (2008). From individual stochastic processes to macroscopic models in adaptive evolution. Stoch. Models 24, 2–44.Google Scholar
[4] Chigansky, P. and Liptser, R., (2010). Moderate deviations for a diffusion-type process in a random environment. Theory Prob. Appl. 54, 29–50.CrossRefGoogle Scholar
[5] Dawson, D. A., (1993). Measure-valued Markov processes. In École d'Eté de Probabilités de Saint-Flour XXI} (Lecture Notes Math. 1541, Springer, Berlin, pp. 1–260.Google Scholar
[6] Ethier, S. N. and Kurtz, T. G., (1986). Markov Processes. John Wiley, New York.Google Scholar
[7] Freidlin, M. I. and Wentzell, A. D., (1998). Random Perturbations of Dynamical Systems. Springer, New York.Google Scholar
[8] Geritz, S. A. H., Kisdi, É., Meszéna, G. and Metz, J. A. J., (1998). Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecology 12, 35–57.Google Scholar
[9] Haccou, P., Jagers, P. and Vatutin, V. A., (2005). Branching Processes: Variation, Growth, and Extinction of Populations. Cambridge University Press.CrossRefGoogle Scholar
[10] Jagers, P., (1975). Branching Processes with Biological Applications. Wiley-Interscience, London.Google Scholar
[11] Jagers, P., (1992). Stabilities and instabilities in population dynamics. J. Appl. Prob. 29, 770–780.CrossRefGoogle Scholar
[12] Jagers, P. and Klebaner, F. C., (2000). Population-size-dependent and age-dependent branching processes. Stoch. Process. Appl. 87, 235–254.CrossRefGoogle Scholar
[13] Klebaner, F. C., (2005). Introduction to Stochastic Calculus with Applications, 2nd. edn. Imperial College Press, London.Google Scholar
[14] Klebaner, F. C., et al. (2011). Stochasticity in the adaptive dynamics of evolution: the bare bones. J. Biol. Dynamics 5, 174–162.Google Scholar
[15] Méléard, S. and Tran, V. C., (2009). Trait substitution sequence process and canonical equation for age-structured populations. J. Math. Biol. 58, 881–921.CrossRefGoogle ScholarPubMed
[16] Métivier, M., (1987). Weak convergence of measure valued processes using Sobolev imbedding techniques. In Stochastic Partial Differential Equations and Applications} (Trento, 1985; Lecture Notes Math. 1236, Springer, Berlin, pp. 172–183.Google Scholar
[17] Oelschläger, K., (1990). Limit theorems for age-structured populations. Ann. Prob. 18, 290–318.Google Scholar
[18] Tran, V. C., (2008). Large population limit and time behaviour of a stochastic particle model describing an age-structured population. ESAIM Prob. Statist. 12, 345–386.CrossRefGoogle Scholar
[19] Waxman, D. and Gavrilets, S., (2005). 20 Questions on adaptive dynamics. J. Evol. Biol. 18, 1139–1154.CrossRefGoogle ScholarPubMed