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Limits of large metapopulations with patch-dependent extinction probabilities

Part of: Stochastic processes Limit theorems Markov processes

Published online by Cambridge University Press:  01 July 2016

R. McVinish  and
P. K. Pollett
R. McVinish*
Affiliation:
University of Queensland
P. K. Pollett*
Affiliation:
University of Queensland
*
∗ Postal address: Department of Mathematics, University of Queensland, Brisbane, QLD 4072, Australia.
∗ Postal address: Department of Mathematics, University of Queensland, Brisbane, QLD 4072, Australia.
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Abstract

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We propose a model for the presence/absence of a population in a collection of habitat patches. This model assumes that colonisation and extinction of the patches occur as distinct phases. Importantly, the local extinction probabilities are allowed to vary between patches. This permits an investigation of the effect of habitat degradation on the persistence of the population. The limiting behaviour of the model is examined as the number of habitat patches increases to ∞. This is done in the case where the number of patches and the initial number of occupied patches increase at the same rate, and for the case where the initial number of occupied patches remains fixed.

Keywords

Chain binomial modelfixed pointweak convergencepoint process

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Secondary: 60G55: Point processes 60F05: Central limit and other weak theorems

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 4 , December 2010 , pp. 1172 - 1186
Copyright
Copyright © Applied Probability Trust 2010 

References

Abramowitz, M. and Stegun, I. A. (eds) (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York.Google Scholar
Andersson, H. and Djehiche, B. (1998). A threshold limit theorem for the stochastic logistic epidemic. J. Appl. Prob. 35, 662–670.CrossRefGoogle Scholar
Andersson, M. (1999). The asymptotic final size distribution of multitype chain-binomial epidemic processes. Adv. Appl. Prob. 31, 220–234.Google Scholar
Arrigoni, F. (2003). Deterministic approximation of a stochastic metapopulation model. Adv. Appl. Prob. 35, 691–720.Google Scholar
Buckley, F. M. and Pollett, P. K. (2010). Analytical methods for a stochastic mainland-island metapopulation model. Ecol. Modelling 221, 2526–2530.Google Scholar
Buckley, F. M. and Pollett, P. K. (2010). Limit theorems for discrete-time metapopulation models. Prob. Surveys 7, 53–83 Google Scholar
Chan, C. K. (2009). Limiting conditional distribution of continuous time Markov chains and their application to metapopulation model. , University of Queensland.Google Scholar
Cornell, S. J. and Ovaskainen, O. (2008). Exact asymptotic analysis for metapopulation dynamics on correlated dynamic landscapes. Theoret. Pop. Biol. 74, 209–225.Google Scholar
Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Daley, D. J., Gani, J. and Yakowitz, S. (2000). An epidemic with individual infectivities and susceptibilities. Math. Comput. Modelling 32, 155–167.Google Scholar
Darling, R. W. R. and Norris, J. R. (2008). Differential equation approximations for Markov chains. Prob. Surveys 5, 37–79.Google Scholar
Gani, J. and Stals, L. (2004). The spread of a viral infection in a plantation. Environmetrics 15, 555–560.Google Scholar
Hanski, I. and Gilpin, M. E. (1997). Metapopulation Biology: Ecology, Genetics and Evolution. Academic Press, San Diego.Google Scholar
Hanski, I. and Ovaskainen, O. (2003). Metapopulation theory for fragmented landscapes. Theoret. Pop. Biol. 64, 119–127.Google Scholar
Hill, M. F. and Caswell, H. (2001). The effects of habitat destruction in finite landscapes: a chain-binomial metapopulation model. Oikos 93, 321–331.Google Scholar
Moilanen, A. (2004). SPOMSIM: software for stochastic patch occupancy models of metapopulation dynamics. Ecol. Modelling 179, 533–550.Google Scholar
Ovaskainen, O. (2001). The quasistationary distribution of the stochastic logistic model. J. Appl. Prob. 38, 898–907.Google Scholar
Ovaskainen, O. and Cornell, S. J. (2006). Asymptotically exact analysis of stochastic metapopulation dynamics with explicit spatial structure. Theoret. Pop. Biol. 69, 13–33.CrossRefGoogle ScholarPubMed
Pellet, J. et al. (2007). An empirical evaluation of the area and isolation paradigm of metapopulation dynamics. Biol. Conservation 136, 483–495.Google Scholar
Preston, C. J. (1977). Spatial birth-and-death processes. Bull. Internat. Statist. Inst. 46, 371–391.Google Scholar
Ter Braak, C. J. F. and Etienne, R. S. (2003). Improved Bayesian analysis of metapopulation data with an application to a tree frog metapopulation. Ecology 84, 231–241.Google Scholar
Weiss, G. H. and Dishon, M. (1971). On the asymptotic behavior of the stochastic and deterministic models of an epidemic. Math. Biosci. 11, 261–265.Google Scholar