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The Limiting Behaviour of Hanski's Incidence Function Metapopulation Model

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  19 February 2016

R. McVinish  and
P. K. Pollett
R. McVinish*
Affiliation:
University of Queensland
P. K. Pollett*
Affiliation:
University of Queensland
*
∗ Postal address: School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia. Email address: r.mcvinish@uq.edu.au.
∗∗ Email address: pkp@maths.uq.edu.au.
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Abstract

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Hanski's incidence function model is one of the most widely used metapopulation models in ecology. It models the presence/absence of a species at spatially distinct habitat patches as a discrete-time Markov chain whose transition probabilities are determined by the physical landscape. In this analysis, the limiting behaviour of the model is studied as the number of patches increases and the size of the patches decreases. Two different limiting cases are identified depending on whether or not the metapopulation is initially near extinction. Basic properties of the limiting models are derived.

Keywords

Extinction thresholdpoint processstochastic patch occupancy modelSPOMweak convergence

MSC classification

Primary: 92D40: Ecology
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 2 , June 2014 , pp. 297 - 316
Copyright
© Applied Probability Trust 

References

Ball, F. (1983). The threshold behaviour of epidemic models. J. Appl. Prob. 20, 227–241.CrossRefGoogle Scholar
Ball, F. and Clancy, D. (1993). The final size and severity of a generalised stochastic multitype epidemic model. J. Appl. Prob. 25, 721–736.CrossRefGoogle Scholar
Ball, F. and Donnelly, P. (1995). Strong approximations for epidemic models. Stoch. Proc. Appl. 55, 1–21.CrossRefGoogle Scholar
Barbour, A. D. (2010). Coupling a branching process to an infinite dimensional epidemic process. ESAIM Prob. Statist. 14, 53–64.CrossRefGoogle Scholar
Buckley, F. M. and Pollett, P. K. (2010). Limit theorems for discrete-time metapopulation models. Prob. Surveys, 7, 53–83.CrossRefGoogle Scholar
Courchamp, F., Berec, L. and Gascoigne, J. (2008). Allee Effects in Ecology and Conservation. Oxford University Press, New York.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. 2, Springer, New York.CrossRefGoogle Scholar
Dautray, R. and Lions, J.-L. (1990). Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 3. Springer, Berlin.Google Scholar
Durrett, R. (1999). Stochastic spatial models. SIAM Rev. 41, 677–718.CrossRefGoogle Scholar
Hanski, I. (1994). A practical model of metapopulation dynamics. J. Anim. Ecol. 63, 151–162.CrossRefGoogle Scholar
Hanski, I., Alho, J. and Moilanen, A. (2000). Estimating the parameters of survival and migration of individuals in metapopulations. Ecology 81, 239–251.CrossRefGoogle Scholar
Hanski, I. and Ovaskainen, O. (2003). Metapopulation theory for fragmented landscapes. Theoret. Pop. Biol. 64, 119–127.CrossRefGoogle ScholarPubMed
Hirsch, M. W. and Smith, H. (2005). Monotone maps: a review. J. Difference Equat. 11, 379–398.CrossRefGoogle Scholar
Istratescu, V. I. (1981). Fixed Point Theory: An Introduction. D. Reidel, Dordrecht.CrossRefGoogle Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Levins, R. (1969). Some demographic and genetic consequences of environmental heterogeneity for biological control. Bull. Entomol. Soc. Amer. 15, 237–240.Google Scholar
MacCluer, B. D. (2009). Elementary Functional Analysis, Springer, New York.CrossRefGoogle Scholar
McVinish, R. and Pollett, P. K. (2010). Limits of large metapopulations with patch-dependent extinction probabilities. Adv. Appl. Prob. 42, 1172–1186.CrossRefGoogle Scholar
McVinish, R. and Pollett, P. K. (2012). The limiting behaviour of a mainland-island metapopulation. J. Math. Biol. 64, 775–801.CrossRefGoogle ScholarPubMed
McVinish, R. and Pollett, P. K. (2013). The limiting behaviour of a stochastic patch occupancy model. J. Math. Biol. 67, 693–716.CrossRefGoogle ScholarPubMed
Moilanen, A. (2004). SPOMSIM: software for stochastic patch occupancy models of metapopulation dynamics. Ecol. Model. 179, 533–550.CrossRefGoogle Scholar
Moyal, J. E. (1962). Multiplicative population chains. Proc. R. Soc. London A 266, 518–526.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
Ovaskainen, O. and Hanski, I. (2001). Spatially structured metapopulation models: global and local assessment of metapopulation capacity. Theoret. Pop. Biol. 60, 281–302.CrossRefGoogle ScholarPubMed
Penrose, M. D. (2008). Existence and spatial limit theorems for lattice and continuum particle systems. Prob. Surveys, 5, 1–36.CrossRefGoogle Scholar
Petit, S., Moilanen, A., Hanski, I. and Baguette, M. (2001). Metapopulation dynamics of the bog fritillary butterfly: movements between habitat patches. Oikos 92, 491–500.CrossRefGoogle Scholar
Ranga Rao, R. (1962). Relations between weak and uniform convergence of measures with applications. Ann. Math. Statist. 33, 659–680.Google Scholar