Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-11T01:38:57.625Z Has data issue: false hasContentIssue false
  • English
  • Français

Deterministic approximation of a stochastic metapopulation model

Part of: Probability theory on algebraic and topological structures Infinite-dimensional dissipative dynamical systems Markov processes Differential equations in abstract spaces

Published online by Cambridge University Press:  01 July 2016

Francesca Arrigoni
Francesca Arrigoni*
Affiliation:
University of Trento
*
∗ Postal address: Department of Mathematics, University of Trento, via Sommarive 14, 38050 Povo, Trento, Italy. Email address: arrigoni@science.unitn.it
Get access Rights & Permissions [Opens in a new window]

Abstract

We analyse the limit behaviour of a stochastic structured metapopulation model as the number of its patches goes to infinity. The sequence of probability measures associated with the random process, whose components are the proportions of patches with different number of individuals, is tight. The limit of every convergent subsequence satisfies an infinite system of ordinary differential equations. The existence and the uniqueness of the solution are shown by semigroup methods, so that the whole random process converges weakly to the solution of the system.

Keywords

MetapopulationMarkov chainweak convergence

MSC classification

Primary: 37L05: General theory, nonlinear semigroups, evolution equations 92D40: Ecology 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60B12: Limit theorems for vector-valued random variables (infinite-dimensional case) 34G20: Nonlinear equations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 3 , September 2003 , pp. 691 - 720
Copyright
Copyright © Applied Probability Trust 2003 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Arrigoni, F. (2001). Deterministic approximation of stochastic metapopulation model. Doctoral Thesis, University of Trento.Google Scholar
[2] Barbour, A. D. (1980). Density dependent Markov population processes. In Biological Growth and Spread, (Lecture Notes Biomath. 38), Springer, Berlin, pp. 36–49.CrossRefGoogle Scholar
[3] Barbour, A. D. and Kafetzaki, M. (1993). A host parasite model yielding heterogenous parasite loads. J. Math. Biol. 31, 157–176.Google Scholar
[4] Berman, A. and Plemmons, R. J. (1994). Nonnegative Matrices in the Mathematical Sciences, revised edn (Classics Appl. Math. 9). SIAM, Philadelphia.Google Scholar
[5] Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
[6] Busenberg, S. N., Iannelli, M. and Thieme, H. R. (1991). Global behaviour of an age-structured epidemic model. Siam J. Math. Anal. 22, 1065–1080.Google Scholar
[7] Capasso, V., Morale, D. and Oelschläger, K. (1998). A rigorous derivation of a nonlinear integro-differential equation from a system of stochastic differential equations for an aggregation model. Preprint 98–38 (SFB 359), IWR, Universität Heidelberg.Google Scholar
[8] Casagrandi, R. and Gatto, M. (1999). A mesoscale approach to extinction risk in fragmented habitats. Nature 400, 560–562.CrossRefGoogle Scholar
[9] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence. John Wiley, New York.Google Scholar
[10] Gyllenberg, M. and Hanski, I. (1992). Single-species metapopulation dynamics: a structured model. Theoret. Pop. Biol. 42, 35–61.Google Scholar
[11] Gyllenberg, M. and Silvestrov, D. (1994). Quasi-stationary distributions of a stochastic metapopulation model. J. Math. Biol. 33, 35–70.Google Scholar
[12] Hadeler, K. P. and Dietz, K. (1984). Population dynamics of killing parasites which reproduce in the host. J. Math. Biol. 21, 45–65.Google Scholar
[13] Hamza, K. and Klebaner, F. C. (1995). Conditions for integrability of Markov chains. J. Appl. Prob. 32, 541–547.CrossRefGoogle Scholar
[14] Hanski, I. A. and Gilpin, M. E. (eds) (1997). Metapopulation Biology. Ecology, Genetics, and Evolution. Academic Press, San Diego, CA.Google Scholar
[15] Kretzschmar, M. (1989). A renewal equation with a birth–death process as a model for parasitic infections. J. Math. Biol. 27, 191–221.CrossRefGoogle Scholar
[16] Levins, R. (1969). Some demographic and genetic consequences of environmental heterogeneity for biological control. Bull. Entomol. Soc. Amer. 15, 237–240.Google Scholar
[17] Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar
[18] Luchsinger, C. J. (1999). Mathematical models of a parasitic disease. Doctoral Thesis, University of Zürich.Google Scholar
[19] Metz, J. A. J. and Gyllenberg, M. (2001). How should we define fitness in structured metapopulation models? Including an application to the calculation of evolutionarily stable dispersal strategies. Proc. R. Soc. London B 268, 499–508.Google Scholar
[20] Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations (Appl. Math. Sci. 44). Springer, New York.Google Scholar
[21] Pollett, P. K. (1999). Modelling quasi-stationary behaviour in metapopulations. Math. Comput. Simul. 48, 393–405.CrossRefGoogle Scholar
[22] Reuter, G. E. H. (1957). Denumerable Markov processes and the associated contraction semigroups on. Acta Math. 97, 1–46.Google Scholar
[23] Stevens, A. (2000). The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J. Appl. Math. 61, 183–212.CrossRefGoogle Scholar
[24] Windisch, G. (1989) M-Matrices in Numerical Analysis (Teubner Texts Math. 115). Teubner, Leipzig.Google Scholar