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Local approximation of a metapopulation’s equilibrium


Barbour, A D; McVinish, R; Pollett, P K (2018). Local approximation of a metapopulation’s equilibrium. Journal of Mathematical Biology, 77(3):765-793.

Abstract

We consider the approximation of the equilibrium of a metapopulation model, in which a finite number of patches are randomly distributed over a bounded subset Ω of Euclidean space. The approximation is good when a large number of patches contribute to the colonization pressure on any given unoccupied patch, and when the quality of the patches varies little over the length scale determined by the colonization radius. If this is the case, the equilibrium probability of a patch at z being occupied is shown to be close to q1(z), the equilibrium occupation probability in Levins’s model, at any point z∈Ω not too close to the boundary, if the local colonization pressure and extinction rates appropriate to z are assumed. The approximation is justified by giving explicit upper and lower bounds for the occupation probabilities, expressed in terms of the model parameters. Since the patches are distributed randomly, the occupation probabilities are also random, and we complement our bounds with explicit bounds on the probability that they are satisfied at all patches simultaneously.

Abstract

We consider the approximation of the equilibrium of a metapopulation model, in which a finite number of patches are randomly distributed over a bounded subset Ω of Euclidean space. The approximation is good when a large number of patches contribute to the colonization pressure on any given unoccupied patch, and when the quality of the patches varies little over the length scale determined by the colonization radius. If this is the case, the equilibrium probability of a patch at z being occupied is shown to be close to q1(z), the equilibrium occupation probability in Levins’s model, at any point z∈Ω not too close to the boundary, if the local colonization pressure and extinction rates appropriate to z are assumed. The approximation is justified by giving explicit upper and lower bounds for the occupation probabilities, expressed in terms of the model parameters. Since the patches are distributed randomly, the occupation probabilities are also random, and we complement our bounds with explicit bounds on the probability that they are satisfied at all patches simultaneously.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Agricultural and Biological Sciences (miscellaneous), Modelling and Simulation, Applied Mathematics
Language:English
Date:1 September 2018
Deposited On:15 Nov 2018 13:39
Last Modified:18 Apr 2019 00:00
Publisher:Springer
ISSN:0303-6812
Additional Information:This is a post-peer-review, pre-copyedit version of an article published in Journal of Mathematical Biology. The final authenticated version is available online at: https://doi.org/10.1007/s00285-018-1231-0
OA Status:Green
Publisher DOI:https://doi.org/10.1007/s00285-018-1231-0

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