Journal of Mathematical Biology

, Volume 77, Issue 3, pp 765–793 | Cite as

Local approximation of a metapopulation’s equilibrium

  • A. D. Barbour
  • R. McVinishEmail author
  • P. K. Pollett


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 \(\Omega \) 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 \(q_1(z)\), the equilibrium occupation probability in Levins’s model, at any point \(z \in \Omega \) 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.


Incidence function model Spatially realistic Levins model Equilibrium Fixed point Metapopulation 

Mathematics Subject Classification

92D40 60J10 60J27 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Universität ZürichZürichSwitzerland
  2. 2.University of QueenslandBrisbaneAustralia

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