Journal of Mathematical Biology

, Volume 62, Issue 5, pp 605–653

A multiscale maximum entropy moment closure for locally regulated space–time point process models of population dynamics



The prevalence of structure in biological populations challenges fundamental assumptions at the heart of continuum models of population dynamics based only on mean densities (local or global). Individual-based models (IBMs) were introduced during the last decade in an attempt to overcome this limitation by following explicitly each individual in the population. Although the IBM approach has been quite useful, the capability to follow each individual usually comes at the expense of analytical tractability, which limits the generality of the statements that can be made. For the specific case of spatial structure in populations of sessile (and identical) organisms, space–time point processes with local regulation seem to cover the middle ground between analytical tractability and a higher degree of biological realism. This approach has shown that simplified representations of fecundity, local dispersal and density-dependent mortality weighted by the local competitive environment are sufficient to generate spatial patterns that mimic field observations. Continuum approximations of these stochastic processes try to distill their fundamental properties, and they keep track of not only mean densities, but also higher order spatial correlations. However, due to the non–linearities involved they result in infinite hierarchies of moment equations. This leads to the problem of finding a ‘moment closure’; that is, an appropriate order of (lower order) truncation, together with a method of expressing the highest order density not explicitly modelled in the truncated hierarchy in terms of the lower order densities. We use the principle of constrained maximum entropy to derive a closure relationship for truncation at second order using normalisation and the product densities of first and second orders as constraints, and apply it to one such hierarchy. The resulting ‘maxent’ closure is similar to the Kirkwood superposition approximation, or ‘power-3’ closure, but it is complemented with previously unknown correction terms that depend mainly on the avoidance function of an associated Poisson point process over the region for which third order correlations are irreducible. This domain of irreducible triplet correlations is found from an integral equation associated with the normalisation constraint. This also serves the purpose of a validation check, since a single, non-trivial domain can only be found if the assumptions of the closure are consistent with the predictions of the hierarchy. Comparisons between simulations of the point process, alternative heuristic closures, and the maxent closure show significant improvements in the ability of the truncated hierarchy to predict equilibrium values for mildly aggregated spatial patterns. However, the maxent closure performs comparatively poorly in segregated ones. Although the closure is applied in the context of point processes, the method does not require fixed locations to be valid, and can in principle be applied to problems where the particles move, provided that their correlation functions are stationary in space and time.


Birth–dispersal–death stochastic processes Hierarchies of moment equations Spatially structured plant population dynamics Biologically driven spatial pattern formation Model reduction 

Mathematics Subject Classification (2000)

92B99 60K35 82B31 


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Michael Raghib
    • 1
    • 2
  • Nicholas A. Hill
    • 1
  • Ulf Dieckmann
    • 3
  1. 1.Department of MathematicsUniversity of GlasgowGlasgowUK
  2. 2.Program in Applied and Computational Mathematics, Department of Ecology and Evolutionary BiologyPrinceton UniversityPrincetonUSA
  3. 3.Ecology and Evolution ProgramInternational Institute for Applied Systems AnalysisLaxemburgAustria

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