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

Raghib M, Hill NA, & Dieckmann U (2011). A multiscale maximum entropy moment closure for locally regulated space-time point process models of population dynamics. Journal of Mathematical Biology 62 (5): 605-653. DOI:10.1007/s00285-010-0345-9.

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Abstract

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 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 nonlinearities 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.

Item Type: Article
Uncontrolled Keywords: Birth-dispersal-death stochastic processes; Hiearchies of moment equations; Spatially structured plant population dynamics; Biologically driven spatial pattern formation; Model reduction
Research Programs: Evolution and Ecology (EEP)
Bibliographic Reference: Journal of Mathematical Biology; 62(5):605-653 (May 2011) (Published online 06 May 2010)
Depositing User: IIASA Import
Date Deposited: 15 Jan 2016 08:45
Last Modified: 24 Feb 2016 14:49
URI: http://pure.iiasa.ac.at/9614

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