Characteristic length scales of spatial models in ecology via fluctuation analysis

A technique of fluctuation analysis is introduced for the identification of characteristic length scales in spatial models, with similarities to the recently introduced methods using correlations. The identified length scale provides the optimal size to extract non-trivial large-scale behaviour in such models. The method is demonstrated for three biological models: genetic selection, plant competition and a complex marine system; the first two are coupled map lattices and the last one is a cellular automaton. These cover the three possibilities for asymptotic (long time) dynamics: fixation (the system converges to a fixed point); statistical fixation (the spatial statistics converge to fixed values); and complex statistical structure (the statistics do not converge to fixed values). The technique is shown to have an additional use in the identification of aggregation or dispersal at various scales. The method is rigorously justifiable in the cases when the system under analysis satisfies the FKG (Fortuin-Kasteleyn-Ginibre) property and has a fast decay of correlations. We also discuss the connection between the fluctuation analysis length scale and hydrodynamic limits methods to derive large scale equations for ecological models.