A scale-invariant method for quantifying the regularity of environmental spatial patterns

Abstract

Spatial patterns of alternating high and low biomass occur in a wide range of ecosystems. Patterns can improve ecosystem productivity and resilience, but the particular effects of patterning depend on their spatial structure. The spatial structure is conventionally classified as either regular, when the patches of biomass are of similar size and are spaced in similar intervals, or irregular. The formation of regular patterns is driven by scale-dependent feedbacks. Models incorporating those feedbacks generate highly regular patterns, while natural patterns appear less regular. This calls for a more nuanced quantification beyond a binary classification. Here, we propose measuring the degree of regularity by the maximum of a pattern’s spectral density, based on the observation that the density of highly regular patterns consists of a narrow and high peak, while the density of highly irregular patterns consists of a low and wide lobe. We rescale the density to make the measure invariant with respect to the characteristic length-scale of a pattern, facilitating the comparison of patterns observed or modelled under different conditions. We demonstrate our method in a metastudy determining the regularity of natural and model-generated patterns depicted in previous studies. We find that natural patterns have an intermediate degree of regularity, resembling random surfaces generated by stochastic processes. We find that conventional deterministic models do not reproduce the intermediate regularity of natural patterns, as they generate patterns which are much more regular and similar to periodic surfaces. We call for appreciating the stochasticity of natural patterns in systems with scale-dependent feedbacks.