Unravelling the spatial structure of regular environmental spatial patterns

Spatial patterns where patches of high biomass alternate with bare ground occur in many resource-limited ecosystems. Especially fascinating are regular patterns, which are self-similar at a lag distance corresponding to the typical distance between patches. Regular patterns are understood to form autogenously through self-organization, which can be generated with deterministic reaction-diffusion models. Such models generate highly regular patterns, which repeat at the characteristic wavelength and are therefore periodic. Natural patterns do not repeat, as they are noisy and as the patch size and spacing vary. Natural patterns are therefore usually perceived as perturbed periodic patterns. However, the self-similarity of natural patterns decreases at longer lag distances, which indicates that their spatial structure is not a perturbed periodic structure originating through deterministic processes. Here, we provide an overview of our recent work on the spatial structure and formation of natural environmental spatial patterns as a basis for discussion: First, we develop a statistical periodicity test and compile a large dataset of more than 10,000 regular environmental spatial patterns. We find that neither isotropic (spotted) nor anisotropic (banded) patterns are periodic. Instead, we find that their spatial structure can be well described as random fields originating through stochastic processes. Second, we recognize the regularity as a gradually varying property, rather than a dichotomous property of being periodic or not. We develop a method for quantifying the regularity and apply it in a metastudy to a set of natural and model-generated patterns found in the literature. We find that patterns generated with deterministic reaction-diffusion models do not well reproduce the spatial structure of environmental spatial structure, as they are too regular. Third, we develop an understanding of pattern formation through stochastic reaction-diffusion processes, which incorporate random environmental heterogeneities. We find that regular patterns form through filtering of the environmental heterogeneities and identify stochastic processes which reproduce both isotropic and anisotropic patterns.