A metaanalysis of the regularity of environmental spatialpatterns and a theory relating them to stochastic processes

Fascinating spatial patterns are found in many ecosystems. For example, patterns in Dryland ecosystems often consist of vegetation patches which alternate with bare soil. The patterns appear strikingly regular, when their formation is driven by scale-dependent feedbacks. Because of their regularity, such patterns are conceptually understood to be periodic. The formation of periodic patterns has been reproduced with idealized numerical models and the vulnerability of pattern forming ecosystems to environmental pressure has been preferentially studied with such models. However, natural patterns appear far from periodic. So does the distance between and the size of the patches vary systematically and the fringes of the patches are ragged. Previously, we revisited tests for periodicity and demonstrated that the large majority of regular patterns found in nature are not periodic. We also introduced a method to quantify the regularity of patterns, and found that natural patterns are of intermediate regularity, in-between uncorrelated noise and periodic functions, which can best be described by stochastic processes, where the irregularity is intrinsic to the pattern and not due to added noise. Here, we corroborate our previous results with a comprehensive metastudy, where we analyze natural and computer-generated patterns found in the leading literature. Furthermore, we extend our theory for the formation of stochastic patterns with arbitrary regularity to two dimensions. We find that our theory captures well the spectral properties of both isotropic, i.e. spotted, labyrinthic and gapped, as well as of anisotropic, i.e. banded patterns.

Figure 1: a) Normalized spectral density averaged over the natural and model generated patterns found in the literature. The density of the natural patterns consists of a wide and low lobe, while the density of the model generated patterns consists of a narrow and high peak. b) Median regularity and interquartile range for the natural and model generated patterns. The modelled patterns are 3-5 times as regular as the natural patterns. Number of samples indicated next to the median.