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    <pageNumber>13</pageNumber>
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    <title language="eng">A scale-invariant method for quantifying the regularity of environmental spatial patterns</title>
    <abstract language="eng">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.</abstract>
    <parentTitle language="eng">Ecological Complexity</parentTitle>
    <identifier type="doi">10.1016/j.ecocom.2024.101104</identifier>
    <identifier type="issn">1476-945X</identifier>
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      <firstName>Christoph</firstName>
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    <title language="eng">Unravelling the spatial structure of regular dryland vegetation patterns</title>
    <abstract language="eng">Many resource-limited ecosystems exhibit spatial patterns where patches of biomass alternate with bare ground. Patterns can enhance ecosystem functioning and resilience, depending on their spatial structure. Particularly conspicuous are regular patterns, where patches are of similar size and spaced in similar intervals. The spatial structure of regular patterns is often described to be periodic. This has been corroborated by statistical testing of natural patterns and generation of periodic patterns with deterministic reaction–diffusion models. Yet, natural regular patterns appear conspicuously erratic compared to periodic patterns. So far, this has been attributed to perturbations by noise, varying patch size and spacing. First, we illustrate by means of an example that the spatial structure of regular vegetation patterns cannot be reproduced by perturbing periodic patterns. We then compile a large dataset of regular dryland patterns and find that their spatial structure systematically differs from periodic patterns. We further reveal that previous studies testing for periodicity overlook two aspects which dramatically inflate the number of false positives and result in the misclassification of patterns as periodic. We amend the test procedure by accounting for both aspects, finding that regular natural patterns have no significant periodic components. Lastly, we demonstrate that stochastic processes can generate regular patterns with similar visual appearance, spatial structure and frequency spectra as natural regular patterns. We conclude that new methods are required for quantifying the regularity of spatial patterns beyond a binary classification and to further investigate the difference between natural and model generated patterns.</abstract>
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    <title language="eng">A 3D-spatial approach for modeling soil hydraulic property distributions on the artificial Huehnerwasser catchment</title>
    <abstract language="eng">Knowledge of catchment 3D spatial heterogeneity is crucial for the assessment and modeling of eco-hydrological processes. Especially during the initial development phase of a hydro-geo-system, the primary structural properties have the potential to determine further development pathways. Small-scale heterogeneity (cm to m scale) may have significant effects on processes on larger spatial scales, but is difficult to measure and quantify. The Hühnerwasser (Chicken Creek) catchment offers the unique opportunity to study early ecosystem development within an initial structural setup that is well-known, from the plot up to the catchment scale.&#13;
Based on information on the open-cast mining technology, catchment boundaries and sediment properties, we developed a structure generator program for the process-based modeling of specific dumping structures and sediment property distributions on the catchment. The structure generator reproduces the trajectories of spoil ridges and can be conditioned to reproduce actual sediment distributions according to remote sensing and soil sampling data. Alternatively, sediment distribution scenarios can be generated based on geological data from the excavation site, or can be distributed stochastically. Using pedotransfer functions, the effective hydraulic van-Genuchten parameters are then calculated from sediment texture and bulk density. The main application of the 3D catchment model is to provide detailed 3D-distributed flow domain information for hydrological flow modeling. Observation data are available from catchment monitoring are available for determining the boundary conditions (e.g., precipitation), and the calibration / validation of the model (catchment discharge, ground water). The analysis of multiple sediment distribution scenarios allows to evaluate the effect of initial conditions on hydrological behavior development. Generally, the modeling approach can be used to pinpoint the influx of specific soil structural features on ecohydrological processes across spatial scales.</abstract>
    <parentTitle language="eng">Geophysical Research Abstracts</parentTitle>
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    <title language="eng">Formation of spatial vegetation patterns in heterogeneous environments</title>
    <abstract language="eng">Functioning of many resource-limited ecosystems is facilitated through spatial patterns. Patterns can indicate ecosystems productivity and resilience, but the interpretation of a pattern requires good understanding of its structure and underlying biophysical processes. Regular patterns are understood to form autogenously through self-organization, for which exogenous heterogeneities are negligible. This has been corroborated by reaction-diffusion models which generate highly regular patterns in idealized homogeneous environments. However, such model-generated patterns are considerably more regular than natural patterns, which indicates that the concept of autogenous pattern formation is incomplete. Models can generate patterns which appear more natural when they incorporate exogenous random spatial heterogeneities (noise), such as microtopography or spatially varying soil properties. However, the mechanism through which noise influences the pattern formation has not been explained so far. Recalling that irregular patterns can form through stochastic processes, we propose that regular patterns can form through stochastic processes as well, where spatial noise is filtered through scale-dependent biophysical feedbacks. First, we demonstrate that the pattern formation in nonlinear reaction-diffusion models is highly sensitive to noise. We then propose simple stochastic processes which can explain why and how random exogenous heterogeneity influences the formation of regular and irregular patterns. Finally, we derive linear filters which reproduce the spatial structure and visual appearance of natural patterns well. Our work contributes to a more holistic understanding of spatial pattern formation in self-organizing ecosystems.</abstract>
    <parentTitle language="eng">PLOS One</parentTitle>
    <identifier type="doi">10.1371/journal.pone.0324181</identifier>
    <identifier type="issn">1932-6203</identifier>
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Regular patterns are understood to form autogenously through self-organization, for which exogenous heterogeneities are negligible. This has been corroborated by reaction-diffusion models which generate highly regular patterns in idealized homogeneous environments. However, such model-generated patterns are considerably more regular than natural patterns, which indicates that the concept of autogenous pattern formation is incomplete. Models can generate patterns which appear more natural when they incorporate exogenous random spatial heterogeneities (noise), such as microtopography or spatially varying soil properties. However, the mechanism through which noise influences the pattern formation has not been explained so far. Recalling that irregular patterns can form through stochastic processes, we propose that regular patterns can form through stochastic processes as well, where spatial noise is filtered through scale-dependent biophysical feedbacks. First, we demonstrate that the pattern formation in nonlinear reaction-diffusion models is highly sensitive to noise. We then propose simple stochastic processes which can explain why and how random exogenous heterogeneity influences the formation of regular and irregular patterns. Finally, we derive linear filters which reproduce the spatial structure and visual appearance of natural patterns well. 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    <title language="eng">Topography and rainfall variability shaping dryland vegetation self-organisation : insights from a numerical modelling study</title>
    <abstract language="eng">The coevolution of hydrological and vegetation dynamics in semi-arid regions often leads to vegetation self-organisation (VSO). While numerous hypotheses on the ecohydrological processes driving VSO have been explored through mathematical models, these have struggled to capture the multiscale complexity emerging from short-term surface runoff over heterogeneous topographies under variable rainfall. This limitation hinders understanding of how natural topography and rainfall variability shape long-term vegetation patterns. &#13;
Previous studies suggest that intra-storm water redistribution at the hillslope scale – controlled by topography and storm intensity – plays a key role in VSO. However, these factors have rarely been considered together due to methodological constraints in numerical solvers. We argue that accurately representing these processes is essential to investigate their interactions. &#13;
This study systematically examines the effects of hillslope topography and intra-annual rainfall distributions on vegetation band formation using a physically based model that couples the Zero-Inertia (Diffusive Wave) approximation of the shallow water equations with the HilleRisLambers–Rietkerk vegetation model. Idealised 30-year simulations were conducted at second-scale hydrodynamic resolution across different hillslope forms (plane, convex, concave), slopes, and rainfall regimes along a semi-arid gradient. &#13;
Results show that both topography and rainfall variability strongly influence band formation through their control on water redistribution and hydrological balance. Steeper slopes enhance runoff over infiltration, reducing water availability and altering band geometry and migration. Concave hillslopes exhibit distinct runoff convergence and redistribution patterns compared to plane or convex slopes. Rainfall intermittency interacts with topography to further affect pattern stability and morphology. While both drivers shape pattern characteristics differently, their joint effects mainly influence band migration without providing a strong stabilising mechanism. &#13;
These results demonstrate the feasibility of long-term, physically based ecohydrological simulations, paving the way for more comprehensive models including sediment transport and geomorphic feedbacks.</abstract>
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      <firstName>Yolanda</firstName>
      <lastName>Pueyo</lastName>
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      <firstName>Christoph</firstName>
      <lastName>Hinz</lastName>
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    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Banded vegetation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Hillslope shape</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Vegetation self-organisation</value>
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      <language>eng</language>
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      <value>Ecohydrology</value>
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      <language>eng</language>
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      <value>Water-limited ecosystem</value>
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