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In a previous paper (2019 Eur. J. Phys. 40 035005) we showed how to design a discrete brachistochrone with an arbitrary number of segments. We have proved, numerically and graphically, that in the limit of a large number of segments, N ≫ 1, the discrete brachistochrone converges into the continuous brachistochrone, i.e. into a cycloid. Here we show this convergence analytically, in two different ways, based upon the results we obtained from investigating the characteristics of the discrete brachistochrone. We prove that at any arbitrary point, the sliding bead has the same velocity on both the continuous and discrete paths, and the radius of the curvature of both paths is the same at corresponding points. The proofs are based on the well-known fact that the curve of a cycloid is generated by a point attached to the circumference of a rolling wheel. We also show that the total acceleration magnitude of the bead along the cycloid is constant and equal to g, whereas the acceleration vector is directed toward the center of the wheel, and it rotates with a constant angular velocity. [ABSTRACT FROM AUTHOR]

So‐called fairy circles (FCs) comprise a spatially periodic gap pattern in arid grasslands of Namibia and north‐west Western Australia. This pattern has been explained with scale‐dependent ecohydrological feedbacks and the reaction‐diffusion, or Turing mechanism, used in process‐based models that are rooted in physics and pattern‐formation theory. However, a detailed ecological test of the validity of the modelled processes is still lacking.Here, we test in a spinifex‐grassland ecosystem of Western Australia the presence of spatial feedbacks at multiple scales. Drone‐based multispectral analysis and spatially explicit statistics were used to test if grass vitality within five 1‐ha plots depends on the pattern of FCs that are thought to be a critical extra source of water for the surrounding matrix vegetation. We then examined if high‐ and low‐vitality grasses show scale‐dependent feedbacks being indicative of facilitation or competition. Additionally, we assessed facilitation of grass plants for different successional stages after fire at fine scales in 1‐m2 quadrats. Finally, we placed soil moisture sensors under bare soil inside the FC gap and under plants at increasing distances from the FC to test if there is evidence for the 'infiltration feedback' as used in theoretical modelling.We found that high‐vitality grasses were systematically more strongly associated with FCs than low‐vitality grasses. High‐vitality grasses also had highly aggregated patterns at short scales being evidence of positive feedbacks while negative feedbacks occurred at larger scales. Within 1‐m2 quadrats, grass cover and mutual facilitation of plants was greater near the FC edge than further away in the matrix. Soil moisture after rainfall was lowest inside the FC with its weathered surface crust but highest under grass at the gap edge, and then declined towards the matrix, which confirms the infiltration feedback.Synthesis. The study shows that FCs are a critical extra source of water for the dryland vegetation, as predicted by theoretical modelling. The grasses act as 'ecosystem engineers' that modify their hostile, abiotic environment, leading to vegetation self‐organization. Overall, our ecological findings highlight the validity of the scale‐dependent feedbacks that are central to explain this emergent grassland pattern via the reaction‐diffusion or Turing‐instability mechanism. [ABSTRACT FROM AUTHOR]

Residential segregation in urban areas is an area of study for scientists and so a mathematical model is proposed that associates segregation with instabilities of spatially uniform mixed population states. The model predicts that uneven invasions of one population group into another might result from interface instabilities rather than from urban heterogeneities.

Abstract: Geodiversity has recently been attracting increasing attention as a measure of diversity for the physical components of natural environments. It has shown positive relations with biodiversity, as well as with several ecosystem services. Yet, so far, geodiversity studies have focused on relatively large spatial scales, ranging between hillslope, basin, and landform scales. It is proposed that either natural‐induced or anthropogenic‐induced, small‐scale (centimeter‐scale to few decimeter‐scale) geodiversity has a large impact on the hydrological connectivity and overall functioning of semi‐arid rangelands and other shrubby and woody drylands. It is further proposed that greater small‐scale geodiversity increases the on‐site retention of water and soil resources, decreasing the vulnerability of rangelands to prolonged droughts and climatic changes. Particularly, positive impact of moderate grazing intensity on rangelands functioning is demonstrated by the formation of livestock trampling routes, which transect hillslopes, increase ecosystem geodiversity, and modify the spatial redistribution of scarce water and soil resources at the patch scale. Numerical simulations of a mathematical model for vegetation patterns in water‐limited systems show that the trampling routes increase the survivability of vegetation patches under prolonged droughts. In practical terms, the concept of small‐scale geodiversity is relevant for the determination, monitoring, and assessment of land degradation, as well as for restoration projects of eroded lands and degraded ecosystems. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this article, the author discusses vegetation self-organisation and spatial patterns for the fairy circles of Namibia. The author highlights statistic of spatial pattern analysis, homogeneity and ecohydrological feedbacks. The author mentions vegetation interaction with abiotic aeolian or erosion effects.

In this article, we develop a unifying framework for the understanding of spatial vegetation patterns in heterogeneous landscapes. While much recent research has focused on self-organised vegetation the prevailing view is still that biological patchiness is mostly due to top-down control by the physical landscape template, disturbances or predators. We suggest that vegetation patchiness in real landscapes is controlled both by the physical template and by self-organisation simultaneously, and introduce a conceptual model for the relative roles of the two mechanisms. The model considers four factors that control whether vegetation patchiness is emerged or imposed: soil patch size, plant size, resource input and resource availability. The last three factors determine the plant-patch size, and the plant-to-soil patch size ratio determines the impact of self-organisation, which becomes important when this ratio is sufficiently small. A field study and numerical simulations of a mathematical model support the conceptual model and give further insight by providing examples of self-organised and template-controlled vegetation patterns co-occurring in the same landscape. We conclude that real landscapes are generally mixtures of template-induced and self-organised patchiness. Patchiness variability increases due to source-sink resource relations, and decreases for species of larger patch sizes. [ABSTRACT FROM AUTHOR]

As a result of their inherent differences in stability, sand ripples and megaripples exhibit variations in terms of their wavelengths and grain-size distributions (unimodal for sand ripples and bimodal for megaripples). While sand ripples form almost straight lines, mega-ripples have greater sinuosity due to their transverse instability, a property that causes small megaripple undulations to grow with time. The origin of the instability is due to variations in megaripple height, variations that do not diminish over time, and due to the inverse dependence of ripple drift velocity on the height. Thus, the taller regions of ripples will move more slowly than the adjacent, shorter portions, an outcome that promotes further perturbation growth. We show an example based on field work of the transverse instability of megaripples. The instability growth rate depends on the difference between the heights of the different segments of the megaripple. In contrast to the underlying instability of megaripples, normal sand ripples are essentially stable and are not affected by transverse perturbations, instead reacting quickly to the wind, which tends to smooth ripple height irregularities. The transverse instability of megaripples derives from the composition of their crests, which comprise coarse particles that allow initial perturbations in ripple height to grow further. The results suggest a physical mechanism for the transverse instability of megaripples and new insight into the spatial patterns of sand ripples. [ABSTRACT FROM AUTHOR]

Abstract: Vegetation band patterns on hill slopes are studied using a mathematical model. The model applies to drylands, where the limiting resource is water, and takes into account positive feedback effects between biomass and water. Multiple band patterns coexisting in wide precipitation ranges are found. For given precipitation and slope conditions band patterns with higher wavenumbers are more biologically productive. High-wavenumber patterns, however, are less resilient to environmental changes. [Copyright &y& Elsevier]

• Dune bistability is characterized by a bimodal distribution of vegetation coverage, which peaks around 5% and 40% in the Mu Us dune field. • UAV technique and remote sensing indices derived from Landsat images can be used to detect dune bistability and monitor its changes. • Several lines of evidence confirm the occurrence and persistence of bistable dune systems under a wide range of semi-arid environments. Sand dunes could have bistable states based on the observation that bare dunes and vegetation-stabilized dunes can coexist in the same area under the same environmental conditions. So far, more quantitative evidence on such an interesting phenomenon has remained elusive. In this study, the vegetation coverage and spatial distribution of dunes in the semi-arid Mu Us dune field, north-central China, were investigated by different remote sensing indices derived from the Landsat images using the Google Earth Engine platform, and the results were verified using the aerial images. Frequency distribution of vegetation coverage in many sub-regions (5 km × 5 km) across the dune field clearly shows two dominant peaks, one around 5% and the other around 40%, representing active and stabilized dune states, respectively. The boundaries between the patches of these two states are sharp, and have hardly shifted during the last twenty years. Such coexistence of bistable states is mostly distributed across a precipitation gradient from 200 to 400 mm in the study area. The relative portion of active dunes in total is reduced with increasing precipitation, while the dominant peaks of vegetation coverage for two dune states remain largely unchanged. All these lines of evidence are in accordance with the theory of alternative stable states and model predictions. Because the reversal would be difficult once the dunes shifted into an undesired state from the standpoint of environmental management, detecting and monitoring these transitions, which are often abrupt, is important for better process-based understanding of the mechanisms involved and anticipating future transitions. [ABSTRACT FROM AUTHOR]

A letter to the editor is presented in response to the article "Fairy circles or ghosts of termitaria? Pavement termites as alternative causes of circular patterns in vegetation of desert Australia," by F. J. Walsh and colleagues in the 2016 issue.

The fairy circles of Namibia form a remarkable gap pattern in arid grassland along the Namib Desert. The origin of the fairy circles is subject to an ongoing debate. Solving the mystery of the fairy circles (FCs) requires the right timing in fieldwork after rainfall, as the newly appearing grasses complete their life cycle within only a few weeks. Here we followed the rains along the Namib between 2020 and 2022 and assessed the cause of the grass death within FCs at different time intervals after grass-triggering rainfall. To assess whether termite herbivory was the cause, we used grass excavations and observations on the roots and shoots. To test if edaphic differences may explain the grass death in FCs, we undertook infiltration measurements in 10 FC-hotspot regions. Finally, we used continuous soil-moisture measurements from the dry into the rainy seasons to examine how the newly emerging grasses affect the soil-water content in space and time. Generally, in study plots that received grass-triggering rainfall most recently, the roots of the dead grasses in FCs were in 100 % of the cases undamaged, root-shoot ratios were significantly greater, and the roots were as long or even longer as those of the surrounding matrix grasses outside of the FCs. This indicates that drought stress caused grasses in the FCs to invest resources into roots to reach the percolating water. The results also show that the cause of the grass death in fairy circles was not induced by termite herbivory. Also, we found no systematic differences in the rate of water infiltration between FCs and the matrix, hence the plant wilting cannot result from quicker percolation within FCs. However, the soil-moisture measurements indicate that the matrix grasses strongly depleted the upper soil water of the FCs after rainfall, which explains why most grasses cannot establish and quickly die in the FCs. The research shows that grass death in fairy circles occurs immediately after rainfall due to plant water stress but not due to termite activity. Our results conform with previous fieldwork, pattern analysis and theoretical modeling, suggesting that Namibia's fairy circles are a self-organized vegetation phenomenon induced by ecohydrological feedbacks. • Namibia's fairy circles (FCs) were investigated in 10 regions along the Namib Desert. • After grass-triggering rainfall, new grasses within FCs die instantly due to wilting. • Termite herbivory is not causing this grass death as the plants had undamaged roots. • Soil-moisture data indicate that the matrix grasses severely depleted the water in FCs. • Ecohydrological biomass-water feedbacks and plant self-organization likely cause the FCs. [ABSTRACT FROM AUTHOR]

Nebkhas, also known as nebkha dunes or coppice dunes, are a unique biogeomorphological aeolian landform that is common in arid and semi-arid regions. They are often regarded as a signal of regional desertification and could be potential dust sources due to their relatively large content of fine sediments, but they also serve as "fertile islands" that enhance biodiversity at a micro-habitat. How the interactions between plant growth and aeolian sand deposition affect the formation and development of nebkhas has not been well documented, though. In this study, the chronology and morphological features of nebkhas in the Mu Us dune field, north-central China were studied using radionuclide (137Cs) dating and unmanned aerial vehicle (UAV) photogrammetric survey. The results show that the nebkhas are mainly distributed around lake basins and dry river valleys with relatively flat topography. Most nebkhas have heights around 0.5–1.5 m, and according to a new integrated age-height model, they mostly formed in the last fifty years. Widespread nebkha growth over that time is interpreted as a response to both desiccation of lake beds and river valleys and a pronounced decline in strong wind frequency. The regular rounded shape of the nebkhas in the study area may reflect the growth form of prostrate, extensively branched Nitraria tangutorum , the dominant formative shrub, and response to the low-energy wind environment. A nebkha's height and horizontal length increase synchronously in the early stage of the development, while the increase in horizontal length and projected area accelerates when the dunes become larger. Feedbacks involving both aeolian sand transport and shrub growth can explain a decline in growth rate with increasing height and an apparent height limit of about 4 m. This study also implies that nebkhas are important biogeomorphological features demonstrating how biotic and abiotic interactions can shape landforms and influence ecological functioning in arid and semi-arid environments. • Nebkhas are a unique biogeomorphological landform. • Nebkhas can have rounded or elongated shape under different wind regimes. • Most nebkhas in the Mu Us dune field formed during the last fifty years. • The growth limit of nebkhas is controlled by plant-aeolian sand feedbacks. [ABSTRACT FROM AUTHOR]

Geodiversity effects on ecosystem productivity were investigated using remotely sensed data and a mathematical model, by introducing a spatially non-uniform surface-water infiltration. Two hillslope types, with different geodiversity levels, as determined by the soil depth and stoniness, were selected for this study. Low-geodiversity hillslopes ('homogenous hillslopes') were identified by their dense herbaceous vegetation cover, and lack of rock fragments cover. High-geodiversity hillslopes ('heterogeneous hillslopes') are characterized by a more sparse cover of herbaceous vegetation, and relatively high rock fragment cover (>20%). Calculating Soil Water Index (SWI) from Landsat-5 TM, Landsat-7 ETM+ (SLC-ON) and Landsat-8 satellite images for the years 1994–2017 shows that during most of these years, SWI was higher at the heterogeneous hillslopes. This was particularly prominent after 2009 when mass mortality of shrubs took place on the homogenous hillslopes but not in the heterogeneous ones. Numerical simulations show that after a prolonged drought, the vegetation in the homogenous system dried out, and the system did not recover even after the yearly rainfall amounts returned to normal. In the heterogeneous system, the vegetation patches survived the drought episode and then recovered. This demonstrates the crucial role played by hillslope geodiversity in determining the durability of vegetation under prolonged droughts in semiarid regions. • The study area includes two hillslope types that differ by their geodiversity. • Soil Water Index (SWI) was calculated from satellite images throughout 1994–2017. • During most of these years, SWI was higher at the heterogeneous hillslopes. • Geodiversity, expressed by stoniness, increases water availability for shrubs. • Numerical simulations of vegetation patches subjected to droughts support this view. [ABSTRACT FROM AUTHOR]

Drylands are pattern-forming systems showing self-organized vegetation patchiness, multiplicity of stable states and fronts separating domains of alternative stable states. Pattern dynamics, induced by droughts or disturbances, can result in desertification shifts from patterned vegetation to bare soil. Pattern formation theory suggests various scenarios for such dynamics: an abrupt global shift involving a fast collapse to bare soil, a gradual global shift involving the expansion and coalescence of bare-soil domains and an incipient shift to a hybrid state consisting of stationary bare-soil domains in an otherwise periodic pattern. Using models of dryland vegetation, we address the question of which of these scenarios can be realized. We found that the models can be split into two groups: models that exhibit multiplicity of periodic-pattern and bare-soil states, and models that exhibit, in addition, multiplicity of hybrid states. Furthermore, in all models, we could not identify parameter regimes in which bare-soil domains expand into vegetated domains. The significance of these findings is that, while models belonging to the first group can only exhibit abrupt shifts, models belonging to the second group can also exhibit gradual and incipient shifts. A discussion of open problems concludes the paper.

Vegetation patches in drylands are localized structures of biomass and water. We study these structures using a mathematical modeling approach that captures biomass-water feedbacks. Biomass-water structures are found to differ in their spatial forms and ecological functions, depending on species type, soil conditions, precipitation range, and other environmental factors. Asymptotic spot structures can destabilize to form ring structures, expanding in the radial direction, or crescent structures, migrating uphill. Stable spot structures can differ in their soil-water distributions, forming water-enriched patches or water-deprived patches. The various biomass-water structures are expected to function differently in the context of a plant community, forming landscapes of varying species diversity.