Bridging ecology and physics: Australian fairy circles regenerate following model assumptions on ecohydrological feedbacks

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.

. It is obvious that such exceptionally ordered patterns can only result from strong interactive processes, because in the absence of interaction, the probability of creating order in a noisy environment is very unlikely (Saha & Galic, 2018).
Fairy circles (FCs) are subject to an ongoing controversy about their origin (Sahagian, 2017). More recent theories have focused on vegetation self-organization and argued that FCs are an emergent vegetation phenomenon that reflects a population-level response to aridity stress where ecohydrological biomass-water feedbacks lead to strictly geometric patterns (Cramer & Barger, 2013;Cramer, Barger, & Tschinkel, 2017;Getzin et al., 2015a,b;Getzin, Yizhaq, Cramer, et al., 2019;Ravi, Wang, Kaseke, Buynevich, & Marais, 2017;Zelnik, Meron, & Bel, 2015). The periodically ordered pattern is thus an expression that there is not enough water to sustain uniform vegetation coverage at the landscape scale and the distinct, so-called 'wavelength' of the gap pattern reflects the spatial scale at which water is most limiting to the plants (Meron, 2016).
One point of criticism, however, is that the support of this hypothesis was mainly rooted in physics and mathematical modelling of pattern formation while there are currently too little field-based studies to verify the validity of such models (Vlieghe & Picker, 2019). Indeed, there is a strong imbalance between the theoretical vegetation models, their a priori assumptions and the scarcity of empirical proof that the modelled processes are correct from an ecological point of view. This imbalance can be attributed to the fact that the recognition of periodically ordered vegetation in drylands has primarily drawn the interest of physicists to replicate the observed patterns via partial-differential equation modelling (Klausmeier, 1999;Lefever & Lejeune, 1997;Rietkerk et al., 2002;von Hardenberg, Meron, Shachak, & Zarmi, 2001;Yizhaq & Bel, 2016). Very few such studies in arid environments, however, have empirically linked measured processes and ecological properties in the field with the theoretical modelling of the assumed ecohydrological feedbacks and resultant vegetation self-organization. crust but highest under grass at the gap edge, and then declined towards the matrix, which confirms the infiltration feedback.
With regard to the FCs, the major empirical support for the vegetation models currently comes from analyses of higher-level components and so-called coarse-graining, hence from aggregating finer-scale processes or patterns to larger-scale summaries (Newman, Kennedy, Falk, & McKenzie, 2019). For example, the modelled FC patterns have been statistically matched with FC snap-shot patterns in aerial imagery (Getzin et al., 2015a(Getzin et al., , 2016 or the rainfall-dependent appearance and disappearance of FCs, as observed in satellite images, has been well replicated with dynamic modelling (Zelnik et al., 2015). In context with pattern-oriented modelling (Grimm et al., 2005;Wiegand, Saltz, Ward, & Levin, 2008), these vegetation models that are based on theoretically assumed processes of plant competition for soil water have, in principle, the ability to replicate the coarse-grained FC spatial structure. Similar successful matching of modelled and real-world patterns has also been demonstrated for completely different systems such as the Everglades in Florida (Acharya, Kaplan, Casey, Cohen, & Jawitz, 2015) or Mediterranean seagrass meadows (Ruiz-Reynés et al., 2017).
This successful matching, however, is not a verification that the modelled processes and mechanisms of plant self-organization are indeed right and happening in the real world to cause the observed patterns (Borgogno et al., 2009). Within a framework of pattern-process inference, these coarse-graining methods are mainly useful to identify unlikely processes for the cause of the pattern and to narrow down the most plausible working hypotheses (Getzin, Yizhaq, Cramer, et al., 2019;McIntire & Fajardo, 2009;Schurr, Bossdorf, Milton, & Schumacher, 2004). Consequently, if the extensive achievements of the mathematical process-based models are to find greater acceptance in ecology, then it is necessary not only to reconstruct the higher-level structures but also to demonstrate true evidence of the validity of the modelled lower-level processes assumed a priori. This can be achieved with more fine-grained and detailed ecological analyses in the field, including experimental testing (Tschinkel, 2015), plant-and soil-related ecohydrological measurements (Cramer et al., 2017;Getzin et al., 2016;Ravi et al., 2017) and high-resolution vegetation mapping employing modern drones (Getzin, Wiegand, & Schoening, 2012;Getzin, Yizhaq, Munoz-Rojas, et al., 2019).
Process-based FC vegetation models use nonlinear partialdifferential equations to reconstruct the periodic FC ordering via so-called Turing-like instabilities, also called Turing mechanisms, and principles of pattern-formation theory (Borgogno et al., 2009;Meron, 2012Meron, , 2015Turing, 1952). This Turing mechanism is also known as reaction-diffusion mechanism or activator-inhibitor principle, and a key to forming spatially periodic Turing patterns is the presence of positive and negative feedback interaction at different spatial scales (Figure 1a; Rietkerk & van de Koppel, 2008). In context with modelling self-organized vegetation patterns in drylands, ecohydrological feedback loops with positive short-range plant interaction (facilitation) but negative long-range inhibition (competition) are central to explain the periodically ordered FCs ( Figure S1; Getzin et al., 2016;Meron, 2018;Tlidi, Lefever, & Vladimirov, 2008). Such models generally predict two uniform stable states-uniform vegetation under high precipitation and baresoil desert at lowest rainfall. Between these two stable states, periodically ordered states such as FCs, vegetation spots, stripes and labyrinths can form and the various proportions of bare soil are an expression that there is not enough water to sustain a continuous or uniform vegetation layer (Figure 1b-d;Meron, 2018).
Fundamental to these processes is that various modes of water transport are capable of inducing pattern-forming feedbacks in water-limited vegetation (Meron, 2012(Meron, , 2016 so that resource concentration leads to self-organized patchiness and a local aggregation of biomass (Rietkerk, Dekker, de Ruiter, & van de Koppel, 2004).
For Namibian FCs, which form primarily on deep aeolian sands, the mode of soil-water diffusion towards spatially confined grass roots has been identified as the dominant feedback mechanism to model the FCs (Zelnik et al., 2015). For Australian FCs, which form as hard weathered clay crusts, overland-water flow and an 'infiltration feedback' have been successfully used to model the process of local water uptake by the spinifex grasses of the genus Triodia (Figure 1e, Figure S1; Getzin et al., 2016). Here, the a priori assumption is that the roots of the Triodia plants induce an infiltration contrast and thereby trigger the positive short-range feedback loop where more vital and larger plants gain more water and thus have a disproportionally greater benefit than weaker neighbouring plants ( Figure 1f).
Irrespective of the differences between these proposed feedback mechanisms in Namibia and Australia, it is assumed that the emergent FCs contribute a critical extra source of water for the surrounding matrix grasses that grow between the FCs and that the most vital grasses should therefore grow around the FC perimeters. By forming periodic vegetation-gap patterns, the grasses benefit from the additional water resource provided by the FCs, and thereby keep the ecosystem functional at precipitation values lower than those required for uniform vegetation (Meron, 2018). In this way, the grasses act as 'ecosystem engineers' because their cooperative action modifies the abiotic environment, they redistribute resources and they facilitate the growth of their own species in their canopy neighbourhood (Borgogno et al., 2009;Gilad, von Hardenberg, Provenzale, Shachak, & Meron, 2004).
Despite the intriguing periodic vegetation patterns in global drylands, there are still only very few studies where field-based data have been directly linked to the theoretical modelling of reaction-diffusion processes and the a priori assumed feedback mechanisms Dunkerley, 2018;Yizhaq et al., 2019). This current lack of bridging between ecology and physics is no surprise given that most of the papers dealing with such periodic dryland vegetation are based on modelling, but there is a gap between the relative mathematical simplicity of the models and the complexity of the real world (Kondo & Miura, 2010). Moreover, while the reaction-diffusion mechanism is now widely accepted for Turing patterns at the subcellular or cellular levels (e.g. pigmentation in zebrafish), for the organismal level such as dryland vegetation, empirical evidence is still rare (Saha & Galic, 2018).
So far, there is mostly no conclusive experimental evidence suggesting that the organized spatial configurations of vegetation observed in nature do emerge from Turing-like dynamics (Borgogno et al., 2009).
Besides the aforementioned field studies, empirical support for the validity of Turing dynamics does partially also exist with regard to fairy circles. For example, in Namibia it has been demonstrated that the deep aeolian sands form a hydraulically connected edaphic environment where water is highly mobile and may laterally flow more than seven metres within a short time (Cramer et al., 2017).
These scales of horizontal water movement equal biomass gradients where the strongest and most vital grasses can be found around the FCs while less vital grasses grow metres away from the FCs. The high mobility of water in deep aeolian sand therefore justifies using fast soil-water diffusion as the specific feedback mechanism to model the Namibian FCs (Zelnik et al., 2015).
For the Australian FCs, fieldwork has demonstrated that water infiltration in unvegetated soil inside the FCs is significantly lower than in soils between the surrounding matrix vegetation. Preliminary experimental testing has also shown that despite the flatness of the landscape, excess surface water moves from the FCs towards the matrix due to strongly reduced infiltration within the FC, which resulted in using the infiltration feedback as the specific mechanism to model the Australian FCs (Figure 1e; Getzin et al., 2016). Based on more than 150 soil excavations it has also been demonstrated that these FCs have no causal link to termite activity but that their high clay contents and soil compaction results from mechanical abiotic F I G U R E 1 Modelling the fairy circles of Australia based on principles of pattern-formation theory. A short-distance positive feedback and a long-distance negative feedback is central to generate a Turing pattern (a, after Rietkerk & van de Koppel, 2008). A bifurcation diagram (b), used for modelling the Australian FCs and visual agreement of modelled (c) and real-world (d) patterns of Triodia plants. The black and red solid lines in the diagram (b) show stable bare soil and uniform vegetation, respectively, and the blue and green lines show two examples of periodic vegetation patterns that the system can take on (details in Getzin et al., 2016). The model is based on a specific pattern-forming feedback, which is overland-water flow due to infiltration contrast (e). This modelled mechanism leads to lowest soil moisture inside the FC but highest soil moisture and the most vigorous plant growth at the FC edge, which then declines towards the matrix (f, after Palmer, 2016; see also Figure S1) weathering in this climatically harsh environment (Getzin, Yizhaq, Munoz-Rojas, et al., 2019). These findings are so far in support of the modelled infiltration contrast, but they do not yet constitute a detailed ecological test of the validity of the model assumptions and associated feedback mechanisms. Indeed, identification and verification of the specific dynamics of the reaction-diffusion system is critical to showing the applicability of the Turing mechanism to the formation of a given pattern (Kondo & Miura, 2010

| Drone survey
Between the 7th and 25th of July 2017, we undertook a drone (unmanned aerial vehicle, UAV) survey where we mapped five FC plots with a Microdrone md4-1000 quadcopter (Figure 2a

| Preparation of drone data
OneButton software (www.icaros.us) was used to stitch the RGB photos and the multispectral images into geo-referenced orthophotos. The RGB photos were directly fed into the software. The with 10,000 cells per hectare and grid cells of 1 m × 1 m resolution ( Figure S2c), using QGIS-2.18 software (www.qgis.org). Finally, three NDVI classes for the 1 m 2 grid cells with x,y-coordinates were used for spatial statistical analyses: bare-soil cells (−0.37 to ≤−0.26), low-vitality grasses (−0.15 to −0.11) and high-vitality grasses (−0.10 to 0.10).
We then identified for the five 100 m × 100 m plots all FCs with a minimum threshold diameter of 2 m, using manual segmentation techniques as applied previously (Getzin et al., 2016;Getzin, Yizhaq, Munoz-Rojas, et al., 2019). Thus, for each FC we created with QGIS software one shapefile with geo-referenced information on the circle's x,y-coordinate and diameter. Smaller FCs were not considered because it is difficult to identify them as genuine FCs. The mean FC diameter in Australia is 4 m (Figure 2c), and it is these large FCs that are considered to function as an extra source of water for the surrounding vegetation.

| Spatial association of grass vitality with FCs and with all bare-soil gaps
The FCs are thought to create a hydraulically connected landscape and thereby affect the vitality of the grasses not only on a local scale but also as a large-scale landscape pattern of water availability (Cramer et al., 2017;Getzin et al., 2016;Tschinkel, 2015). We therefore used the Berman test (Berman, 1986) Figure S3).
In the Berman (1986)

| Spatial patterns assessed with univariate random labelling
Finally, we were interested if the spatial patterns of high-vitality grasses differ from the patterns of low-vitality grasses and if these patterns show aggregated (clustered) or segregated (regular) distributions that correspond to the modelled positive and negative feedbacks at small and larger scales (Figure 1a). To test this formally, we applied the pair-correlation function (or short: g-function) and the null model of univariate random labelling (Wiegand & Moloney, 2004) once to the high-vitality and once to the low-vitality grasses. The g(r) is a neighbourhood-density function that describes clumping and regularity at a given radius r, using a standardized density. It is the expected density of points at a given distance r of an arbitrary point, divided by the intensity λ of the pattern (Stoyan & Stoyan, 1994). Under complete spatial randomness, CSR, g(r) = 1, aggregation is indicated by g(r) > 1, while regularity has values of g(r) < 1. The function can be extended to describe point patterns with two types of points (e.g. low-vitality grasses = pattern 1, highvitality grasses = pattern 2): the bivariate pair-correlation function g 12 (r) is the expected density of points of pattern 2 at distance r of an arbitrary point of pattern 1, divided by the intensity λ 2 of pattern 2.
Under random thinning the g-functions are invariant, hence under the null model of random labelling g 12 (r) = g 21 (r) = g 11 (r) = g 22 (r). We used univariate random labelling as null model and the function g 11 (r) to explore if the low-vitality grasses (n 1 ) are a random subset of the overall grass pattern, i.e. the combined pattern of low-vitality (n 1 ) and high-vitality (n 2 ) grasses. The same was done for the highvitality grasses (n 2 ), using the function g 22 (r). The test was applied by computing the function g 11 (r) from the observed data, then randomly re-sampling sets of low-vitality grasses (or high-vitality grasses using g 22 (r), respectively) from the joined pattern of low-vitality and highvitality grasses to generate the simulation envelopes. If the functions g 11 (r) or g 22 (r) show significant positive or negative deviations from the null model, this indicates that the low-or high-vitality grasses, respectively, have an additional clustered or regular spatial structure, conditional on the overall distribution of all grasses within the study plot.
Univariate random labelling is an ideal null model to explore this question because it also accounts for first-order heterogeneity, i.e. non-relevant 1 m 2 grid cells with land-cover classes of bare soil or of trees and shrubs do not affect the spatial outcome of the analysis. Therefore, the null model envelopes reflect the first-order properties and, unlike for a CSR null model, are not parallel to a value of g(r) = 1. The spatial correlation functions g 11 (r) and g 22 (r) were tested for significant deviations from the null model using the fifth-lowest and fifth-highest values of 199 Monte Carlo simulations for constructing approximately 95% simulation envelopes (Baddeley et al., 2014). All spatial analyses were done in R-software using the packages 'spatstat' (Baddeley & Turner, 2005) and 'landscape metrics' (Hesselbarth et al., 2019), as well as in the software Programita (Wiegand & Moloney, 2004).

| Quadrat-based mapping of post-fire succession
On 28th  In each of the selected plots, 10 FCs that were connected along a transect were selected and the quadrat was placed at the north, west, south and east side of the peripheral vegetation, as well as at four neighbouring locations in the matrix vegetation, which were between 2 and 3 m away from the FC gap edge ( Figure S4).
The quadrat was subdivided into 10 cm × 10 cm subunits. For each quadrat, the number of individual, physically separated, grass plants per square metre was recorded (count), as well as their total crown cover within the square, which allowed to assess the cover precisely to 1%. The formula cover/count × 100 will

| Drone-based spatial analyses
The number of digitized FCs in the five 1-ha plots ranged between 81 and 105 (Table 1). Their diameters ranged between 2.0 and 5.9 m, with an overall mean diameter of 3.8 m. Of all the 10,000 cells per hectare, the number of cells representing bare soil ranged between 1,122 and 2,053 or 11.2% and 20.5% respectively. Generally, there were more low-vitality than high-vitality grasses in the plots, except Low-vitality grasses always had lower Z 1 scores and thus a lower spatial association with the 81-105 FCs than high-vitality grasses (Table 1; Figure 3a). The density of high-vitality grasses was therefore more positively affected by the distribution of FCs than the density of low-vitality grasses. These results were robust because they were the same when we used in an additional analysis not the median but the mean NDVI value (−0.10) of all five plots to differentiate between low-and high-vitality grasses ( Figure S5a).  Table 1. High-vitality grasses had consistently higher Z 1 scores than low-vitality grasses, thus demonstrating a stronger association with the FCs. An opposite relationship is shown for results with the adjacency matrix (b), which considered all small gaps up to a minimum size of 1 m 2 , revealing that the proportion of lowvitality grasses bordering a bare-soil gap was higher than for the high-vitality grasses matrix, the percentage of high-vitality grasses bordering a bare-soil cell was consistently lower than for the low-vitality grasses (  In contrast, all low-vitality grasses did not show positive deviations from the null model. Here, mostly regularity was detected and this regularity was just marginally significant at the smallest scales of r ≤ 2 m (Figure 4a,b,e, inset). In the youngest plot FC-C2, low-vitality grasses showed even random distributions at the smallest scales.  Figure S6).

| Quadrat-based analyses of post-fire succession
Grass cover in the 1 m 2 sample quadrats was significantly smaller in the matrix locations than at the FC edge in all three age groups F I G U R E 4 Spatial patterns of grasses in the five 1-ha plots assessed with the null model of univariate random labelling, once shown for the high-vitality grasses, denoted g 22 , and once for the low-vitality grasses, g 11 (inset figure). The pattern is regular or aggregated at circular neighbourhood distances if the red line of the g-function is either below the lower or above the upper grey lines of the simulation envelopes respectively. Envelopes were constructed using the 5th lowest and 5th highest value of 199 Monte Carlo simulations of the null model of 3.5, 4.5 and >15 years after fire. At the FC edge, grass cover increased significantly with post-fire age (  Figure 5b). The best measure to describe the merging of small individual grasses into self-organized patches (i.e. large hummocks) is cover/count × 100 because it would give a value of 100 for complete coverage with one plant. This measure was significantly smaller in the matrix locations than at the FC edge in all age groups (Table 2). Overall, it also increased with post-fire age (Figure 5c).

| Weather station data
The highest recorded air temperature between 4 August 2017 and 6 April 2018 was 48.2°C on 6 January 2018. Seven more days occurred TA B L E 2 Details of the quadrat-based measurements, shown in Figure 5. The location effects within plots, edge versus matrix, were examined with one-tailed t tests to analyse if cover and cover/count were significantly greater at the edge or if count was significantly smaller at the edge. The age effects among the three plots were first assessed with analysis of variance (ANOVA), and then with post-hoc Tukey tests to analyse if cover, count or cover/count differ significantly for different post-fire ages.

F I G U R E 5
Results from the quadrat-based fieldwork for the three plots with different post-fire age. The plots FC-F1, FC-C2 and FC-F3 burnt 3.5, 4.5 and >15 years ago respectively. Cover increased with post-fire age and was highest at the FC edge (a). Similarly, the number of individual plants declined (b). The statistic count/cover shows that grasses were increasingly merging together with increased post-fire age and at the FC edge (c)

F I G U R E 6
Measured soil temperature at 2 cm depth under alive grass at the FC gap edge and in bare soil of the gap interior, and corresponding air temperature (a). Volumetric water content at the six soil moisture sensors within and around the FCs (b). The lines of the legend are sorted according to the positions of the sensors, starting from the FC gap interior outwards to bare soil in the matrix. The black arrow indicates that soil water of only the sensor in the FC gap interior (red line) showed decreased soil water in mid-January, which can be attributed to physical weathering and the re-establishment of the soil crust during the rainfall events in the months before. Note that soil moisture values had different absolute values during non-rainfall times due to differences in the initial calibration

| D ISCUSS I ON
With this study, we are quantitatively testing whether the theoretically assumed, scale-dependent feedback mechanisms and modes of water transport are valid for modelling the emergent patterns of Australian FCs. Generally, it is not enough to show that the modelled high-order components, such as the spatially periodic FCs patterns, are statistically well matched by the real-world patterns observed in aerial imagery. This is because several modes of water transport and different feedback mechanisms of plant self-organization can lead to periodic vegetation patterns (Meron, 2015(Meron, , 2016Rietkerk & van de Koppel, 2008). If the process-based partial-differential equation models that are rooted in physics and in pattern-formation theory shall find greater acceptance in ecology, then it is necessary to verify beyond these coarse-graining properties that the a priori assumed

| Landscape-scale feedbacks of grasses and their association with FCs-Hypothesis 1
According to pattern-formation theory, the periodically ordered FCs are not a random event but are inherently linked to the demography of the grass population. In Triodia grasslands with climax vegetation, healthy grasses of high vitality should therefore show a stronger spatial association with the FCs than low-vitality grasses since they should benefit from surface water flow from the gap interior to the edges. We examined this hypothesis with a drone survey during the driest month of the year with zero rainfall in July and June, and only 11 mm in May 2017 (data F I G U R E 7 Drone image of the weather station and the positions of the soil moisture sensors (a). Sum of the peak soil-water content 2, 12, and 24 hr after rainfall, accumulated over 15 rainfall events (b). Soil-water content was highest under the alive grass plant at the gap edge but lowest in the FC gap interior (a) (b) for Newman Airport, Australian Government Bureau of Meteorology, 2018). As a consequence and based on the NDVI, we found generally more low-vitality than high-vitality grasses in the plots. Only in the very young plot FC-C2, which burnt 2 years and 8 months prior to the drone survey in July 2017, we found an opposite relationship because the smaller plants were likely not strongly constrained by intraspecific competition and resultant mortality. Given that the other drone-mapped plots represent typical climax stages of Triodia grassland, it is not surprising that a considerable proportion of the 1-ha plots consisted of bare soil-litter transitions with variable amounts of dead lignified grasses. The perennial grass species T. basedowii grows permanently until neighbourhood competition, resource depletion or the central collapse of a hummock leads to plant death or ring formation. Hence, if fire is absent for a long time, considerable amounts of dead leaf material within hummocks accumulate (Grigg et al., 2008) and this increase of combustible material increases the probability of fire (Haydon, Friar, & Pianka, 2000). The research plots FC-L1, FC-L2 and FC-C5 that had not burnt for >15 years, were likely in a terminal phase of this successional stage because their entire grass layers burnt in April 2018 due to a wildfire.
In these plots with climax vegetation, but even in the young plot FC-C2, high-and low-vitality grasses responded systematically to the pattern of FCs. Based on the spatially explicit Berman test, we found that the high-vitality grasses of all five investigated plots were consistently more strongly associated with the surrounding FCs than the low-vitality grasses (Figure 3a). This is a strong indication that the grasses that grow closer to the FCs must have benefited from these large gaps, which have mean diameters of nearly 4 m in the study region, equalling an area of about 12 m 2 (Getzin et al., 2016). Since this result is based on a spatially explicit test, it reflects the positive response of all high-vitality grasses in the 1-ha study plot, no matter whether they grow near the periphery of the FC gap or at distances of some metres away from the FCs. These results were also robust when we used the mean NDVI value rather than the median as cutoff value to differentiate between low-and high-vitality grasses. Our analysis also shows that the Epanechnikov kernel is more suitable than a Gaussian kernel to account for the relationship between FCs and grass vitality because its parabolic shape depicts more clearly the direct neighbourhood effect of FCs on the surrounding vegetation half way to the next FC. Expectedly, kernel radii larger than 5 m include too much noise effects from other nearby FCs and therefore result in inconsistent and non-significant results ( Figure S5).
In contrast, if the adjacency matrix is assessed, which considers all FCs and all bare-soil gaps in the area down to a minimum size of 1 m 2 , then an opposite relation can be found. In this case, the percentage of low-vitality grasses bordering a bare-soil gap was consistently higher than for the high-vitality grasses. This indicates that not every small gap is necessarily important for the demography of the Triodia grasses. In other words, the potential positive effect of less neighbourhood competition from growing randomly next to a small bare-soil gap could be functionally unimportant, relative to the overall constraint that the demographics of this highly water-limited grassland system is governed by the wavelength of the FCs at the landscape-scale and the geometric rules of pattern-formation theory Getzin et al., 2016;Meron et al., 2004;Rietkerk & van de Koppel, 2008). Also, small gaps which occur in the matrix are likely those places which receive less water from the FCs. Consequently, continuous plant cover cannot be sustained in the matrix and small gaps form.
The FCs are distributed in a spatially periodic pattern and, as predicted by pattern-formation theory, form a grid-like hexagonal array (Getzin, Yizhaq, Cramer, et al., 2019). Our empirical results thus support the theoretical model assumptions based on the reactiondiffusion principle and scale-dependent feedbacks where the hardened FC gaps function as an important additional source of water for the surrounding vegetation matrix (Getzin et al., 2016;Meron, 2018).

| Scale-dependent feedbacks, facilitation and competition of grasses-Hypothesis 2
Grasses of the matrix in-between FCs have already been investigated in previous studies (Cramer et al., 2017;Ravi et al., 2017;Tarnita et al., 2017), but the spatially explicit analysis of their vitality status is a novel approach. The Australian outback is an ideal ecosystem to study scale-dependent feedbacks of grasses because T. basedowii is a long-lived perennial that grows in size for years and decades until fire destroys the vegetation (Levin et al., 2012). High-vitality grasses which primarily exert positive short-range feedbacks should have clustered distributions and increasingly merge at the FC gap edge to increase their benefit from higher water availability. But low-vitality grasses which suffer from long-range negative feedbacks, should not have clustered but segregated or scattered random distributions because the lack of water does only support the survival of separate individuals. Our results support this model assumption based on univariate random labelling and fine-scale quadrat sampling.
High-vitality grasses were in all five plots significantly aggregated and the small-scale clustering at 1 m neighbourhood distance was particularly strong in the four long unburnt plots with g 22 values ranging between 1.65 and 1.86. This aggregation is indicative of a strong positive short-range feedback where the more vital grasses not only benefit from the nearby FC gaps but also from a clustered growth pattern and resultant self-organized patch formation due to facilitation (Rietkerk et al., 2004). The reason for the clustered pattern of vital grasses can be found, for example, in a reduction of temperature under grass canopy versus bare soil. In our study area, soil surface temperature under grass shade was up to 25°C lower than in unshaded bare soil (Figure 6a). Also, enhanced trapping of sediments or organic detritus may trigger the increase in local density of grasses and thereby the increase in water-trapping efficiency among the grasses (Dunkerley & Brown, 1999). In such densely vegetated areas, the protection against evapotranspiration and against soil-crust formation enhances surface water infiltration which, in turn, favours vegetation growth. Thereby, the grasses redistribute the water resources, they modulate the abiotic environment and thus function as 'ecosystem engineers' to better cope with the hostile habitat (Borgogno et al., 2009;Gilad et al., 2004). Generally, such facilitative patterns are well known to become increasingly important under increasing xeric conditions and in this case the positive effects of nurse shade outweigh competitive negative effects (Holmgren, Scheffer, & Huston, 1997 With the assessment of the g 11 -function and its low absolute deviations from the lower null-model envelopes (Figure 4), we found that low-vitality grasses showed a weaker feedback strength than high-vitality grasses with their strong positive deviations from the null model. has accounted for the growth pattern of these typical hummocks via the 1 m 2 grid resolution, we also mapped the post-fire grass recovery of young hummocks within 1 m 2 sample quadrats that were subdivided into 10 cm × 10 cm subunits.
Our ground mapping supports those drone-based findings because we found a spatio-temporal trend that plant cover bordering the FC edge was higher than for the matrix grasses at distances of 2-3 m away from the gap edge. This trend became more prominent with increasing post-fire age from 3.5 years towards >15 years (Figure 5a), indicating that the increase in plant cover around the FCs follows a demographic rule: the highest coverage forms directly around the FC edges, while off the FCs the plant cover is lower and the probability of gap formation between the hummocks increases. Corresponding to these findings, the number of individual grass hummocks declined in a similar way and the cover/count statistic shows that the cooperative merging of grass individuals increased towards the FC edge but also with post-fire age. These results provide additional evidence that the grasses are overall benefiting from the FCs that function as an important extra source of water and that the trend of facilitation via the merging of seed-regenerating grass individuals was positively associated with the decreasing distance to the FC gaps (Getzin et al., 2016;Meron, 2018). With their population-level response to aridity stress the Triodia grasses act as ecosystem engineers and the morphological plasticity in their growth patterns are adaptations to maximize water harvesting ( Figure S4g-j). Such morphological adaptations of grass plants are well known from a number of water-limited environments and include, for example, the formation of ring-like shapes (Sheffer, Yizhaq, Gilad, Shachak, & Meron, 2007;Yizhaq et al., 2019), spiral-like (Fernandez-Oto, Escaff, & Cisternas, 2019) or banded grass patterns (Dunkerley, 2018;Dunkerley & Brown, 1999) and high grass biomass of the perennial belt around FCs in Namibia (Cramer et al., 2017;Fernandez-Oto, Tlidi, Escaff, & Clerc, 2014).

| Spatio-temporal data recording of a weather station and the infiltration feedback-Hypothesis 3
Central to the process-based modelling of the FCs is a Turing-like instability that is triggered by a specific feedback mechanism and mode of water transport which depend on the prevailing soil type and dominant plant architecture. While in Namibia on deep aeolian sand it is the 'uptake-diffusion feedback' that is assumed to induce the instability, in Australia on hardened clay crusts and associated overland-water flow it is the 'infiltration feedback' (Getzin et al., 2016;Zelnik et al., 2015). Consequently, the pivotal process for modelling the emergent pattern of the Australian FCs is enhanced soil-water infiltration under alive plants growing next to the hardened FCs gaps. The positive short-range feedback loop is thus triggered by the growth of roots that loosen the soil crusts and increase vegetation growth via locally taking up more water. Here we provide empirical evidence that this assumed mechanism of water transport is indeed existing in the studied grassland system. In this regard, the results of our installed data loggers show two important findings: First, soil water only at the sensor in the FC gap without protective plant cover strongly decreased over time during the very hot January, after several major rainfall events happened during the months before ( Figure 6b). This demonstrates that the soil crusts in the FCs originate from particle dispersion during heavy rainfall events and subsequent mechanical weathering of the soil surface under the extremely hot temperatures, which ultimately prevent plant establishment (Getzin et al., 2016). With more than 150 excavations in the Australian FCs, it has been previously shown that pavement termitaria are not responsible for the bare soil in FCs, but it is the high clay content and associated high soil compaction, induced by mechanical weathering, that is causing the absence of vegetation in FCs and similarly in nearby large bare-soil areas (Getzin, Yizhaq, Munoz-Rojas, et al., 2019), just like in many more Australian drylands (Dunkerley, 2002).
Second, the modelled infiltration feedback is supported by the observation that the peak soil-water content 2, 12 and 24 hr after rainfall was highest under the alive plant at the gap edge but lowest in the gap interior. Soil-water then decreased with larger scales from the alive plant at the FC edge towards the matrix locations about 2-2.5 m distance away. This shows further that the FC gaps function as an additional source of water for the matrix grasses and that it is the plants growing close to the FCs that are benefiting the most from this water source. These findings are in line with previous infiltration measures and an irrigation experiment within a FC (Getzin et al., 2016) and the field results support the assumed in-phase spatial profiles of biomass and soil water of the model ( Figure S1). The results can also explain why the grasses around the gap edge are merging together because a nearly closed barrier around the FCs minimizes the outflow of water from the gap towards the matrix plants. This population-level response to water stress and the positive net-effect of facilitation also explains why the high-vitality grasses are strongly associated with the FCs and why they can afford to have clustered distributions, while at the same time the low-vitality grasses at further distances are struggling as segregated or randomly scattered entities.

| CON CLUS IONS
Bridging empirical ecology and physics is one of the most challenging, but also most interesting, endeavours in the natural sciences.
Such multi-disciplinary research efforts usually take time due to different types of commonly used language or terminology but also due to finding a general acceptance of each other within the two fields of science. Here, we demonstrate exemplarily based on the striking Australian fairy circles, how modelling and a priori assumed processes can be verified using detailed ecological fieldwork.
Our findings support the presence of ecohydrological feedbacks at the landscape-scale and at the scale of the immediate plant neighbourhood that are central to explain this emergent grassland gap pattern via Turing dynamics and pattern-formation theory. We have provided multiple-scale evidence that the grasses with their cooperative growth dynamics function as ecosystem engineers. Thereby, they modulate the abiotic environment to better cope with the permanent shortage of water in this arid ecosystem. The empirically found shortrange positive and long-range negative feedbacks between the plants, the relative feedback strengths and the infiltration contrast all agree with the theoretical assumptions used for modelling the FCs based on Turing dynamics and the reaction-diffusion mechanism.
We conclude that by forming periodic gap patterns, the vegetation benefits from the additional water resource provided by the FC gaps, and thereby keeps the ecosystem functional at lower precipitation values compared with uniform vegetation (Meron, 2018). More field studies are needed for a more complete validation of the mathematical models of FCs and to enable a more realistic parameterization. Especially investigating the time scales of pattern formation in this dryland deserves more research because such patterns may be very long lived and form over hundreds of years (Caviedes-Voullième & Hinz, 2020). In this way, mutual collaboration between empiricists and modellers, for example with backgrounds in physics and ecology, will lead to a deep understanding of multi-scale feedbacks in complex ecological systems.

ACK N OWLED G EM ENTS
We thank Scott Duffy and the CASA/Aviation Group for permitting us to undertake the drone flights. We are grateful to the drone

PE E R R E V I E W
The peer review history for this article is available at https://publo ns.com/publo n/10.1111/1365-2745.13493.

DATA AVA I L A B I L I T Y S TAT E M E N T
All data used for the presented results are provided as CSV files and can be downloaded from Zenodo https://doi.org/10.5281/ze nodo.3979233 (Getzin et al., 2020). Other raw data are confidential, as they are subject to ongoing publications.