A connectivity threshold between grass patches amplifies coastal dune formation

Domain: biology
Published: 2026-03-19
Editorial outcome: major_revision

Abstract
Biogeomorphic systems, key providers of ecosystem services, emerge from self-reinforcing feedbacks between landscape-building biota and geomorphic processes. Typically, these feedbacks are considered to operate at an individual patch scale, yet it remains unclear how interactions between patches shape landscapes at larger scales. Here we show how dune-building grasses form functional clusters of interacting patches that strongly amplify engineering capacity. By analyzing a decade of morphological development in an establishing coastal dune system, we discover that dune height is primarily driven by the initial density of neighboring patches, rather than individual patch size. We identify an S-shaped relationship consistent with a spatial percolation threshold: increasing local patch density triggers an abrupt shift from isolated sand-trapping patches to functionally connected clusters that enhance dune growth. This work reveals an important yet overlooked spatial dimension of ecosystem engineering, one that can be harnessed to inform future restoration designs and enhance ecosystem resilience.

Introduction
Reciprocal organism-landform interactions – so-called biogeomorphic feedbacks – are increasingly recognized as important drivers of landscape formation and ecosystem functioning1,2. Well-known examples of biogeomorphic ecosystems are fluvial landscapes where vegetation controls channel meandering, and peatlands where vegetation retains water and builds landforms by peat accretion3,4. Such systems are particularly common in the coastal zone5. Reefs built by corals, mussels, or oysters provide structurally complex hard substrate in sandy environments, filter the water and attenuate hydrodynamics6. Similarly, coastal vegetation in mangrove forests, salt marshes, seagrass beds, and coastal dunes attenuates physical flow, leading to the trapping of water- or airborne particles within their canopies and stabilizing sediment through their roots7–10. As these interactions engineer the landscape, they generate important ecosystem services, including coastal defense, carbon storage, freshwater provisioning, and biodiversity sustainment11,12.

An important commonality of all biogeomorphic ecosystems is that the feedbacks between biota and geomorphic processes are self-reinforcing (i.e., positive), meaning that organisms engineer landforms and facilitate each other through local positive density-dependent relationships1,5. For example, mussels, corals, and oysters create hard substrates that promote their own recruitment, while in vegetated systems, plant patches trap and stabilize sediment, enhancing conditions for further plant establishment and growth6–8. Local self-reinforcing feedbacks in combination with a more distant constraint, such as resource limitation or physical stress, have been found to give rise to spatial self-organization13,14. Spatial self-organization is a second commonality of biogeomorphic systems and can be observed in a range of coastal landscapes: mussel beds15, mangroves9, nearshore coral reefs16, seagrass beds10, coastal dunes8, mudflats17, and salt marshes7.

In many study systems, local self-reinforcing feedbacks lead to spatial self-organization into regular or periodic patterns, i.e., patterns with a characteristic wavelength13. However, not all self-organized ecosystems stabilize at a fixed scale. Classic scale-dependent feedback produces stationary bands, spots, or labyrinths whose spacing reflects a balance between short-range facilitation and long-range inhibition, for example in mussel beds15 or seagrass meadows10. By contrast, other systems exhibit irregular patches that increase in size over time, as observed in for example coral reefs6 and salt marshes18. Theory suggests that such ‘coarsening’ patterns can arise when long-range inhibition is relatively weak compared with local facilitation, or when inhibition operates at a different scale, for example, by effectively acting as a global constraint19–21. Therefore, understanding why some biogeomorphic systems stabilize while others keep growing hinges on quantifying both the relative strength and spatial extent of inhibitory versus facilitative processes.

In biogeomorphic coastal ecosystems, local self-reinforcing feedbacks are typically suggested to emerge from positive density dependence generated within patches of aggregated individuals10,17,22–24. However, previous work found that the range of influence of local facilitative effects are not merely restricted to the patch itself, but may extend far beyond14. In establishing biogeomorphic systems, this spatially extended facilitation could become particularly important when the influence zones of neighboring pioneer patches overlap. In this case, physically separated patches can become functionally connected, for instance, by modulating flow as a collective rather than as individual patches25. Consequently, functional clusters of patches can be formed within the system, whose joint engineering capacity may exceed that of individual patches. We therefore suggest that such a spatially extended effect of local self-reinforcing feedbacks could amplify the landscape-building capacity of ecosystem engineers and represents a thus far overlooked role in the formation of biogeomorphic landscapes.

Coastal dunes are an iconic example of biogeomorphic landscapes. About one third of the world’s coastline consists of sandy shores that are protected by coastal dunes26. Similar to other establishing vegetated biogeomorphic systems (Fig. 1a), developing coastal dunes have been proposed to function as self-organizing systems8,27. Coastal dune formation starts with pioneer dune grasses colonizing the beach28. By trapping airborne sediment, these dune grasses become buried, stimulating plants to develop additional shoots and expand, thus further enhancing sand trapping. As a result of this local self-reinforcing feedback, small embryonic or nebkha dunes are formed by the individual patches of dune grass8,28. In time, with abundant sediment present, single embryonic dunes will continue to grow, eventually coalescing and maturing to form coastal foredunes28,29. Recent work demonstrated that the spatial organization of shoots within dune grass patches significantly influences sand trapping efficiency and subsequent dune formation at the patch scale30,31. However, at scales beyond individual patches, it remains unclear how spatial interactions among patches influence sand trapping and thus affect the strength and spatial extent of self-reinforcing feedbacks driving dune formation.

Here, we explore how spatially extended interactions among engineering patches, rather than the size of individual patches alone, influence engineering capacity in embryonic (i.e., establishing) coastal dune landscapes. Specifically, we empirically quantify whether the spatial organization of physically separated dune grass patches functionally connects them to amplify dune formation. We hypothesize that, in establishing dunes, collective sand trapping by neighboring patches amplifies early dune formation beyond what individual patches can achieve alone. Following spatial percolation principles32, we expect that when patches lie close enough for their zones of influence to overlap, they form functionally connected clusters whose joint engineering capacity outperforms that of isolated patches. Moreover, we predict that the initial spatial arrangement of patches sets a template for how interactions unfold, making early patch organization a strong predictor of future landscape morphology. In practice, this means that because functional clusters of patches collectively alter airflow33,34, they amplify embryo dune growth by enhancing sand deposition and trapping (Fig. 1c). By contrast, isolated patches do not have that advantage, resulting in slower sand capture and reduced dune growth (Fig. 1b). To test these predictions, we used an extensive dataset of annual high-resolution aerial imagery (0.1 × 0.1m) spanning over a decade of development in a large (90 ha), establishing dune system. We focused on a 12-ha subsection beyond direct wave impact, where dune development emerged under relatively homogeneous and near-synchronous conditions. Within this subsection, we identified over 4000 dune grass patches and generated detailed elevation maps using photogrammetry, capturing the transition from an embryonic dune field to a mature dune landscape. By linking patch configuration with local morphological change through time, we determine the dominant scale at which ecosystem engineering emerges and assess whether functional connectivity between patches amplifies coastal dune formation.

Results
Between-patch interactions amplify dune formation
Our results support the hypothesis that dune height development is not primarily driven by within-patch facilitation, indicated by the size of a patch, but instead arises from interactions between neighboring patches that jointly amplify dune growth. As a first step, we tested whether initial patch size correlates with dune height one year later, but no clear relationship was found (Fig. 2a; Spearman’s ρ=0.18, p<0.001). We then examined how the local density of neighboring patches influences dune height (within a 7m radius of each focal patch, see “Methods” and Supplementary Fig. 1). The results revealed that dune height much better predicted by local patch density (Fig. 2b; spearman’s ρ=0.57, p<0.001). Similar relationships can be found for Δ dune height (Supplementary Fig. 2). Finally, we checked whether local patch density could be a product of underlying environmental factors such as initial topography or distance to the shoreline within our study area. Although initial topography shows a significant correlation, neither is are strong drivers of patch density (Spearman’s ρ=0.097, p<0.001 and ρ=0.019, p=0.199 respectively) or future dune height (Supplementary Table 1 and Fig. 3). These findings show that dune formation is not an isolated outcome of individual patch performance, but rather an emergent property of spatially clustered patches. This implies that by building a shared dune body, physically separated patches become functionally connected.

Between-patch interactions create a template for long-term dune landscape formation
Long term analyses of dune landscape development highlight that between-patch interactions not only shape dune height in the short term (Fig. 2b) but also have a lasting effect as the system continues to develop over multiple years (Fig. 3a). Specifically, we found that the initial patch density in 2013 has an evident positive effect on dune height for 8 consecutive years from the period 2014–2023 (Fig. 3a). Indeed, for all years, dune height follows an S-shaped curve as function of initial patch density, with limited dune growth for a patch density of below 0.03 patch m−2, a steep increase around 0.05 patch m−2, after which it flattens off from 0.10 patch m−2. This S-shaped relationship can be best described by a Hill function (see Eq. (2) in the “method” section and Supplementary Table 2 for model fitting). This long-term, nonlinear response shows how early patch density leaves a lasting imprint on landscape development, with dune growth accelerating once a critical density threshold is surpassed.

The effect of initial patch density on dune height becomes increasingly nonlinear over time, yielding a sharper more threshold-like transition in early patch density beyond which dunes rapidly build up. This is reflected in the increasing sigmoidicity exponent h of the S-shaped curves (Fig. 3b). The combined height of the system builds up relatively rapid at first and stabilizes in the final years of the analysis, as indicated by the spacing between the S-curves (Fig. 3a; see for long term dune development Supplementary Fig. 7). Interestingly, the inflection point k stays relatively constant over time (0.054±0.003 patch m−2; Fig. 3c). Together, these analyses indicate that dune formation is governed by an initial patch-density threshold whose influence intensifies over time.

It is striking that the spatial distribution of vegetation patches in the earliest year (2013) best predicts dune height up to a decade later. In support of previous analyses, the explanatory power of patch density on dune height declines as dune grass patches of this initial embryonic system grow and expand over time (see for the effect of patch density from 2017 and 2021 Supplementary Fig. 8 and Table 3). This further highlights that it is predominantly the initial patch distribution and the associated patch densities that shape future dune development. The strength of this early effect can also be visually recognized by comparing the 2013 vegetation map with elevation change over time (Fig. 3d). Together, these results highlight that it is the initial organization of vegetation, rather than its later expansion, that sets the development trajectory of dune growth.

A connectivity-based threshold drives nonlinear dune growth
The threshold-like relationship between patch density and dune growth observed in Fig. 3a shows a striking parallel with the percolation threshold that is well known from statistical physics32. In our case, the inflection point k of the S-shaped relationship (Fig. 3a) marks the patch density around which small changes lead to disproportionately large deviations in dune height, a hallmark sign of systems undergoing a connectivity-driven transition35. Indeed, interpreting the inflection point k of the S-shaped relationship as a percolation-like threshold suggests that, around this density, a critical transition occurs: patches become functionally connected forming large “spanning” clusters (Fig. 4a). Below the percolation zone around k, patch densities are too low to form such clusters. Above the percolation zone, the system is “percolated” – i.e., functional clusters have formed – implying that from that point onward additional patches will yield diminishing returns due to saturation of connectivity (Fig. 4a). Assuming each patch affects its surroundings within a circular range of influence, using continuum (disk) percolation36, we can calculate the minimal radius of this range of influence required to form a spanning cluster at the patch density of k (Fig. 3a k ≈ 0.05 patch m−2; see “Methods” for details). This yields a range of influence with a 2.27m radius and a maximum interaction distance between patches of 4.54m (to connect their ranges of influence). Figure 4b illustrates how this estimated 2.27m allows the range of influence of patches to overlap, forming a functionally connected cluster that uses its joint engineering capacity to build a single dune body. Thus, applying the percolation framework on our empirical data exposes a mechanistic understanding of the scale at which individual patches engineer their environment and how connectivity between patches amplifies this process to shape the entire dune landscape.

To test whether the nonlinearity in Fig. 3a indeed reflect a connectivity mechanism, we performed three complementary analyses. First, we tested whether the maximum interaction distance predicted by percolation theory (~4.54m) appears independently in our data. When relating mean nearest-neighbor distance to dune height, we find an S-shaped relation for which amplified dune growth happens within a given distance (4.12, 4.48 and 5.11m; (Fig. 5d bc). These distances closely match the 4.54m maximum interaction distance. Second, using that same maximum interaction distance, we quantified “spanning”-cluster formation as a function of patch density (Fig. 5d). Indeed, the results yield the expected sigmoidal increase. The fraction of patches being part of large clusters rises most steeply near the inflection point k and saturates where the dune-height curves level off (Fig. 3a). Third, because our inflection point k is estimated within a finite window (the local neighborhood), we examined how k shifts with window size (L) (Supplementary Fig. 10). In the case k represents a connectivity threshold, percolation theory predicts a characteristic scaling for k: larger windows require less density to connect (more alternative paths), so k(L) decreases and converges. For two-dimensional percolation, k(L) yields a linear relation with L−0.75 (see Eq. (6))32. Consistent with this, Fig. 5e shows that k systematically decreases with window size and then stabilizes, thereby it obeys the hallmark L−0.75 scaling for a connectivity threshold (Fig. 5f). Importantly, two alternative explanations would show a very different scaling trend: (i) if the spatial patch distribution alone (short-range clustering; see Supplementary Fig. 9) was responsible, k logically should increase and then plateau with window size; and (ii) if dune height depended only on total cover (no connectivity), the threshold would stay constant across window sizes.

Together, these tests show that once patches are close enough to interact, functional connectivity emerges and dune growth accelerates, consistent with a connectivity-based threshold.

Methods
Study site
To test our hypothesis, we required a naturally formed embryonic dune field that recently transitioned into a mature landscape. Since our study investigates landscape-building processes, an ideal location would experience minimal limitations in accommodation space, sediment availability, and seasonal erosion54–56. Additionally, for this dune system high-resolution aerial imagery capturing the main development of the system needed to be available. The Hors on Texel, the Netherlands (52°59'43.1“N 4°43'46.0“E, Fig. 6), turned out to be a very suitable location. The Hors consists of an unmanaged, actively developing dune system (~90ha in 2024), progressing south-westward over recent decades. Flood erosion at the Hors is minimized by an extensive beach plain fronting the embryonic dunes, which attenuates wave impact57,58. This beach plain also provides sufficient accommodation space for dune expansion and serves as a large fetch area for wind-driven sediment transport56,59. Wind directions vary throughout the year, with South-Western winds being most common (Supplementary Fig. 11). For our analysis, we focused on a 12-ha subsection of the Hors, representing an embryonic dune field establishing around 2011–2012, with clearly identifiable dune grass patches and embryo dunes in the imagery from 2013 onward (Fig. 6). This subsection is oriented parallel to the tideline and was part of a significant colonization event. By 2013, pioneer grasses had formed distinct embryonic dunes, with minimal spatial or temporal gradients, ensuring uniform age and environmental conditions across the area. At that time, dune grass patches covered 8.5% of this area, comprising 4,349 embryo dunes with an average patch size of 2.3m². Two species dominate dune formation here: Elytrigia juncea (Sand couch) and Calamagrostis arenaria (formerly Ammophila arenaria, Marram grass)60. E. juncea acts as the pioneer species, facilitating landscape modification and subsequent colonization by C. arenaria, which is the superior dune builder61. Although E. juncea likely initiated landscape changes at the study site, the distinct embryo dunes visible from 2013 onwards were predominantly formed by C. arenaria.

Data
Aerial imagery of the Hors was provided by the municipality of Texel and Beeldmateriaal, both of which conduct annual high-resolution airborne surveys. We used pre-processed orthomosaics from the national airborne survey (0.1 × 0.1m), to detect and extract dune grass patches (Beeldmateriaal 2010–2021; Beeldmateriaal HRL 2021–2023, CC BY 4.0). Digital elevation models (DEMs) were generated from stereo photos provided by the municipality of Texel using Agisoft Metashape, covering the years 2014, 2015, 2017–2020, and 2022–2023, with horizontal resolutions ranging from 0.05 to 0.07m. All imagery was captured in early spring, prior to the main growing season (February–April). To evaluate the beach topography prior to our main study period (2008–2013), we used preprocessed elevation data from Hoogtebestand Kust (5×5m). We extracted dune grass patches using a geospatial deep learning platform (Picterra). A custom model combining object detection and feature masking enabled us to delineate both the location and spatial extent (surface area and perimeter) of individual patches (see for examples feature masking Fig. 6, and for more detail on model approach: www.picterra.ch).

Data analysis
Between-patch interactions amplify dune formation. We first examined the relationship between the size of dune grass patches in the initial embryonic dune field (2013) and their corresponding dune height in the subsequent year. For each patch, dune height was extracted from all available DEMs using its centroid coordinates. Next, we tested whether local patch density could explain future dune height. Local patch density was calculated by counting the number of neighboring patches within a circular search radius (r search) around each patch center, assuming isotropic interactions for simplicity.

To determine an ecologically relevant neighborhood radius, we related dune height to neighbor counts for radii from 2 to 30m using Spearman’s rank correlation coefficient (ρ) and applied the elbow criterion for scale-dependent patterns62. The elbow location was identified with piecewise (broken-stick) regression (segmented package in R)63 as the smallest radius at which ≥95% of the maximum ρ was reached, and further increases in radius yielded only marginal gains in explanatory power. This procedure identified 7m as the neighborhood radius (see Supplementary Fig. 1), which we used in all subsequent analyses. A patch was considered a neighbor if its centroid lay within this radius (r search). Local patch density was defined as:

Patch density = N patches / (π r search 2) (1)

Underlying environmental factors. To test whether the observed relationship between patch density and dune height reflects underlying environmental gradients, we quantified distance to shoreline and initial elevation and evaluated their associations with local patch density and future dune height. We computed distance to the shoreline as the planar distance from each patch centroid to the mapped shoreline for the year 2013; initial topography was extracted at each patch centroid from the DEM of the year 2010 (in this year vegetation cover was neglectable, see Supplementary Fig. 7). For the correlation between initial topography, distance to shoreline and patch density we used Spearman’s rank correlation. Moreover, for the relation of these variables to future dune height, we used a standardized multiple linear model with future dune height as the response and patch density, distance to shoreline, initial topography, and patch size as predictors (see for details Supplementary Table 1 and Supplementary Fig. 3). To support these tests, Supplementary Fig. 7 presents aerial imagery of system development from a bare beach to a mature dune field (2008–2023), together with time series of total dune volume and vegetation cover, illustrating the onset of dune formation following vegetation establishment.

Between-patch interactions create a template for long-term dune landscape formation. To determine the best functional fit between patch density and future dune height, we tested multiple candidate models: linear, power-law, negative exponential, and a generalized Hill function64. Based on the Akaike Information Criterion (AIC), the Hill function consistently provided the best fit across all elevation years (see Supplementary Table 2 for model fitting details). The generalized Hill function can be defined as:

y = y min + (y max - y min) * xh / (kh + xh) (2)

Within the context of this study, the parameters of Eq. 2, can be best described as follows. Dune height (m) is represented by y. With y min being the minimum dune height predicted at very low patch densities, indicating the baseline dune growth by isolated patches. Y max stands for the maximum dune height achievable as patch density increases. k is the inflection point of the curve, marking a key threshold in the relationship between patch density and dune development at the steepest point of the curve. The Hill coefficient h, controls the steepness, or sigmoidicity, of the curve around k. A higher value of h indicates a sharper, more threshold-like transition in dune growth.

A connectivity-based threshold drives nonlinear dune growth. To test whether the observed nonlinearity between patch density and dune height reflects a connectivity mechanism, we interpreted the inflection point k of the Hill function (Eq. 2) as the midpoint of a percolation transition. We then translated k into an interaction length by calculating the range of influence around each patch centroid search needed to form a spanning cluster at the observed patch density for k. We treated each vegetation patch as the center of a disc of influence with radius r, such that two patches interact when their disks just touch. The expected number of neighbors within distance r of a focal patch is:

N(r) = ρ π r2 G(r) (3)

Where ρ is patch density (patches m−2), π r2 is the area of the neighborhood, and G(r) is the pair-correlation function describing the degree of clustering relative to a random spatial distribution (estimated from a pair-correlation function of the system; Supplementary Fig. 9). Following continuum disk percolation for a two-dimensional system, a spanning cluster appears when a patch is directly connected to, on average, λ c ≈ 1.128 neighbors36. At the percolation point we therefore require:

N(r) = λ c = 1.128 (4)

Combining Eq. 3 and eq. 4 gives:

r = sqrt( λ c / (k π G(r)) ) (5)

Since G(r) is based on r, a few iterations have to be calculated to find r that fits with the clustered spatial distribution of the system. We then defined the maximum interaction distance between patch centroids as d max = 2r.

Using d max, we performed three complementary analysis. First, we related dune height to the mean nearest-neighbor distance per patch (averaged over the 2, 3, and 5 closest neighbors; centroid-to-centroid). We fitted Hill functions (Eq. 2) to these relations to estimate distance thresholds. For this, we used the value of the inflection point k, which we later compared to the theoretically derived d max. Second, we treated patches as connected when their centroids were d max apart and identified connected sets of patches. We classified a set as “spanning” when it bridged a distance needed to connect opposite sides of our 7m radius neighborhood window. We then quantified, across patch density, the fraction of patches belonging to such spanning components.

Third, we performed a finite-size scaling analysis to test whether the density threshold k behaves as expected for a 2-D percolation transition. We re-estimated k for different window sizes (L) with radii (3-24m; see Supplementary Fig. S10) as the inflection point of the patch density versus dune height curves, computing local patch density as in Eq. (1). In finite systems the apparent percolation threshold depends on the window size (L), interpretating k as the percolation threshold, finite-size scaling theory predicts:

k(L) = k c + a L−1/ν (6)

Where k c is the density threshold in an infinite system, a is a constant, and ν is the correlation-length exponent. The universal value for ν in 2D percolation32 is ν = 4/3, so k(L) should vary approximately linearly with L−0.75. Therefore, to test whether the observed finite-size dependence of our patch density versus dune-height threshold is consistent with 2-D percolation, we regressed k(L) against L−0.75. Unless noted otherwise, analyses above included all patches present in 2013 (N = 4349 patches). All analyses descripted here were performed in R (v 2024.04.2).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors study the spatial evolution of coastal ecosystems, focusing on how dune height increases over time as a function of the initial patch size and initial patch density. They find that the initial patch size has no measurable impact on dune height, whereas the initial patch density does. This suggests that dune height is well predicted by local patch density and is therefore an emergent property of clustered patches. They further report that the relationship between dune height and initial patch density follows an S-shaped curve, best described by a Hill function, indicating a nonlinear dependency. Finally, the authors interpret these results within a percolation framework, arguing that above a critical density threshold, patches form a percolating cluster that drives the nonlinear increase.\n\nThe data analysis is carefully executed, and the results are both novel and interesting. However, I have two major concerns that should be addressed before I can recommend publication:\n1. Causal explanation and evidence for connectivity:\nThe proposed explanation that the nonlinear relation arises from functional connectivity between patches, as captured by percolation theory, is not sufficiently supported by evidence in its current form. It is plausible that higher initial patch density simply reflects more favorable environmental conditions, which themselves promote faster dune growth, independent of any connectivity effects. In other words, the observed nonlinearity need not imply percolation or even functional connectivity between patches.\nIf the available data allow, the authors should provide direct evidence of functional connectivity between patches. If such evidence cannot be derived from the current dataset, they should instead cite relevant studies that have demonstrated such connectivity, or else develop a mechanistic model of nonlinear dynamics to support the claim (see Comment 2 below).\n2. Role of dynamics vs. pure percolation:\nClassical percolation theory is purely structural and does not yield a Hill coefficient. The Hill coefficient typically arises in dynamical systems, such as gene regulation or ecological processes involving nonlinear feedback (see Gao, J., Barzel, B., & Barabási, A.-L. (2016). Universal resilience patterns in complex networks. Nature, 530, 307–312). While a Hill function may empirically fit the data well, the explanation for the observed phenomenon should incorporate a dynamical component in addition to structural percolation.\nI strongly recommend that the authors develop a nonlinear dynamical model to explain their findings. Such a model could be fitted both to the temporal evolution of dune height and to the final-state relationship with initial density. This approach would not only strengthen the interpretation but also allow the authors to explicitly compare two scenarios: one incorporating functional connectivity and one without it. Demonstrating that the connectivity-based model fits better would substantially reinforce their central hypothesis.", "reviewer_2": "In the manuscript 'A percolation threshold between grass patches triggers coastal dune formation' the authors present field data showing a persistent correlation between the local height of coastal dunes and the local density of vegetation patches at the beginning of the observation period. From the sigmoidal shape of the relation, and the fact that closely spaced patches trap more sand, the authors extrapolate the existence of a 'percolation threshold' for patch density above which sand is continuously captured by functionally connected patches and dunes form.\n\nThe focus on the role of patches, instead of individual plants, is certainly interesting, as well as the central argument regarding the relation between patch connectivity, dune growth and percolation theory. However, as discussed in detail below, it seems to me that the authors are in fact just describing how dunes grow instead of providing the causal mechanism for coastal dune formation as they claim in the title. Therefore, I don't recommend publication in the current form.\n\nMain points:\n1. There is no proof of causality. The central claim is that dunes will only form (in that particular location at least) wherever patch densities are above a given threshold. For that to hold, patch density cannot itself depend on elevation, at least initially (otherwise there is no clear way to disentangle what leads to what), and the initial data should ideally show a wide enough range of patch densities with similar elevations. However, the data presented already shows the trend of higher elevations for denser patches from the beginning. How can the authors prove that this trend is not just a sub-dependence of patch density on elevation?\nThe authors should analyze a subset of their initial data containing only patches with similar elevation. Given the wide dispersion of the elevation data in Fig.2b, it should be possible to find a wide range of patch densities with similar elevations in 2014 and track their elevation over time.\n2. The null hypothesis (that patch density is not relevant for dune formation or final size) is not really tested because regions with low patch density are also closer to the shore (from looking at the study site in google earth) and have lower elevation, and therefore are more exposed to flooding and water-driven erosion. This could explain the lack of mature dune formation on those regions: as a combination of more erosion and less vegetation growth.\nAlong these lines, an alternative explanation for the authors' findings is that patch density affects the growth rate of the dunes, but not their formation, nor their final height. Instead, dune formation is ultimately controlled by purely physical factors such as sand supply and water-driven erosion, and thus distance from the shoreline (for erosion) and distance to sand sources. The dependency of dune growth on patch density reported by the authors would just be a consequence of patch density being affected by seawater flooding and erosion and thus dependent on local elevation and distance from shoreline.\nConsequently, the authors should carefully consider the influence of physical factors in their analysis, in particular, the local elevation of the inter-patch areas (i.e. excluding proto-dunes and potentially dune crests), the distance to the shoreline and the main direction of the sand supply. Ideally, they should isolate the effect of patch density by randomizing the physical factors, which right now seems to be highly correlated. One way to visualize the effects of physical parameters could be to plot a color map of: (1) the average dune height vs. patch density and distance from shoreline in 2013 (properly defined); and (2) the average dune height vs. patch density and average elevation of the inter-patch regions (in 2013).\n3. By using the word 'trigger' in the title, the authors argue that a coastal dune can form only in those conditions (patch density above a threshold). However, this is far from obvious because they are ignoring the effect of lateral growth (by rhizomes) of dune-building plants, where in theory a single initial patch in a highly favorable location can trap sand and expand laterally in a positive feedback that would lead to a dune ridge (foredune). In fact, it has been argued that this lateral dune growth driven by the lateral propagation of dune vegetation (as a single, large, patch) could be the main mechanism for post-storm dune recovery from a washover (e.g. Goldstein et al. Earth Surf. Dyn, 2017).\n\nMinor point:\n4. The authors didn't find a correlation between dune height and patch size, but what about between dune height and patch area (as defined in the manuscript) over time? I would expect a positive correlation for all years, where taller dunes have larger patch areas (eventually saturating at 100% cover).\n5. How can the authors explain the large dispersion in dune height data for a given patch density (sigma/mean ~ 25%) in the context of percolation theory, where the states before and after the transition are well-defined, i.e. deterministic? Assuming sand supply is relatively uniform, a patch density above the threshold should produce a relatively well defined dune height if the percolation argument holds (once patches are connected they should uniformly trap most of the incoming sand).\n6. In Fig.3a, the authors should show the full standard deviation. The dispersion of the data for a given patch density is important in the evaluation of a causal explanation.\nOrencio Durán", "reviewer_3": "“A percolation threshold between grass patches triggers coastal dune formation”\nBerghuis et al.\nReviewer Report – DTR\nI would like to thank the journal for allowing me to review this work. I thoroughly enjoyed reading the manuscript and I believe that the authors should be commended for their hard work, ideas and excellent writing. Below I have grouped my comments into the sections listed on this page https://www.nature.com/ncomms/for-reviewers/writing-your-report\nI hope my comments and suggestions are taken in the spirit in which they are intended.\nBest wishes,\nDom Robson\n\nKey results:\nThe authors present novel analyses suggesting that the primary control on dune growth in the formation of coastal dune systems is the initial number density of vegetated patches rather than the size of the initial patches. The authors demonstrate that there is an intermediate range of patch densities at which dune growth increases dramatically with patch density. The lower and upper boundaries of this range define the minimum required patch density to form dunes, and the case where increasing patch density no longer helps to facilitate dune growth. These limits may have important implications for restoration and management projects in coastal dune systems, something which may be increasingly important due to anthropogenic climate change.\n\nValidity (interpretation and conclusions):\nOverall, I think the authors’ interpretation of the data and the conclusions drawn are reasonable. However, there are some limitations that I feel need to be addressed before the manuscript is publishable. Additional discussion may be sufficient to address some of these, however some extra data analyses may also be required.\n\nFirstly, a very minor point and easily addressed, the authors should explicitly state in the discussion that the exact numerical values of e.g. the relevant spatial scale, infection point, and range of influence are specific to the study site and not necessarily universal. Presumably, these precise values would depend on plant-type, granulometry, wind regime etc, and this should be made clear by the authors.\n\nSecondly, a more major concern, the authors have proposed a mechanism which explains the apparent dependence of dune size/growth on patch density based upon the connectivity of the patches i.e. whether they form spanning clusters. The authors claim that the inflexion point on the S-shaped curves represents the point at which spanning clusters are formed; however, I do not see that this is fully justified in the analyses presented. The authors could explicitly determine (using their range of influence) whether each cluster does “span” it’s 7m radius neighbourhood. They could then plot the proportion of spanning clusters as a function of patch density to support their claim.\n\nWithout this, there is an alternative explanation for the data as presented; it could simply be the total vegetation in a cluster that controls dune growth regardless of whether or not it spans the space.\n\nThe authors show in S3 that the patch density and total vegetation area are strongly linked. This would mean that a similar S-shaped relation would exist between dune size and total vegetation area in a cluster. From a qualitative visual inspection of figure 3d, is appears that the largest individual patches (e.g. ~40% up near the LHS) occur as isolated individuals while clusters consist of multiple similarly-sized patches. If this is correct, and if dune growth depends on local vegetation cover then the shape of the curve in figure 2a would be explained. At small patch size, most patches are in clusters of similarly sized patches. Increasing patch size therefore increases total vegetation cover leading to an increase in dune growth. For large patches, dune size decreases with patch size since the largest patches are isolated, meaning that total local vegetation is reduced even if the individual patch is larger.\n\nUnder this interpretation, the S-shape that would presumably be apparent would be explained by there being a minimum threshold of vegetation presence to facilitate dune growth and a maximum threshold above which sediment flux, rather than vegetation, limits the growth of the dunes.\n\nThe authors should either explain why this cannot be the correct interpretation or provide explicit results on connectivity vs. density which support their claimed mechanism. However, this would not change the more general argument that the patches are interacting; it would simply change the exact manner of that interaction.\n\nOn a related note, if the spanning mechanism is accurate, then, depending on the wind regime, there may be preferred shapes of spanning clusters. For instance, if sand fluxes predominantly occur in a particular direction, then a cluster which spans the flux-perpendicular dimension would presumably capture a different amount of sediment than one that was spanning in the flux-parallel dimension. Depending on the wind regime and sediment availability at the study site, this might call into question whether defining circular neighbourhoods really makes sense.\n\nFinally, the authors show, in S6, that the initial patch density is far more important control on dune height than intermediate densities. The question, therefore, becomes: “What determines the initial density of patches?”. It could be imagined that topographical features or high sediment availability could be contributing factors to the initial distribution of pioneer patches. For example, if grasses are more likely to form initially in an area where there is a high deposition of sediment (and seeds?), then that same high rate of deposition would inherently lead to faster-growing dunes than an area with a lower deposition rate (and so also a lower initial density of vegetation). Thus, it could simply be a coincidence that both the grasses and dunes prefer certain areas of the system to others. Discussion along these lines should also be included.\n\nSignificance:\nThe novel conclusions presented in the work and the increasing need for effective management and restoration of coastal landscapes make this a significant piece of work. Additionally, the links drawn between ecology, geomorphology, and statistical physics concepts make this interesting to a wider audience.\n\nData and methodology:\nThe conclusions presented in this work are based upon a reasonably large spatiotemporal dataset comprising a decade of data pertaining to around 4000 patches in a formation-stage dune field. The dataset provides the authors with dune growth measurements for a range of approximately one order of magnitude of the explanatory variables (patch size/density) which is a suitable range from which to draw the type of conclusions made by the authors.\n\nAs I have already stated, additional remarks should be included to make it explicit that the precise quantitative results apply only to the study site in question and may not be applicable in other locations.\n\nA major consideration in defining density is the spatial scale considered, which is fully justified by the authors in the Supplementary Materials (S1).\n\nIn the Supplementary Materials, the authors also do a good job of demonstrating the curve fitting methodology as well as the derivation of key findings such as the range of influence.\n\nAnalytical Approach:\nI felt that the authors’ approach was sound and, with the exceptions I discussed in the validity section above, the analyses were valid and comprehensive as demonstrated both in the manuscript itself and the Supplementary Materials included.\n\nThe justification of the choice of scale is particularly good, however I do believe that the authors should also justify the choice of a circular neighbourhood. This could be done by e.g. providing a wind or sediment flux rose showing significant directional variability.\n\nSuggested Improvements:\nOverall, I felt that this was a well-considered manuscript and I found it interesting to read. I have already described the main areas which I found lacking and I believe improvements are required either in the form of additional discussion or new analyses. Specifically:\n1. The authors should make clear that quantitative results pertain solely to this site rather than being global values.\n2. The authors should provide additional justification for why their proposed mechanism of connectivity is more valid than one that relies solely on the total vegetation in a cluster.\n3. The authors should discuss what might control the initial distribution of patches.\n\nIn addition to these I have some very minor additional suggestions:\n1. In figures 2 and S2 the captions describe the binning of the patch size but do not state whether patch density has been binned to produce the figures (and all of the other figures showing patch density). With regularly spaced points it appears that these data have also been binned and the authors should state the width of said bins. Additionally, it would be good to see the raw (unbinned) plots in the supplementary material.\n2. Percolation appears in the title and the summary paragraph but is not mentioned again until line 197. The authors may want to consider mentioning it briefly in the introduction.\n3. Also on percolation, when the authors introduce the topic they cite some examples of its recognition in ecological settings (line 202). Percolation has also been applied in a study related to clustering in systems of desert dunes in the following work: Génois, Mathieu, et al. \"Out-of-equilibrium stationary states, percolation, and subcritical instabilities in a fully nonconservative system.\" Physical Review E 94.4 (2016): 042101. Including this additional reference might encourage a wider geomorphological audience.\n\nClarity and context:\nI found the manuscript to be clear and accessible and generally very well written. I enjoyed reading through the work. I felt the figures contained a good amount of information while being visually appealing and easy to interpret.\n\nBin widths for patch density should be included in the Figure 2 and S2 captions.\n\nReferences:\nI am more familiar with desert aeolian geomorphology and forest ecology literature so I cannot fully comment on this. However, reference have been provided by the authors for additional information and justification where necessary.\n\nI suggest that the authors include the following reference:\nGénois, Mathieu, et al. \"Out-of-equilibrium stationary states, percolation, and subcritical instabilities in a fully nonconservative system.\" Physical Review E 94.4 (2016): 042101.\nThis would connect the work to percolation in desert dunes.\n\nMy expertise:\nI have expertise in both aeolian geomorphology and ecology, however my experience of coastal systems is not as complete. Other reviewers may have a better grasp on precisely where this fits in with that field."}, "author_response": "Dear reviewers,\nWe wish to thank you for assessing our manuscript now entitled: \"A connectivity threshold between grass patches amplifies coastal dune formation\" (NCOMMS-25-56579A). Your constructive comments and valuable suggestions enabled us to strengthen our work substantially.\nTwo important concerns raised were shared by all three reviewers: (1) the role of environmental conditions and physical drivers affecting dune growth and (2) the (lack of) evidence for connectivity-based percolation as the cause of the observed nonlinearity. We first summarized our response to these two issues below, followed by a detailed point-by-point reply to all comments.\nFirst, the reviewers asked whether environmental gradients confounded our results. Because our central aim was to assess the effect of (interacting) grass patches on dune growth, we predefined a study zone beyond wave impact where dune development is as homogeneous and near-synchronous as possible. This design minimized major gradients (distance to shoreline, initial topography), allowing dune grass effects to be evaluated without obvious confounding factors. However, as pointed out by the reviewers we did not document and test these confounds sufficiently. We now include an extended time record of elevation plus additional vegetation data. These show an initially (almost) flat beach and an increase of elevation only after vegetation established; consistent with vegetation-driven causality. An additional statistical model further indicates that initial elevation does not predict patch density and dune height and shows no strong relation with shoreline distance.\nSecond, the reviewers questioned the evidence supporting connectivity-based percolation as a driver of nonlinear dune growth. In the discussion, we hypothesized that patch density affects dune growth via functional connectivity of dune grass patches, yielding an S-shaped relation indicative of a percolation threshold. We now strengthen this argument with three additional analyses that we show in a new section of the results. First, using the maximum interaction distance derived from continuum disk percolation theory, we confirm that cluster formation as a function of density follows the expected S-shaped curve. Second, we show that the maximum (nearest) neighbor distance for which dune height growth is amplified, closely matches the maximum interaction distance independently derived from percolation theory. Lastly, we find that the inflection point scales with window size (neighborhood radius) exactly as expected for a connectivity threshold. Together, these results indicate that a density-driven transition in functional connectivity could indeed explain the observed nonlinearity: once patches are close enough to interact, connectivity emerges and dune growth accelerates.\nWe are confident we addressed the concerns raised in a satisfactory way and that this significantly improved the manuscript.\nSincerely,\nPaul Berghuis and co-authors\n\nReviewer #1 (Remarks to the Author):\nThe authors study the spatial evolution of coastal ecosystems, focusing on how dune height increases over time as a function of the initial patch size and initial patch density. They find that the initial patch size has no measurable impact on dune height, whereas the initial patch density does. This suggests that dune height is well predicted by local patch density and is therefore an emergent property of clustered patches. They further report that the relationship between dune height and initial patch density follows an S-shaped curve, best described by a Hill function, indicating a nonlinear dependency. Finally, the authors interpret these results within a percolation framework, arguing that above a critical density threshold, patches form a percolating cluster that drives the nonlinear increase.\nThe data analysis is carefully executed, and the results are both novel and interesting. However, I have two major concerns that should be addressed before I can recommend publication:\nCausal explanation and evidence for connectivity:\nComment 1:\nThe proposed explanation that the nonlinear relation arises from functional connectivity between patches, as captured by percolation theory, is not sufficiently supported by evidence in its current form. It is plausible that higher initial patch density simply reflects more favorable environmental conditions, which themselves promote faster dune growth, independent of any connectivity effects. In other words, the observed nonlinearity need not imply percolation or even functional connectivity between patches.\nReply: We thank Reviewer #1 for their helpful suggestions. We selected a study area (within our study system) where both the initial topography and the external forcing exhibited minimal heterogeneity (see method section p. 21 L509-534), we now also highlight this already in the introduction.\np. 6 L114-116: “We focused on a 12-ha subsection beyond direct wave impact, where dune development emerged under relatively homogeneous and near-synchronous conditions. Within this subsection…..”\nHowever, we agree that this was insufficiently quantified. To address this issue, we now include complementary aerial imagery and elevation data, and performed additional statistical analyses:\n(i) First, we added additional aerial imagery and elevation maps, to better document initial topography and development of the study site over time. These new data highlight the homogenous flat initial conditions, and the onset of dune development due to vegetation establishment now included in Supplementary Fig. 7.\np. 24 L586-589: “To support these tests, Supplementary Figure 7 presents aerial imagery of system development from a bare beach to a mature dune field (2008–2023), together with time series of total dune volume and vegetation cover, illustrating the onset of dune formation following vegetation establishment.”\nSupplementary Fig. 7: “Supplementary Fig. 7 | Evolution of the study site from bare beach to mature dunes. Top row: aerial imagery of the study system (2010–2021) with the analyzed area outlined in black, showing (i) a bare, flat beach (2010), (ii) full colonization by dune-grass patches (2013), and (iii) a mature, vegetated dune system (2021). Bottom panel: time series of average dune height within the study area (solid orange line, left axis; 2006–2023) extracted from yearly elevation maps of the Dutch coast (5 × 5 m spatial resolution), with a shaded envelope indicating spatial variability in elevation through time. The dashed green curve (right axis) shows total vegetation cover (%), computed as the summed area of vegetation patches divided by the study-area extent. Aerial imagery: Dutch Cadaster orthophotos (0.10 × 0.10 m). Elevations: Hoogtebestand Kust Nederland, National Geo-register (www.nationaalgeoregister.nl).”\n(ii) To emphasize that patch density shapes dune height, and not the other way around, we added the data of the initial topography (2010) in figure 3a as a baseline and adjusted the caption accordingly.\np. 9 L173-174: “The orange dashed line indicates the base line topography prior to vegetation establishment (2010; see Supplementary Fig. 7).”\n(i) Finally, we tested whether the relation between patch density and future dune height is explained by underlying favorable environmental conditions, rather than by patch density itself. First, we tested the correlation between local patch density and the underlying environmental factors (initial topography and distance to shoreline) using Spearman's rank correlation. Moreover, we used a standardized linear model to quantify the correlation of patch density, initial topography and distance to shoreline with future dune height.\np. 23 L575-586: “Underlying environmental factors\nTo test whether the observed relationship between patch density and dune height reflects underlying environmental gradients, we quantified distance to shoreline and initial elevation and evaluated their associations with local patch density and future dune height. We computed distance to the shoreline as the planar distance from each patch centroid to the mapped shoreline for the year 2013; initial topography was extracted at each patch centroid from the DEM of the year 2010 (in this year vegetation cover was neglectable, see Supplementary Fig. 7). For the correlation between initial topography, distance to shoreline and patch density we used Spearman’s rank correlation. Moreover, for the relation of these variables to future dune height, we used a standardized multiple linear model with future dune height as the response and patch density, distance to shoreline, initial topography, and patch size as predictors (see for details Supplementary Table 1 and Supplementary Fig. 3”\np. 7 L138-143: “Finally, we checked whether local patch density could be a product of underlying environmental factors such as initial topography or distance to the shoreline within our study area. Although, initial topography shows a significant correlation, neither are strong drivers of patch density (Spearman’s ρ = 0.097, p < 0.001 and ρ = 0.019, p = 0.199 respectively) or future dune height (Supplementary Table 1 and Fig. 3).”\nSupplementary Table 1: “Supplementary Table 1 | Model performance for explaining future dune height (2023) from environmental conditions and patch properties.\nStandardized multiple linear models compare the effects of patch density (PD), environmental variables (ENV: distance to shoreline and initial topography), and patch size (PS), alone and in combination. For each model, the table reports the R² and Akaike’s Information Criterion (AIC), together with changes by adding components (ΔR², ΔAIC). Patch density alone explains substantially more variation in future dune height (R² = 0.26; AIC = 11012) than environmental conditions alone (R² = 0.02; AIC = 12244). Adding patch density to the environmental model yields an increase in explanatory power and an improvement in model fit (ENV + PD: ΔR² = +0.29; ΔAIC = −1506), adding patch size provides a smaller but still positive gain (ENV + PD + PS: ΔR² = +0.02; ΔAIC = −149). Values in parentheses correspond to linear models in which quadratic terms for distance to shoreline and patch density were included; these models show similar patterns, with higher R² and lower AIC overall. Distance to shoreline, patch density and patch size were log-transformed prior to analysis. All predictors and the response were standardized using z-scores. N = 4349.”\npatches\nModel R² ΔR² AIC ΔAIC\nPD 0.26 (0.27) / 11012 (11001) /\nENV 0.02 (0.12) -0.24 (-0.15) 12244 (11788) +1232 (+787)\nENV + PD 0.31 (0.38) +0.29 (+0.26) 10738 (10274) -1506 (1514)\nENV + PD + PS 0.33 (0.40) +0.02 (+0.02) 10589 (10124) -149 (150)\nSupplementary Figure 3: “Supplementary Fig. 3 | Standardized effects of environmental conditions and patch properties on future dune height (2023) derived from the models in Supplementary Table 1A. Points show standardized regression coefficients (β), interpreted as the change (in SD units) in dune height associated with a 1 SD increase in each predictor, and horizontal lines indicate cluster-robust 95% confidence intervals. The left panel shows coefficients from the linear models including only main effects, and the right panel shows coefficients from the corresponding models that also include quadratic terms capturing the non-linear relations for distance to shoreline and patch density. Across all models, patch density has a consistently large and positive effect on future dune height, whereas the effects of distance to shoreline and initial elevation are weaker, and patch size makes a more modest additional contribution when added to the ENV + PD model. The dashed vertical line marks zero.”\nComment 2:\nIf the available data allow, the authors should provide direct evidence of functional connectivity between patches. If such evidence cannot be derived from the current dataset, they should instead cite relevant studies that have demonstrated such connectivity, or else develop a mechanistic model of nonlinear dynamics to support the claim (see Comment 3 below).\nReply: We agree with reviewer #1 that the argumentation supporting our functional connectivity interpretation needed strengthening. As requested, we now provide further evidence by performing additional analysis on the available data and by better explaining previously presented analyses:\n(i) In figure 2 we show that the surface area of a single patch (patch size) is a poor predictor of future dune height, and that adding the properties of physically separated neighbors, drastically increases the predictability. We now explicitly mention how this implies functional connectivity.\np. 7 L144-145: “This implies that by building a shared dune body physically separated patches become functionally connected.”\n(ii) We performed additional analyses to highlight the role of (functional) connectivity in a direct way. We calculated the average distance for each patch to the nearest neighboring patches and related this distance to future dune height. This resulted in a S-shaped trend (see figure 5abc of the revised manuscript). This time with a maximum distance threshold, i.e. only below this distance dune height is amplified. This shows it is not just general presence of neighboring vegetation, but the ability to functionally connect that drives dune height.\np. 25 L629-633: “First, we related dune height to the mean nearest-neighbor distance per patch (averaged over the 2, 3, and 5 closest neighbors; centroid-to-centroid). We fitted Hill functions (Eq. 2) to these relations to estimate distance thresholds. For this we used the value of the inflection point k, which we later compared to the theoretically derived 𝑑 .”\n𝑚𝑎𝑥\nWe present the results of this analysis in a new part of the result section under the subheading: ‘A connectivity-based threshold drives nonlinear dune growth’. Here we now also describe two additional analyses supporting a connectivity threshold as explanation of the non-linear relationship between patch density and future dune height, see the response to comment 2 of reviewer #3. These additional results are combined in figure 5.\np. 12 L237-240: “First, we tested whether the maximum interaction distance predicted by percolation theory (~ 4.54 m) appears independently in our data. When relating mean nearest-neighbor distance to dune height, we find an S-shaped relation for which amplified dune growth happens within a given distance (4.12, 4.48 and 5.11 m; Fig. 5abc).”\np13. L260-264: “Fig. 5 | Empirical support for the conceptual model in Figure 4. a, b, c) Dune height (2023) as a function of nearest-neighbor distance (2013), averaged over the 2, 3, and 5 nearest neighbors (colors). Trend lines show Hill-function fits (Eq. (1)), where k represents the distance threshold (vertical dashed lines)…..”\n(iii) Finally, to better support our functional connectivity interpretation from a mechanistic perspective, we now refer to experimental and field studies which show the ability of spatially separated vegetation patches to collectively alter air flow regimes (depending on the spatial density) and therefore potentially sediment deposition rates.\np. 6 L107-111: “In practice, this means that because functional clusters of patches collectively alter air flow33,34, they amplify embryo dune growth by enhancing sand deposition and trapping (Fig. 1c). By contrast, isolated patches do not have that advantage, resulting in slower sand capture and reduced dune growth (Fig. 1b).”\n33. Mayaud, J. R., & Webb, N. P. (2017). Vegetation in drylands: effects on wind flow and aeolian sediment transport. Land, 6(3), 64.\n34. Hesp, P. A., Dong, Y., Cheng, H., & Booth, J. L. (2019). Wind flow and sedimentation in artificial vegetation: Field and wind tunnel experiments. Geomorphology, 337, 165-182.\nComment 3:\nRole of dynamics vs. pure percolation:\nClassical percolation theory is purely structural and does not yield a Hill coefficient. The Hill coefficient typically arises in dynamical systems, such as gene regulation or ecological processes involving nonlinear feedback (see Gao, J., Barzel, B., & Barabási, A.-L. (2016). Universal resilience patterns in complex networks. Nature, 530, 307–312). While a Hill function may empirically fit the data well, the explanation for the observed phenomenon should incorporate a dynamical component in addition to structural percolation.\nReply: We agree with reviewer #1 that classical structural percolation (in an infinite system) does not by itself yield a Hill coefficient, and that the varying Hill coefficient emerges from system dynamics, which was not well explained. To clarify the increase of the hill coefficient in time (Fig. 3b), we added the following:\n(i) We now explicitly coupled the structural percolation analogy, which relates to the initial spatial distribution of dune grass patches, with nonlinear dynamics shaping dune development over time.\np. 14 L299-305: “Coupling biogeomorphic feedbacks with the percolation framework, offers an explanation of why the initial spatial arrangement of dune-grass patches is such a strong predictor of future dune height. It is the percolation-like transition of dune-grass patches into functional clusters that sets the stage for future dune development, after which nonlinear plant–sediment feedbacks (Fig. 1b) shape the observed S-curves to become steeper over time (Fig. 3a). The relatively constant k across years (Fig. 3c) is consistent with a structural connectivity threshold, while the increasing h (Fig. 3b) likely reflects strengthening dynamic feedbacks.”\n(ii) Moreover, to prevent confusion, we now mention (already in the results section) the sub-type of percolation applicable to our system (continuum disk percolation), and more clearly mention we work with a finite-size window. For an infinite system the onset of spanning connectivity is abrupt, for a finite-size window there is a probability of spanning (since here spatial configurations matters) resulting in a percolation zone, rather than a strict threshold. We now reflect on this in the discussion.\np. 11 L205-213: “Indeed, interpreting the inflection point k of the S-shaped relationship as a percolation-like threshold suggests that, around this density, a critical transition occurs: patches become functionally connected forming large “spanning” clusters (Fig. 4a). Below the percolation zone around k, patch densities are too low to form such clusters. Above the percolation zone, the system is “percolated” – i.e. functional clusters have formed – implying that from that point onward additional patches will yield diminishing returns due to saturation of connectivity (Fig. 4a). Assuming each patch affects its surroundings within a circular range of influence, using continuum (disk) percolation36,…”\np. 11 L284-298: “Concepts from percolation theory highlight the role of connectivity in the S-shaped relation between dune height and patch density. For percolation in its classical, structural form (site/bond percolation on lattices)32, connectivity emerges abruptly once occupancy crosses the percolation threshold, typical examples are fluid conduction through porous media37 and electric conductivity38. However, our study system more closely follows the principles of continuum (disk) percolation36, where it is not the entities themselves but their ranges of influence that overlap to form functionally connected clusters, e.g. habitat connectivity39. Importantly, our study system is finite, spatially heterogeneous and noisy: e.g. variation in patch size and geometry, non-random clustering, and stochastic disturbances broaden an idealized system scale transition into a percolation zone, blurring an exact percolation threshold but preserving an overall sigmoidal response. Our finite-size scaling of k and the nearest-neighbor distance test indicate that this noise does not erase the underlying mechanism. Instead, once local ranges of influence overlap sufficiently, this functional connectivity accelerates dune building. Beyond this point, adding even more patches yields diminishing returns as functional clusters already have been formed.”\nComment 4:\nI strongly recommend that the authors develop a nonlinear dynamical model to explain their findings. Such a model could be fitted both to the temporal evolution of dune height and to the final-state relationship with initial density. This approach would not only strengthen the interpretation but also allow the authors to explicitly compare two scenarios: one incorporating functional connectivity and one without it. Demonstrating that the connectivity-based model fits better would substantially reinforce their central hypothesis.\nReply: As explained in the response to comment 2, we strengthened our functional connectivity interpretation with additional empirical data and analyses, and feel this provides more direct evidence compared to a non-linear dynamical simulation model.\n\nReviewer #2 (Remarks to the Author):\nIn the manuscript 'A percolation threshold between grass patches triggers coastal dune formation' the authors present field data showing a persistent correlation between the local height of coastal dunes and the local density of vegetation patches at the beginning of the observation period. From the sigmoidal shape of the relation, and the fact that closely spaced patches trap more sand, the authors extrapolate the existence of a 'percolation threshold' for patch density above which sand is continuously captured by functionally connected patches and dunes form.\nThe focus on the role of patches, instead of individual plants, is certainly interesting, as well as the central argument regarding the relation between patch connectivity, dune growth and percolation theory. However, as discussed in detail below, it seems to me that the authors are in fact just describing how dunes grow instead of providing the causal mechanism for coastal dune formation as they claim in the title. Therefore, I don't recommend publication in the current form.\nMain points:\nComment 1:\nThere is no proof of causality. The central claim is that dunes will only form (in that particular location at least) wherever patch densities are above a given threshold. For that to hold, patch density cannot itself depend on elevation, at least initially (otherwise there is no clear way to disentangle what leads to what), and the initial data should ideally show a wide enough range of patch densities with similar elevations. However, the data presented already shows the trend of higher elevations for denser patches from the beginning. How can the authors prove that this trend is not just a sub-dependence of patch density on elevation?\nThe authors should analyze a subset of their initial data containing only patches with similar elevation. Given the wide dispersion of the elevation data in Fig.2b, it should be possible to find a wide range of patch densities with similar elevations in 2014 and track their elevation over time.\nReply: We thank Reviewer #2 for the constructive feedback and suggestions given. First, we would like to clarify that our claim is not that dunes will only form wherever patch densities are above a given threshold. Instead, we propose that the formation of a functional cluster amplifies dune build-up (but below this threshold dune formation also takes place). To clarify this in the manuscript we modified the following sentences in the abstract and results summary of the discussion section:\np. 2 L10-13: “We identify an S-shaped relationship consistent with a spatial percolation threshold: increasing local patch density triggers an abrupt shift from isolated sand-trapping patches to functionally connected clusters that amplify dune growth.”\np. 14 L274-278: “Specifically, we found that in establishing coastal dune systems, spatially separated neighboring patches interact to amplify sand trapping and thus dune growth compared to single, isolated patches.”\nFurthermore, we also believe that our initial title could be confusing in this regard, see the response to comment 3.\nWe also acknowledge the concerns about the proof of causality related to the proposed sub-dependence of patch density on initial topography. The same concern was raised by reviewer #1 and therefore we want to refer to the response on comment 1 of reviewer #1\nComment 2:\nThe null hypothesis (that patch density is not relevant for dune formation or final size) is not really tested because regions with low patch density are also closer to the shore (from looking at the study site in google earth) and have lower elevation, and therefore are more exposed to flooding and water-driven erosion. This could explain the lack of mature dune formation on those regions: as a combination of more erosion and less vegetation growth.\nAlong these lines, an alternative explanation for the authors' findings is that patch density affects the growth rate of the dunes, but not their formation, nor their final height. Instead, dune formation is ultimately controlled by purely physical factors such as sand supply and water-driven erosion, and thus distance from the shoreline (for erosion) and distance to sand sources. The dependency of dune growth on patch density reported by the authors would just be a consequence of patch density being affected by seawater flooding and erosion and thus dependent on local elevation and distance from shoreline.\nConsequently, the authors should carefully consider the influence of physical factors in their analysis, in particular, the local elevation of the inter-patch areas (i.e. excluding proto-dunes and potentially dune crests), the distance to the shoreline and the main direction of the sand supply. Ideally, they should isolate the effect of patch density by randomizing the physical factors, which right now seems to be highly correlated. One way to visualize the effects of physical parameters could be to plot a color map of: (1) the average dune height vs. patch density and distance from shoreline in 2013 (properly defined); and (2) the average dune height vs. patch density and average elevation of the inter-patch regions (in 2013).\nReply: We agree that additional analyses were needed to disentangle important environmental gradients from the effect of patch density in our analysis. Indeed, a similar remark was made by reviewer #1.\nWhile distance to the shoreline in combination with initial topography can determine erosion rates and therefore dune height, these effects are limited in the case of our study site. See the response to comment 1 of reviewer #1, where we disentangle the effect of patch density, initial topography and distance to the shoreline.\nComment 3:\nBy using the word 'trigger' in the title, the authors argue that a coastal dune can form only in those conditions (patch density above a threshold). However, this is far from obvious because they are ignoring the effect of lateral growth (by rhizomes) of dune-building plants, where in theory a single initial patch in a highly favorable location can trap sand and expand laterally in a positive feedback that would lead to a dune ridge (foredune). In fact, it has been argued that this lateral dune growth driven by the lateral propagation of dune vegetation (as a single, large, patch) could be the main mechanism for post-storm dune recovery from a washover (e.g. Goldstein et al. Earth Surf. Dyn, 2017).\nReply: We agree and changed our title to: ‘A connectivity threshold between grass patches amplifies coastal dune formation’\nMinor points:\nComment 4:\nThe authors didn't find a correlation between dune height and patch size, but what about between dune height and patch area (as defined in the manuscript) over time? I would expect a positive correlation for all years, where taller dunes have larger patch areas (eventually saturating at 100% cover).\nReply: The expectation of reviewer #2 is correct and indeed a larger combined path area results in higher dunes. This because there is a strong co-linearity between patch density and total patch area (more patches means higher cover), as well as a positive effect of cover in general.\np. 8 L153-154: “For the relationship between patch density and combined neighbor area, see Supplementary Fig. 4 and Fig. 5.”\nSupplementary Figure 4: “Supplementary Fig. 4 | Relationship between patch density and total neighbor patch cover. For each focal patch (2013; N = 4349), we quantified local patch patches density (patches m⁻²) and total neighbor cover (summed area of neighboring patches, m²) within a 7 m neighborhood. Each point represents one focal patch; darker regions indicate areas where many points overlap. Patch density is strongly and positively associated with total neighbor cover (Pearson correlation coefficient r = 0.83, 95% CI 0.82–0.84, p < 0.001). Logically, higher patch densities correspond to larger combined patch cover.”\nDue to the strong collinearity, and since we have a relatively confined patch size distribution, it is difficult to fully disentangle the effect of both. However, for total patch areas common along a range of patch densities, we can analyze the effect of different patch densities for fixed covers. This analysis shows that for a given total area cover, dune height increases with an increase in patch density. We now added this analysis, together with the patch size distribution, to supplementary material S4.\nSupplementary Figure 5: “5 Supplementary Fig. 5 | Patch size distribution and patch density effects on dune height for a fixed total cover. a) Patch size distribution for all vegetation patches in 2013 (N = 4349). b) Boxplots of dune height (2023) as a function of the number of patches neighboring patches, for neighborhoods that share approximately the same total vegetated area within a 7 m radius (color panels: 16–18, 18–20, and 20–22 m²). Within each fixed-cover bin, median dune height increases significantly as cover is divided over more neighbors (two-way ANOVA: F = 19.4, p < 0.001), indicating that, when total vegetated area is held constant, 3,1041 spreading that cover across a larger number of patches is associated with higher dunes.”\nComment 5:\nHow can the authors explain the large dispersion in dune height data for a given patch density (sigma/mean ~ 25%) in the context of percolation theory, where the states before and after the transition are well-defined, i.e. deterministic? Assuming sand supply is relatively uniform, a patch density above the threshold should produce a relatively well defined dune height if the percolation argument holds (once patches are connected they should uniformly trap most of the incoming sand).\nReply: We agree that there is a significant variation in the data. However, percolation provides a threshold for connectivity, not necessarily a confined post-threshold state. This is particularly true for an open ecosystem where many other, often stochastic, factors affect the system. We now reflect upon this in the discussion section.\np. 11 L284-298: “Concepts from percolation theory highlight the role of connectivity in the S-shaped relation between dune height and patch density. For percolation in its classical, structural form (site/bond percolation on lattices)32, connectivity emerges abruptly once occupancy crosses the percolation threshold, typical examples are fluid conduction through porous media37 and electric conductivity38. However, our study system more closely follows the principles of continuum (disk) percolation36, where it is not the entities themselves but their ranges of influence that overlap to form functionally connected clusters, e.g. habitat connectivity39. Importantly, our study system is finite, spatially heterogeneous and noisy: e.g. variation in patch size and geometry, non-random clustering, and stochastic disturbances broaden an idealized system scale transition into a percolation zone, blurring an exact percolation threshold but preserving an overall sigmoidal response. Our finite-size scaling of k and the nearest-neighbor distance test indicate that this noise does not erase the underlying mechanism. Instead, once local ranges of influence overlap sufficiently, this functional connectivity accelerates dune building. Beyond this point, adding even more patches yields diminishing returns as functional clusters already have been formed.”\nMoreover, we propose a percolation threshold in a finite window, which should not be confused with the abrupt ‘deterministic’ transition well known in theoretical infinite systems. We now clarified this, please see the response to comment 3 of reviewer #1.\nComment 6:\nIn Fig.3a, the authors should show the full standard deviation. The dispersion of the data for a given patch density is important in the evaluation of a causal explanation.\nReply: We agree with the reviewer. To keep figure 3a readable, we added the full standard deviations for key years in supplementary material H. And refer to this in the caption of figure 3.\np. 9 L172-173: “Shaded areas indicate standard deviation (scaled to 0.2 of actual value for clarity; full standard deviations are shown in Supplementary Fig. 6)”\nSupplementary Figure 6: “Supplementary Fig. 6 | Full standard deviation for key years of figure 3a of the main manuscript. Average dune height for 2010, 2014, 2018 and 2022 as a function of local vegetation patch density in 2013. For 2014, 2018 and 2022 curves follow the hill equation of eq. 1. The year 2010 is derived from alternative elevation data as presented supplementary material S7. Shaded areas indicate standard deviation; N = 4349.”\npatches\n\nReviewer #3 (Remarks to the Author):\n“A percolation threshold between grass patches triggers coastal dune formation”\nBerghuis et al.\nThe authors present novel analyses suggesting that the primary control on dune growth in the formation of coastal dune systems is the initial number density of vegetated patches rather than the size of the initial patches. The authors demonstrate that there is an intermediate range of patch densities at which dune growth increases dramatically with patch density. The lower and upper boundaries of this range define the minimum required patch density to form dunes, and the case where increasing patch density no longer helps to facilitate dune growth. These limits may have important implications for restoration and management projects in coastal dune systems, something which may be increasingly important due to anthropogenic climate change.\nValidity (interpretation and conclusions):\nOverall, I think the authors’ interpretation of the data and the conclusions drawn are reasonable. However, there are some limitations that I feel need to be addressed before the manuscript is publishable. Additional discussion may be sufficient to address some of these, however some extra data analyses may also be required.\nComment 1:\nFirstly, a very minor point and easily addressed, the authors should explicitly state in the discussion that the exact numerical values of e.g. the relevant spatial scale, infection point, and range of influence are specific to the study site and not necessarily universal. Presumably, these precise values would depend on plant-type, granulometry, wind regime etc, and this should be made clear by the authors.\nReply: We thank reviewer #3 for their helpful feedback. We agree that the exact numbers for e.g. the patch density at the inflection point are site specific. We now mention this explicitly in the discussion section.\np. 15 L305-308 “The timing and position of functional cluster formation are context dependent, effected by a site-specific range of influence, which likely varies with sediment size and supply, wind regime, and species traits28,29.”\nComment 2:\nSecondly, a more major concern, the authors have proposed a mechanism which explains the apparent dependence of dune size/growth on patch density based upon the connectivity of the patches i.e. whether they form spanning clusters. The authors claim that the inflexion point on the S-shaped curves represents the point at which spanning clusters are formed; however, I do not see that this is fully justified in the analyses presented. The authors could explicitly determine (using their range of influence) whether each cluster does “span” it’s 7m radius neighborhood. They could then plot the proportion of spanning clusters as a function of patch density to support their claim.\nWithout this, there is an alternative explanation for the data as presented; it could simply be the total vegetation in a cluster that controls dune growth regardless of whether or not it spans the space.\nThe authors show in S3 that the patch density and total vegetation area are strongly linked. This would mean that a similar S-shaped relation would exist between dune size and total vegetation area in a cluster. From a qualitative visual inspection of figure 3d, is appears that the largest individual patches (e.g. ~40% up near the LHS) occur as isolated individuals while clusters consist of multiple similarly-sized patches. If this is correct, and if dune growth depends on local vegetation cover then the shape of the curve in figure 2a would be explained. At small patch size, most patches are in clusters of similarly sized patches. Increasing patch size therefore increases total vegetation cover leading to an increase in dune growth. For large patches, dune size decreases with patch size since the largest patches are isolated, meaning that total local vegetation is reduced even if the individual patch is larger.\nUnder this interpretation, the S-shape that would presumably be apparent would be explained by there being a minimum threshold of vegetation presence to facilitate dune growth and a maximum threshold above which sediment flux, rather than vegetation, limits the growth of the dunes.\nThe authors should either explain why this cannot be the correct interpretation or provide explicit results on connectivity vs. density which support their claimed mechanism. However, this would not change the more general argument that the patches are interacting; it would simply change the exact manner of that interaction.\nReply: We agree that the evidence for the role of connectivity in the relation between patch density and future dune height needed strengthening, as also suggested by reviewer #1. In the response to comment 2 of reviewer #1 we focused on further exploring the occurrence of functional connectivity. Here, we build on this argumentation by providing additional analyses linking patch density and functional connectivity to the spanning cluster formation (following percolation principles).\n(i) As proposed by the reviewer, we now used the presented range of influence to calculate the proportion of spanning clusters formed and relate this to local patch density. We present the results of this analysis in figure 5 of the revised manuscript (see also below), and added this analysis to the newly created ‘A connectivity-based threshold drives nonlinear dune growth’ section of the results.\np. 25 L633-637: “Second, we treated patches as connected when their centroids were ≤ 𝑑 apart and identified connected sets of patches. We classified a set as “spanning” when it bridged a distance needed to connect opposite sides of our 7 m radius neighborhood window. We then quantified, across patch density, the fraction of patches belonging to such spanning components.”\n𝑚𝑎𝑥\np. 12 L241-245: “Second, using that same maximum interaction distance, we quantified “spanning”‐cluster formation as a function of patch density (Fig. 5d). Indeed, the results yield the expected sigmoidal increase. The fraction of patches being part of large clusters rises most steeply near the inflection point k and saturates where the dune-height curves level off (Fig. 3a).”\n(ii) Moreover, the S-shaped dependence on nearest-neighbour distance (see (iii) in the response to comment of reviewer #1), results in a distance threshold near ≈ 4.5. This closely matches the maximum interaction distance independently derived from continuum disk percolation, offering additional support for a density driven connectivity threshold.\np. 25 L629-633: “First, we related dune height to the mean nearest-neighbor distance per patch (averaged over the 2, 3, and 5 closest neighbors; centroid-to-centroid). We fitted Hill functions (Eq. 2) to these relations to estimate distance thresholds. For this we used the value of the inflection point k, which we later compared to the theoretically derived 𝑑 .”\n𝑚𝑎𝑥\np. 12 L237-241: “First, we tested whether the maximum interaction distance predicted by percolation theory (~ 4.54 m) appears independently in our data. When relating mean nearest-neighbor distance to dune height, we find an S-shaped relation for which amplified dune growth happens within a given distance (4.12, 4.48 and 5.11 m; Fig. 5abc). These distances closely match the 4.54 m maximum interaction distance.”\n(iii) Finally, since our inflection point k is defined within a finite analysis window (the 7m radius), a connectivity-driven transition makes a specific prediction: as the window radius increases, the inflection point k should systematically decrease following a characteristic scaling. The results of this analysis are presented in figure 5c and 5d, and included as final analysis in the new ‘A connectivity-based threshold drives nonlinear dune growth’ section.\np. 13 L245-258: “Third, we performed a finite‐size scaling analysis to test whether the density threshold k behaves as expected for a 2-D percolation transition. We re-estimated k for different window sizes (L) with radii (3-24 m; see Supplementary Fig. S10) as the inflection point of the patch density versus dune height curves, computing local patch density as in Eq. (1). In finite systems the apparent percolation threshold depends on the window size (L), interpretating k as the percolation threshold, finite-size scaling theory predicts:\n1\n𝑘(𝐿)= 𝑘 +𝑎𝐿 − 𝜈 (6)\n𝑐\nWhere 𝑘 is the density threshold in an infinite system, 𝑎 is a constant, and 𝜈 is the correlation-length exponent. The universal value for 𝜈 in 2D percolation32 is 𝜈 = 4/3, so 𝑘(𝐿) should vary approximately linearly with 𝐿−0.75. Therefore, to test whether the observed finite-size dependence of our patch density versus dune-height threshold is consistent with 2-D percolation, we regressed 𝑘(𝐿) against 𝐿−0.75.”\np. 13 L245-258: “Third, because our inflection point k is estimated within a finite window (the local neighborhood), we examined how k shifts with window size (L) (Supplementary Fig. 10). In the case k represents a connectivity threshold, percolation theory predicts a characteristic scaling for k: larger windows require less density to connect (more alternative paths), so k (L) decreases and converges. For two-dimensional percolation, k (L) yields a linear relation with L-0.75 (see Eq. (6))32. Consistent with this, Fig. 5e shows that k systematically decreases with window size and then stabilizes, thereby it obeys the hallmark L-0.75 scaling for a connectivity threshold (Fig. 5f). Importantly, two alternative explanations would show a very different scaling trend: (i) if the spatial patch distribution alone (short-range clustering; see Supplementary Fig. 9) was responsible, k logically should increase and then plateau with window size; and (ii) if dune height depended only on total cover (no connectivity), the threshold would stay constant across window sizes. Together, these tests show that once patches are close enough to interact, functional connectivity emerges and dune growth accelerates, consistent with a connectivity-based threshold.”\np. 13 L260-270: “Fig. 5 | Empirical support for the conceptual model in Figure 4. a, b, c) Dune height (2023) as a function of nearest-neighbor distance (2013), averaged over the 2, 3, and 5 nearest neighbors (colors). Trend lines show Hill-function fits (Eq. (1)), where k represents the distance threshold (vertical dashed lines). These thresholds closely match the maximum interaction distance predicted from percolation theory (~4.54 m). Shaded bands show the standard deviation. d) Using this interaction distance, we calculated the fraction of patches in “spanning” clusters (%) as a function of local patch density. The spanning fraction increases most steeply near the inflection point k from Fig. 3c (0.054 ± 0.003 patches m⁻²) and saturates where dune-height curves level off. e) Dependence of k on window radius: k decreases with increasing radius and then approaches a window-independent limit. f) Finite-size scaling test: plotting 𝑘(𝐿) against 𝐿−0.75 (x-axis, labelled by the corresponding radii in metres) indeed yields an approximately linear relationship (R² = 0.94).”\nComment 3:\nOn a related note, if the spanning mechanism is accurate, then, depending on the wind regime, there may be preferred shapes of spanning clusters. For instance, if sand fluxes predominantly occur in a particular direction, then a cluster which spans the flux-perpendicular dimension would presumably capture a different amount of sediment than one that was spanning in the flux-parallel dimension. Depending on the wind regime and sediment availability at the study site, this might call into question whether defining circular neighborhoods really makes sense.\nReply: We agree with the reviewer that a dominant wind direction and sediment supply at the study site could influence the functional connectivity described. Although there is a dominant wind direction (SW), there is also a significant amount of variability in the wind direction. Given this, together with the expected turbulence as wind enters the dune system, we consider a circular neighborhood appropriate for this study, while acknowledging that this is a simplification. We have now added a wind rose in Supplementary Fig. 11 and refer to it in the study site description.\np. 23 L563-565: “Local patch density was calculated by counting the number of neighboring patches within a circular search radius (𝑟 ) around each patch center, assuming isotropic interactions for simplicity.”\n𝑠𝑒𝑎𝑟𝑐ℎ\np. 21 L519-520: “Wind directions vary throughout the year, with South-Western winds being most common (Supplementary Fig. 11).”\nComment 4:\nFinally, the authors show, in S6, that the initial patch density is far more important control on dune height than intermediate densities. The question, therefore, becomes: “What determines the initial density of patches?”. It could be imagined that topographical features or high sediment availability could be contributing factors to the initial distribution of pioneer patches. For example, if grasses are more likely to form initially in an area where there is a high deposition of sediment (and seeds?), then that same high rate of deposition would inherently lead to faster-growing dunes than an area with a lower deposition rate (and so also a lower initial density of vegetation). Thus, it could simply be a coincidence that both the grasses and dunes prefer certain areas of the system to others. Discussion along these lines should also be included.\nReply: We thank reviewer #3 for raising this concern (similar to reviewer #1 and #2). Please see the response to comment 1 of reviewer #1.\nMinor comments:\n1. In figures 2 and S2 the captions describe the binning of the patch size but do not state whether patch density has been binned to produce the figures (and all of the other figures showing patch density). With regularly spaced points it appears that these data have also been binned and the authors should state the width of said bins. Additionally, it would be good to see the raw (unbinned) plots in the supplementary material.\nReply: We did not use any binning for the patch density variable. Since we patch density is derived from the amount of neighboring patches (whole numbers) points are regularly spaced.\n2. Percolation appears in the title and the summary paragraph but is not mentioned again until line 197. The authors may want to consider mentioning it briefly in the introduction.\nReply: We agree and now mention percolation in the introduction.\np. 6 L103-105: “Following spatial percolation principles32, we expect that when patches lie close enough for their zones of influence to overlap, they form functionally connected clusters whose joint engineering capacity outperforms that of isolated patches.”\n3. Also on percolation, when the authors introduce the topic they cite some examples of its recognition in ecological settings (line 202). Percolation has also been applied in a study related to clustering in systems of desert dunes in the following work: Génois, Mathieu, et al. \"Out-of-equilibrium stationary states, percolation, and subcritical instabilities in a fully nonconservative system.\" Physical Review E 94.4 (2016): 042101. Including this additional reference might encourage a wider geomorphological audience.\nReply: We thank the reviewer for this suggestion. However, we prefer not to include this reference, as it could imply mechanistic equivalence between vegetated and non-vegetated dunes, which differ fundamentally in their dynamics.\n\nReviewer #1 (Remarks to the Author):\nI thank the authors for their thorough and constructive revisions. They have addressed all of my major concerns with substantial new analyses, clearer explanations, and additional empirical evidence. In particular, the authors now convincingly demonstrate that the observed nonlinear relation between patch density and dune height cannot be explained by underlying environmental gradients alone. The added aerial imagery, topographic baseline, and statistical tests provide strong support for this claim.\nFurthermore, the new analyses on nearest-neighbor distances, together with the clarified interpretation of functional connectivity and the incorporation of relevant literature, significantly strengthen the argument that clustered patches act collectively to influence dune development. The discussion now more clearly integrates structural percolation concepts with nonlinear biogeomorphic dynamics, satisfactorily addressing my concern regarding the dynamical basis of the observed Hill-function behavior.\nOverall, the revisions greatly improve the clarity, rigor, and interpretability of the manuscript. The authors have responded comprehensively to all comments, and I am satisfied that the concerns have been fully resolved.\nI therefore recommend the manuscript for publication.\n\nReviewer #1 (Remarks on code availability):\nThe code is written in R, a programming language I am not familiar with. However, the README file appears comprehensive and provides a reliable description of the code.\n\nReviewer #2 (Remarks to the Author):\nThe authors successfully addressed all my comments and suggestions. I gladly recommend the publication of the current manuscript.\n\nReviewer #3 (Remarks to the Author):\n“A percolation threshold between grass patches triggers coastal dune formation”\nBerghuis et al.\nThe authors present novel analyses suggesting that the primary control on dune growth in the formation of coastal dune systems is the initial number density of vegetated patches rather than the size of the initial patches. All three reviewers raised similar concerns in the first round that the connectivity mechanism proposed by the authors had not been sufficiently proven. In the revised manuscript the authors have significantly improved the manuscript and present a much more rigorous justification of the mechanism they propose. The authors have addressed all of the concerns that were raised in my first round comments and, as such, I can recommend this article for publication.\nReply: We thank all reviewers for their contributions. Your constructive comments and valuable suggestions enabled us to strengthen our work substantially.\n\nReviewer #3 (Remarks on code availability):\nThe code provided produces the data file on which the analyses described could be performed. The README files describe the flow of running the code and the code itself is well-written and easy to follow. However, the preprocessing stages (i.e. getting the DEMs and identifying the patches using deep learning) are not provided.\nSubsequent analyses (e.g. curve fitting etc) are also not provided.\nThe README describes a potential source for DEM data that could be downloaded to run the code. However, this would not be at the same resolution as the model used in the study. Therefore, it is not possible to reproduce all of the analyses using the code provided.\nReply: We thank reviewer 3 for the final comments on code availability. As described in the method section we did not develop our own code for identifying the patches using deep learning, we used a licensed platform picterra (one can get a subscription, but unfortunately I can not share the code). The raw high resolution DEM maps are not publicly available, and we can therefore not upload this data in the data repository. However, we do provide the patches detected with picterra as shapefiles. Moreover all the high resolution elevation data used in the analysis is provided in the source data file. We included the data behind the fitted curves in the source data file, we now also added the code used for curve fitting."}]

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