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Periodic travelling waves (PTW) are a common solution type of partial differential equations. Such models exhibit multistability of PTWs, typically visualised through the Busse balloon, and parameter changes typically lead to a cascade of wavelength changes through the Busse balloon. In the past, the stability boundaries of the Busse balloon have been used to predict such wavelength changes. Here, motivated by anecdotal evidence from previous work, we provide compelling evidence that the Busse balloon provides insufficient information to predict wavelength changes due to a delayed loss of stability phenomenon. Using two different reaction-advection-diffusion systems, we relate the delay that occurs between the crossing of a stability boundary in the Busse balloon and the occurrence of a wavelength change to features of the essential spectrum of the destabilised PTW. This leads to a predictive framework that can estimate the order of magnitude of such a time delay, which provides a novel "early warning sign" for pattern destabilization. We illustrate the implementation of the predictive framework to predict under what conditions a wavelength change of a PTW occurs.

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The Wildland-Urban Interface (WUI), where vegetation and built-up structures intermingle, encompasses a variety of territorial elements that interact spatially, being variable both in space and time. Mapping the WUI at finer scales is paramount to assess wildfire exposure and define tailored mitigation strategies. Our aim was to develop a semi-automated method to map the WUI at municipal level, leveraging recent advances in data and technology. We tested the procedure in four municipalities of mainland Portugal with different fire history, biophysical conditions, and sociodemographic contexts. We considered WUI as either intermix or interface. Our approach integrates both building location data and high-resolution vegetation maps, to calculate the density of buildings and forest cover proportion within different circular moving window sizes. Within each radius, we evaluated the total area and spatial distribution of the WUI types, as well as the number of buildings within WUI and within the fire perimeters recorded between the years 2000 and 2022 and analysed the differences between municipalities. We then compared the mapped WUI with previous WUI mappings for mainland Portugal, to identify common spots and potential spatial divergences. We found that the area mapped as WUI within all four municipalities ranged from about 400 km2 to 1135 km2 depending on the radius size. A distinct distribution for each type of WUI was observed as the radius size increased: the intermix WUI showed a tendency to increase, and the interface WUI increased only between the radius of 100 and 200 m, decreasing gradually in subsequent radii. Between 39.4% and 45.5% of the nearly 200,000 buildings in the study areas were within WUI, depending on radius size and a total of 5436 buildings were within the historic fire perimeter. Although the comparison with other maps showed fair agreement, due to differences in data and methodology, common areas mapped as WUI were found, which suggests that these areas should receive greater attention from decision-makers regarding fire management strategies, since their classification as WUI remains consistent across different methodologies.

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Spatial patterns in plant community structures within stressed ecosystems have drawn much attention in the field of ecology. However, the mechanisms underlying spatial formation and its impact on species coexistence and diversity remain controversial. In this study, we investigated concentric circular vegetation patches in coastal saline land, and analysed the spatial patterning of plant communities and associated soil physicochemical properties. Thereafter, we tested how the soil conditioned by plant communities from different locations within the vegetation patches influence the species growth and inter-specific competition. Our results show soil salinity enlarges in a centrifugal manner in horizontal direction in all patches. Soil salinity decreased and species diversity increased along with the increase of patch size. In addition, we found significant shifts in both the composition of plant communities and in soil physicochemical properties from outer to center. The results indicate that the pioneer species Suaeda salsa facilitated the subsequent species. However Suaeda salsa was inhibited and became inferior competitor in the soil conditioned by the subsequent species. We infer that the less-visible spatial patterns of soil physicochemical properties at small scales create ecological niches for specialized species, allowing them to coexist but not mix. We suggest that a trade-off between tolerance to salt stress and competitive ability under ameliorated conditions may underlie mechanisms of pattern formation in small scale. Our findings lend support to the idea that soil stress constraints community assembly and triggers spatial patterns, which, in turn, buffer the stress on plant communities and enhance species diversity.

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Despite being one of the most pristine regions in the world, Antarctica is currently also one of the most vulnerable to climate change. Antarctic vegetation comprises mostly lichens and bryophytes, complemented in some milder regions of Maritime Antarctica by two vascular plant species. Shifts in the spatial patterns of these three main vegetation groups have already been observed in response to climate change, highlighting the urgent need for the development of comprehensive large-scale ecological models of the effects of climate change. Besides climate, Antarctic terrestrial vegetation is also strongly influenced by non-climatic microscale conditions related to abiotic and biotic factors. Nevertheless, the quantification of their importance in determining vegetation patterns remains unclear. The objective of this work was to quantify the importance of abiotic and biotic microscale conditions in determining the spatial cover patterns of the major functional types, lichens, vascular plants and bryophytes, explicitly determining the likely confinement of each functional type to the microscale conditions, i.e., their ecological niche. Microscale explained >60 % of the spatial variation of lichens and bryophytes and 30 % of vascular plants, with the niche analysis suggesting that each of the three functional types may be likely confined to specific microscale conditions in the studied gradient. Models indicate that the main microscale ecological filters are abiotic but show the potential benefits of including biotic variables and point to the need for further clarification of vegetation biotic interactions' role in these ecosystems. Altogether, these results point to the need for the inclusion of microscale drivers in ecological models to track and forecast climate change effects, as they are crucial to explain present vegetation patterns in response to climate, and for the interpretation of ecological model results under a climate change perspective.

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Vegetation patterns on slopes strongly affect the water cycle processes in a basin, especially the water yield and confluence in arid areas. Quantifying and evaluating the effects of hydrological change on the migration and transformation of pollutants are challenging. Based on 4-year stream water quality data of 13 monitoring sites in the Huangshui River basin, a typical arid watershed of the Chinese Loess Plateau, the redundancy analysis (RDA) and structural equation modeling (SEM) analysis tools were used to quantify its relationship with vegetation patterns. In the study, land use and the enhanced vegetation index (EVI) were used as a metric of vegetation patterns; accordingly, the 13 catchments were divided into three groups via the cluster analysis, including large (over 80%), medium (70 ~ 80%), and small (below 70%) proportion vegetation patterns (LVP, MVP, SVP). The results of the LVP group showed that vegetation patterns negatively affected the contamination of total phosphorus (TP), ammonia nitrogen (NH3-N), permanganate index (CODMn), and biochemical oxygen demand (BOD5) in the stream water, and the contribution rates were - 0.57. While the proportion of urban area positively correlated with stream water quality in the groups of MVP and SVP, the contribution rates were 0.46 and 0.36, respectively. Moreover, the precipitation in the groups of MVP and SVP negatively correlated with pollutants (- 0.24 and - 0.26). Those results revealed the response of stream water quality to vegetation patterns on the slope with the consideration of precipitation, land use, and socio-economic factors for the regional water and land resource allocation. This study has important management implications for vegetation patterns on slope of fragile ecosystems in arid areas.

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Regular spatial patterns of vegetation are a common sight in drylands. Their formation is a population-level response to water stress that increases water availability for the few via partial plant mortality. At the individual level, plants can also adapt to water stress by changing their phenotype. Phenotypic plasticity of individual plants and spatial patterning of plant populations have extensively been studied independently, but the likely interplay between the two robust mechanisms has remained unexplored. In this paper, we incorporate phenotypic plasticity into a multi-level theory of vegetation pattern formation and use a fascinating ecological phenomenon, the Namibian "fairy circles," to demonstrate the need for such a theory. We show that phenotypic changes in the root structure of plants, coupled with pattern-forming feedback within soil layers, can resolve two puzzles that the current theory fails to explain: observations of multi-scale patterns and the absence of theoretically predicted large-scale stripe and spot patterns along the rainfall gradient. Importantly, we find that multi-level responses to stress unveil a wide variety of more effective stress-relaxation pathways, compared to single-level responses, implying a previously underestimated resilience of dryland ecosystems.

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Spatiotemporal patterns of vegetation are a ubiquitous feature of semi-arid ecosystems. On sloped terrain, vegetation patterns occur as stripes perpendicular to the contours. Field studies report contrasting long-term dynamics between different observation sites; some observe slow uphill migration of vegetation bands while some report stationary patterns. In this paper, we show that long-range seed dispersal provides a mechanism that enables the occurrence of both migrating and stationary patterns. We utilise a nonlocal PDE model in which seed dispersal is accounted for by a convolution term. The model represents vegetation patterns as periodic travelling waves and numerical continuation shows that both migrating and almost stationary patterns are stable if seed dispersal distances are sufficiently large. We use a perturbation theory approach to obtain analytical confirmation of the existence of almost stationary patterned solutions and provide a biological interpretation of the phenomenon.

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Model-aided understanding of the mechanism of vegetation patterns and desertification is one of the burning issues in the management of sustainable ecosystems. A pioneering model of vegetation patterns was proposed by C. A. Klausmeier in 1999 (Klausmeier, 1999) that involves a downhill flow of water. In this paper, we study the diffusive Klausmeier model that can describe the flow of water in flat terrain incorporating a diffusive flow of water. It consists of a two-component reaction-diffusion system for water and plant biomass. The paper presents a numerical bifurcation analysis of stationary solutions of the diffusive Klausmeier model extensively. We numerically investigate the occurrence of diffusion-driven instability and how this depends on the parameters of the model. Finally, the model predicts some field observed vegetation patterns in a semiarid environment, e.g. spot, stripe (labyrinth), and gap patterns in the transitions from bare soil at low precipitation to homogeneous vegetation at high precipitation. Furthermore, we introduce a two-component reaction-diffusion model considering a bilinear interaction of plant and water instead of their cubic interaction. It is inspected that no diffusion-driven instability occurs as if vegetation patterns can be generated. This confirms that the diffusive Klausmeier model is the minimal reaction-diffusion model for the occurrence of vegetation patterns from the viewpoint of a two-component reaction-diffusion system.

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The savanna biome is characterised by a continuous vegetation cover, comprised of herbaceous and woody plants. The coexistence of species in arid savannas, where water availability is the main limiting resource for plant growth, provides an apparent contradiction to the classical principle of competitive exclusion. Previous theoretical work using nonspatial models has focussed on the development of an understanding of coexistence mechanisms through the consideration of resource niche separation and ecosystem disturbances. In this paper, we propose that a spatial self-organisation principle, caused by a positive feedback between local vegetation growth and water redistribution, is sufficient for species coexistence in savanna ecosystems. We propose a spatiotemporal ecohydrological model of partial differential equations, based on the Klausmeier reaction-advection-diffusion model for vegetation patterns, to investigate the effects of spatial interactions on species coexistence on sloped terrain. Our results suggest that species coexistence is a possible model outcome, if a balance is kept between the species' average fitness (a measure of a species' competitive abilities in a spatially uniform setting) and their colonisation abilities. Spatial heterogeneities in resource availability are utilised by the superior coloniser (grasses), before it is outcompeted by the species of higher average fitness (trees). A stability analysis of the spatially nonuniform coexistence solutions further suggests that grasses act as ecosystem engineers and facilitate the formation of a continuous tree cover for precipitation levels unable to support a uniform tree density in the absence of a grass species.

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Vegetation patterns are a ubiquitous feature of water-deprived ecosystems. Despite the competition for the same limiting resource, coexistence of several plant species is commonly observed. We propose a two-species reaction-diffusion model based on the single-species Klausmeier model, to analytically investigate the existence of states in which both species coexist. Ecologically, the study finds that coexistence is supported if there is a small difference in the plant species' average fitness, measured by the ratio of a species' capabilities to convert water into new biomass to its mortality rate. Mathematically, coexistence is not a stable solution of the system, but both spatially uniform and patterned coexistence states occur as metastable states. In this context, a metastable solution in which both species coexist corresponds to a long transient (exceeding [Formula: see text] years in dimensional parameters) to a stable one-species state. This behaviour is characterised by the small size of a positive eigenvalue which has the same order of magnitude as the average fitness difference between the two species. Two mechanisms causing the occurrence of metastable solutions are established: a spatially uniform unstable equilibrium and a stable one-species pattern which is unstable to the introduction of a competitor. We further discuss effects of asymmetric interspecific competition (e.g. shading) on the metastability property.

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Currently R is the most popular software for data analyses among biologists. Here, we present bwimage, a package designed to describe patterns from black and white images. The package can be used for a wide range of applications. We implemented functions previously described in the literature to calculate parameters designed originally, but not exclusively, for vegetation structures. Additionally, we propose a new parameter: the aggregation index. We demonstrate applications for field work, providing examples that range from calculation of canopy openness, description of patterns in vertical vegetation structure, to patterns in bird nest structure. We provide advice and illustrated examples of how to produce high quality images for analyses.

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We report for the first time on the formation of spirals like vegetation patterns in isotropic and uniform environmental conditions. The vegetation spirals are not waves and they do not rotate. They belong to the class of dissipative structures found out of equilibrium. Isolated or interacting spirals and arcs observed in South America (Bolivia) and North Africa (Morocco) are interpreted as a result of curvature instability that affects the circular shape of localized patches. The biomass exhibits a dynamical behaviour with arcs that transform into spirals. Interpretation of observations and of the predictions provided by the theory is illustrated by recent measurements of peculiar plant morphology (the alfa plant, or Stipa tenacissima L.) originated from northwestern Africa and the southern part of the Iberian Peninsula.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)'.

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Banded patterns consisting of alternating bare soil and dense vegetation have been observed in water-limited ecosystems across the globe, often appearing along gently sloped terrain with the stripes aligned transverse to the elevation gradient. In many cases, these vegetation bands are arced, with field observations suggesting a link between the orientation of arcing relative to the grade and the curvature of the underlying terrain. We modify the water transport in the Klausmeier model of water-biomass interactions, originally posed on a uniform hillslope, to qualitatively capture the influence of terrain curvature on the vegetation patterns. Numerical simulations of this modified model indicate that the vegetation bands arc convex-downslope when growing on top of a ridge, and convex-upslope when growing in a valley. This behaviour is consistent with observations from remote sensing data that we present here. Model simulations show further that whether bands grow on ridges, valleys or both depends on the precipitation level. A survey of three banded vegetation sites, each with a different aridity level, indicates qualitatively similar behaviour.

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Spatial self-organization of dryland vegetation constitutes one of the most promising indicators for an ecosystem's proximity to desertification. This insight is based on studies of reaction-diffusion models that reproduce visual characteristics of vegetation patterns observed on aerial photographs. However, until now, the development of reliable early warning systems has been hampered by the lack of more in-depth comparisons between model predictions and real ecosystem patterns. In this paper, we combined topographical data, (remotely sensed) optical data, and in situ biomass measurements from two sites in Somalia to generate a multilevel description of dryland vegetation patterns. We performed an in-depth comparison between these observed vegetation pattern characteristics and predictions made by the extended-Klausmeier model for dryland vegetation patterning. Consistent with model predictions, we found that for a given topography, there is multistability of ecosystem states with different pattern wavenumbers. Furthermore, observations corroborated model predictions regarding the relationships between pattern wavenumber, total biomass, and maximum biomass. In contrast, model predictions regarding the role of slope angles were not corroborated by the empirical data, suggesting that inclusion of small-scale topographical heterogeneity is a promising avenue for future model development. Our findings suggest that patterned dryland ecosystems may be more resilient to environmental change than previously anticipated, but this enhanced resilience crucially depends on the adaptive capacity of vegetation patterns.

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Self-organised patterns of vegetation are a characteristic feature of semi-deserts. In this paper, we find the internal competition reaction between plants can lead to the formation of spatial vegetation patterns, which is different from the normal formation mechanism of long-range competition and short-range facilitation. It is also found that the internal competition between biomass has an effect on the characters of banded vegetation. In particular, it is shown that the internal competition reaction leads to the decrease of the migration speed and the increase of the wavelength of the stripe patterns.

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Vegetation maps are models of the real vegetation patterns and are considered important tools in conservation and management planning. Maps created through traditional methods can be expensive and time-consuming, thus, new more efficient approaches are needed. The prediction of vegetation patterns using machine learning shows promise, but many factors may impact on its performance. One important factor is the nature of the vegetation-environment relationship assessed and ecological redundancy. We used two datasets with known ecological redundancy levels (strength of the vegetation-environment relationship) to evaluate the performance of four machine learning (ML) classifiers (classification trees, random forests, support vector machines, and nearest neighbor). These models used climatic and soil variables as environmental predictors with pretreatment of the datasets (principal component analysis and feature selection) and involved three spatial scales. We show that the ML classifiers produced more reliable results in regions where the vegetation-environment relationship is stronger as opposed to regions characterized by redundant vegetation patterns. The pretreatment of datasets and reduction in prediction scale had a substantial influence on the predictive performance of the classifiers. The use of ML classifiers to create potential vegetation maps shows promise as a more efficient way of vegetation modeling. The difference in performance between areas with poorly versus well-structured vegetation-environment relationships shows that some level of understanding of the ecology of the target region is required prior to their application. Even in areas with poorly structured vegetation-environment relationships, it is possible to improve classifier performance by either pretreating the dataset or reducing the spatial scale of the predictions.

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Mountain vegetation is highly specialized to harsh climatic conditions and therefore is sensitive to any change in environment. The rarest and most vulnerable plants occurring in alpine regions are expected to respond rapidly to environmental changes. An example of such a species is Carex lachenalii subsp. lachenalii Schkuhr, which occurs in Poland on only a few isolated sites in the Tatra Mts. The aim of this study was to assess changes in distribution of C. lachenalii in the Tatra Mts over the past 50-150years and the effects of climate change, tourism and historical grazing on the ecological niche of C. lachenalii. We focused on changes in the importance of functional diversity components in shaping plant species composition. Over the past 50-150years, the elevation of the average distribution of C. lachenalii shifted about 178m upward alongside a significant prolongation of the vegetative season by approximately 20days in the last 50-60years. Species composition of plots without C. lachenalii was characterized by competition between plants, whereas on plots with C. lachenalii habitat filtering was the most important component. Our results suggest that climate change was the main factor driving upward shift of C. lachenalii. Moderate trampling enhanced horizontal spread of this plant, whereas cessation of grazing grazing caused decline of C. lachenalii. The three environmental factors studied that determined shifts in distribution of C. lachenalii may also contribute to changes in distribution of other rare mountain plant species causing changes in ecosystem functioning.

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Vegetation in dryland ecosystems often forms remarkable spatial patterns. These range from regular bands of vegetation alternating with bare ground, to vegetated spots and labyrinths, to regular gaps of bare ground within an otherwise continuous expanse of vegetation. It has been suggested that spotted vegetation patterns could indicate that collapse into a bare ground state is imminent, and the morphology of spatial vegetation patterns, therefore, represents a potentially valuable source of information on the proximity of regime shifts in dryland ecosystems. In this paper, we have developed quantitative methods to characterize the morphology of spatial patterns in dryland vegetation. Our approach is based on algorithmic techniques that have been used to classify pollen grains on the basis of textural patterning, and involves constructing feature vectors to quantify the shapes formed by vegetation patterns. We have analysed images of patterned vegetation produced by a computational model and a small set of satellite images from South Kordofan (South Sudan), which illustrates that our methods are applicable to both simulated and real-world data. Our approach provides a means of quantifying patterns that are frequently described using qualitative terminology, and could be used to classify vegetation patterns in large-scale satellite surveys of dryland ecosystems.

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Understanding how desertification takes place in different ecosystems is an important step in attempting to forecast and prevent such transitions. Dryland ecosystems often exhibit patchy vegetation, which has been shown to be an important factor on the possible regime shifts that occur in arid regions in several model studies. In particular, both gradual shifts that occur by front propagation, and abrupt shifts where patches of vegetation vanish at once, are a possibility in dryland ecosystems due to their emergent spatial heterogeneity. However, recent theoretical work has suggested that the final step of desertification - the transition from spotted vegetation to bare soil - occurs only as an abrupt shift, but the generality of this result, and its underlying origin, remain unclear. We investigate two models that detail the dynamics of dryland vegetation using a markedly different functional structure, and find that in both models the final step of desertification can only be abrupt. Using a careful numerical analysis, we show that this behavior is associated with the disappearance of confined spot-pattern domains as stationary states, and identify the mathematical origin of this behavior. Our findings show that a gradual desertification to bare soil due to a front propagation process can not occur in these and similar models, and opens the question of whether these dynamics can take place in nature.

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The minimal ecological requirements for the fomation of regular vegetation patterns in semiarid systems have been recently questioned. Against the general belief that a combination of facilitative and competitive interactions is necessary, recent theoretical studies suggest that, under broad conditions, non-local competition among plants alone may induce patterns. In this paper, we review results along this line, presenting a series of models that yield spatial patterns when finite-range competition is the only driving force. A preliminary derivation of this type of model from a more detailed one that considers water-biomass dynamics is also presented.