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Spatially heterogeneous stressors can alter the performance of indicators of regime shifts

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... While catastrophic shifts have been observed and analyzed for a wide variety of ecological systems, including lakes , peatlands (Swindles et al., 2018), marine systems (Lees et al., 2006), mussel beds (Génin et al., 2018a) forests (Magnuszewski et al., 2015), and rangelands (Kéfi et al., 2007), using a range of different approaches and frameworks at different scales, persistent challenges remain, and different approaches and methods are rarely sufficiently brought together. In particular, there is a need to understand more clearly the thresholds or tipping points that may result in catastrophic shifts, and to identify the early warning signs and overall impacts, such that appropriate measures can be put in place to manage environmental resources more sustainably. ...
... We developed a suite of different vegetation models that contributed to the fundamental understanding of the plant-soil responses of Mediterranean ecosystems to degradation drivers (Mayor et al., 2013(Mayor et al., , 2019Siteur et al., 2014;Schneider and Kéfi, 2016;Génin et al., 2018a;Baudena et al., 2020). Models that incorporate spatially explicit grazing (i.e., that grazers tend to eat at the borders of vegetation patches rather than in their centers), predicted a higher probability and magnitude of catastrophic shifts under increasing stress (drought or grazing intensity) than previous models assuming homogeneous grazing (Schneider and Kéfi, 2016). ...
... the early warning signals at high grazing pressure. This suggests that we need to be cautious regarding the use of early warning signals of ecosystem degradation when the pressure at play has a spatially explicit component (see also Génin et al., 2018a). ...
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One of the most challenging issues in Mediterranean ecosystems to date has been to understand the emergence of discontinuous changes or catastrophic shifts. In the era of the 2030 Sustainable Development Goals, which encompass ideas around Land Degradation Neutrality, advancing this understanding has become even more critical and urgent. The aim of this paper is to synthesize insights into the drivers, processes and management of catastrophic shifts to highlight ways forward for the management of Mediterranean ecosystems. We use a multidisciplinary approach that extends beyond the typical single site, single scale, single approach studies in the current literature. We link applied and theoretical ecology at multiple scales with analyses and modeling of human–environment–climate relations and stakeholder engagement in six field sites in Mediterranean ecosystems to address three key questions: i) How do major degradation drivers affect ecosystem functioning and services in Mediterranean ecosystems? ii) What processes happen in the soil and vegetation during a catastrophic shift? iii) How can management of vulnerable ecosystems be optimized using these findings? Drawing together the findings from the use of different approaches allows us to address the whole pipeline of changes from drivers through to action. We highlight ways to assess ecosystem vulnerability that can help to prevent ecosystem shifts to undesirable states; identify cost-effective management measures that align with the vision and plans of land users; and evaluate the timing of these measures to enable optimization of their application before thresholds are reached. Such a multidisciplinary approach enables improved identification of early warning signals for discontinuous changes informing more timely and cost-effective management, allowing anticipation of, adaptation to, or even prevention of, undesirable catastrophic ecosystem shifts.
... In addition to comparing pattern morphology, features such as the wavelength and orientation of bands relative to the local slope , Penny et al., 2013, the width of bands [Gowda et al., 2018], the local biomass , and arc curvature [Gandhi et al., 2018a], have been identified using satellite or aerial imagery and compared to model predictions. Other work has focused on vegetation patch size distribution [Kéfi et al., 2007] as an indicator of desertification [Berdugo et al., 2017, Génin et al., 2018, Moreno-de las Heras et al., 2011]. ...
... We note that there is an alternative approach to early warning signs, not based in pattern, but instead based on a statistical physics analysis of the scaling distribution of vegetation patch size, and its deviation from a power-law distribution [Kéfi et al., 2007]. Spatial heterogeneity has been noted in this case to impact the performance of early warning indicators [Génin et al., 2018]. For a more comprehensive review of early warning signs in the context of spatial patterns in ecological systems, see [Kefi et al., 2014]. ...
Preprint
This is a book chapter, written as a contribution to a new edition of "Dryland Ecohydrology", edited by Paolo D'Odorico, Amilcare Porporato, and Christiane Runyan, (to appear, Springer 2019). It aims to (1) describe some of the background to conceptual mathematical models of spontaneous pattern formation, in the context of dryland vegetation patterns, and (2) review some of the observational studies of the phenomenon. The chapter also highlights challenges and opportunities associated with the development of the models in light of increasing availability of remote sensing data. This includes both satellite imagery of the patterns and elevation data of the topography. The vast scales, in time and space, associated with the key processes further suggest avenues for improved mathematical modeling paradigms.
... In contrast, rising spatial or temporal autocorrelation and variance sometimes, but not always, predict the onset of tipping points (Carpenter and Brock, 2011;Dakos et al., 2012). In some systems these indicators may even decrease before critical transitions, causing several authors to emphasize the need for caution and good system knowledge before interpreting them G enin et al., 2018). In addition, most of the work evaluating the performance of early-warning indicators has not been performed in agroecosystems, where inherent variability and heterogeneity of cropping patterns may strongly impact the meaning and applicability of such signals (Vandermeer, 2011). ...
... Critically, this means that systems with different resilience to a known disturbance can be compared in terms of baseline biodiversity or statistical properties, and associated resilience indicators either invalidated or confirmed (e.g. Cariveau et al., 2013;G enin et al., 2018;Isbell et al., 2015). Addressing question (i) is essential to increase our ability to predict (anticipate) the consequences of observed environmental change on agroecosystem functions. ...
Article
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Predicting the resilience of biodiversity-driven functions in agroecosystems to drivers of environmental change (EC) is of critical importance to ensure long-term and environmentally safe agricultural production. However, operationalizing resilience of such functions is challenging, because conceptual approaches differ, direct measures are difficult, and the validity and interpretation of existing indicators are unclear. Here, we (1) summarize dimensions of resilience that apply in agroecosystems, and the disturbances they are subject to under EC. We then (2) review indicators of the resilience of biodiversity-driven functions in agroecosystems, and their support in theoretical and empirical studies. (3) Using these indicators, we examine what can be learned for the resilience of these functions to drivers of EC, focussing on the ecosystem services of biological pest control, biological disease control in soil and pollination. We conclude (4) that research into the resilience of these services is still in its infancy, but novel tools and approaches can catalyse further steps to assess and improve the resilience of biodiversity-driven agroecosystem functions under EC.
... Researchers have employed spatial correlates of classic early warning indicators (e.g., variance, autocorrelation, skewness) with limited success [20][21][22][23][24][25][26] due to a number of concerns over inconsistent results and detection accuracy [27][28][29], although artificial intelligence approaches may help overcome some of these problems [30]. In addition, many of these methods require temporal data to document and verify changes in results from spatial approaches [31][32][33]. ...
Article
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Studying ecosystem dynamics is critical to monitoring and managing linked systems of humans and nature. Due to the growth of tools and techniques for collecting data, information on the condition of these systems is more widely available. While there are a variety of approaches for mining and assessing data, there is a need for methods to detect latent characteristics in ecosystems linked to temporal and spatial patterns of change. Resilience-based approaches have been effective at not only identifying environmental change but also providing warning in advance of critical transitions in social-ecological systems (SES). In this study, we examine the usefulness of one such method, Fisher Information (FI) for spatiotemporal analysis. FI is used to assess patterns in data and has been established as an effective tool for capturing complex system dynamics to include regimes and regime shifts. We employed FI to assess the biophysical condition of eighty-five Swedish lakes from 1996–2018. Results showed that FI captured spatiotemporal changes in the Swedish lakes and identified distinct spatial patterns above and below the Limes Norrlandicus, a hard ecotone boundary which separates northern and southern ecoregions in Sweden. Further, it revealed that spatial variance changed approaching this boundary. Our results demonstrate the utility of this resilience-based approach for spatiotemporal and spatial regimes analyses linked to monitoring and managing critical watersheds and waterbodies impacted by accelerating environmental change.
... microbial acclimation to elevated temperatures (Bradford et al., 2008, Kaiser et al., 2014, no longer align with the new peat-surface temperature signature. If spatial changes result in process rates that breech thresholds and tipping points, they could irreversibly shift the balance of key system feedbacks (Belyea, 2009, Rietkerk & van de Koppel, 2008, Waddington et al., 2015 and cause major shifts in system functioning (Génin et al., 2018, Johnstone et al., 2016, Schneider & Kefi, 2016. ...
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Temperatures at the soil–atmosphere interface influence ecosystem function by driving nonlinear terrestrial biogeochemical, ecohydrological, and micrometeorological processes. Whilst climate, soil and vegetation controls on spatially average ecosystem temperatures are recognised, how interacting and heterogeneous ecosystem layers create spatio-temporal complex thermal ecosystems has not been determined. Such thermal hot spots and hot moments may underpin the capability of ecosystems to support biological and biogeochemical diversity and control the likelihood of tipping points in system-regulating feedbacks being locally exceeded. This is of notable importance in peatlands, where soil temperatures control the storage of their associated globally important carbon stocks. Here, through the application of high spatio-temporal resolution surface temperature data and peat thermal modelling, we assess the impact of system heterogeneity (spatio-temporal impact of the following system layers: tree, shrubs, microtopography, groundcover species and sub-surface ice) on surface temperature regimes. We show (a) that peat-surface thermal hotspot intensity and longevity is linked to system heterogeneity and (b) that not all system layers have an equal influence over the peat-surface thermal regime and extreme temperatures; thermal heterogeneity increases up to a maximum of five layers of heterogeneity and decreases thereafter. The results crucially demonstrate that such changes in the spatio-temporal thermal dynamics and extremes may occur without significant changes in median temperatures. This is important to the conceptual understanding of peatland responses and ecosystem resilience to disturbance. It emphasises the need to determine the potential for transitions in magnitude, longevity and locality of small-scale thermal extremes to induce functional transitions that propagate through given ecosystems, and to characterise the impact of such small-scale spatio-temporal complexity on ecosystem scale biogeochemical and ecohydrological function.
... Resilience has an explicit spatial component. It can become encoded in the abiotic and biotic elements of a landscape (Allen et al. 2016;Cumming et al. 2017;Génin et al. 2018). Whereas a single vegetated dune landform can impart resistance, resilience encompasses how its response and recovery is shaped by the configuration of the larger landscape and how history has shaped the assemblage of topography-modifying organisms. ...
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When topography is incorporated into models of barrier dune dynamical states, how it is represented determines the dynamical properties inferred. Bottom-up representations rely on elevation and localized biogeomorphic modification. Top-down representations incorporate constraints imposed by the spatial patterns of topography. These spatial patterns emerge from island morphological context and the extent localized biogeomorphic processes can expand and structure the larger landscape. We compared topographies across 30 sites among seven barrier islands of the Virginia (U.S.A) coast to gauge the importance of elevation, the bottom-up variable often weighted most in dune biogeomorphic models, relative to top-down patch and continuous surface landscape representations of topography. LiDAR-derived digital elevation models of each site were characterized with non-metric multidimensional scaling to assess how these bottom-up and top-down metrics structured dune topographic variability. Multiple response permutation procedures gauged the strength of topographic differences among sites grouped according to island morphology versus groupings defined by clustering of topographic metrics. Elevation was the dominant metric structuring topography for these low relief islands. Spatial structure was weakly developed. Topographic differences were more robust when based on clusters defined largely by elevational properties rather than by island or island morphological type. For the Virginia barrier islands, storm inputs may more directly shape topography and override landscape-extent top-down spatial structure. The dominance of elevation suggests that resistance may be the more relevant dynamical property for this coast. Properties like resilience may be greater on higher islands with longer storm-free intervals in which biogeomorphic elements can configure relief and act as recursive top-down controls.
... Other work has then quantified the effects of stressors on population dynamics in a single field or laboratory population (Hendriks et al. 2005). However, much less is understood about the effects of stressors at the global level (i.e., metapopulation, defined as the sum of abundances of individuals over all local populations), specifically when stressors are heterogeneously distributed over a species range (Spromberg et al. 1998;Fritsch et al. 2010;Genin et al. 2018). ...
Article
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Stressors such as antibiotics, herbicides and pollutants are becoming increasingly common in the environment. The effects of stressors on populations are typically studied in homogeneous, non-spatial settings. However, most populations in nature are spatially distributed over environmentally heterogeneous landscapes with spatially-restricted dispersal. Little is known about the effects of stressors in these more realistic settings. Here, we combine laboratory experiments with novel mathematical theory to rigorously investigate how a stressor's physiological effect and spatial distribution interact with dispersal to influence population dynamics. We prove mathematically that if a stressor increases death rate and/or simultaneously decreases population growth rate and yield, a homogeneous distribution of stressor leads to a lower total population size than if the same amount of stressor was heterogeneously distributed. We experimentally test this prediction on spatially-distributed populations of budding yeast, Saccharomyces cerevisiae. We find that the antibiotic, cycloheximide, increases yeast death rate but reduces growth rate and yield. Consistent with our mathematical predictions, we observe that a homogeneous spatial distribution of cycloheximide minimizes the total equilibrium size of experimental metapopulations, with the magnitude of the effect depending predictably on dispersal rate and geographic pattern of antibiotic heterogeneity. Our study has implications for assessing population risk posed by pollutants, antibiotics, and global change, and in the rational design of strategies for employing toxins to control pathogens and pests.
... Their utility is being confirmed in empirical studies [24][25][26][27], but their trends may not be consistent in self-organized, patterned spatial systems because environmental changes other than an impeding regime shift may be driving trends in the indicator [28]. Spatially heterogeneous stressors also appear to confound the detection of a CSD-signal [29,30]. ...
Article
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Given the intensity and frequency of environmental change, the linked and cross-scale nature of social-ecological systems, and the proliferation of big data, methods that can help synthesize complex system behavior over a geographical area are of great value. Fisher information evaluates order in data and has been established as a robust and effective tool for capturing changes in system dynamics, including the detection of regimes and regime shifts. The methods developed to compute Fisher information can accommodate multivariate data of various types and requires no a priori decisions about system drivers, making it a unique and powerful tool. However, the approach has primarily been used to evaluate temporal patterns. In its sole application to spatial data, Fisher information successfully detected regimes in terrestrial and aquatic systems over transects. Although the selection of adjacently positioned sampling stations provided a natural means of ordering the data, such an approach limits the types of questions that can be answered in a spatial context. Here, we expand the approach to develop a method for more fully capturing spatial dynamics. The results reflect changes in the index that correspond with geographical patterns and demonstrate the utility of the method in uncovering hidden spatial trends in complex systems.
... In Chapter 2 of this work (Génin et al., 2018a), we focused on indicators of ecosystem degradation. These often rely on the measurement of some kind of aggregation pattern (e. g. spatial autocorrelation), or the size and distribution of vegetation patch sizes, which arise in plant communities because 5 of facilitation (Génin et al., 2018b;Schneider & Kéfi, 2016). An interesting lead for interaction research is their use in conservation. ...
Thesis
Les systèmes écologiques sont complexes car composés d'une multitude d'éléments en interaction. Ces interactions, entre espèces par exemple, forment des réseaux qui présentent des propriétés structurelles déterminantes pour la réponse du système écologique entier aux perturbations.Pour mieux identifier cette réponse, il est donc important de cartographier les interactions présentes dans les communautés écologiques et de comprendre leurs variations dans le temps et l'espace.Dans ce travail, nous avons utilisé les communautés de plantes comme systèmes écologiques modèles afin (i) d'identifier à partir de patrons spatiaux certains motifs présents dans les réseaux d'interaction écologiques (les boucles de rétroaction) et (ii) de cartographier les réseaux d'interactions (tels que mesurés par les associations spatiales entre plantes)le long de gradients de stress. Pour ce faire, nous avons utilisé deux jeux de données documentant des communautés de plantes de clairières subalpines (Etats-Unis) et méditerranéennes (La Crau, France). Nos résultats montrent que les boucles de rétroaction peuvent être inferées à partir des patrons spatiaux présents dans les communautés de plantes, permettant ainsi d'identifier des communautés pouvant répondre de manière abrupte aux perturbations. Les interactions entre plantes (déduites de leurs associations spatiales) dépendent fortement du stress appliqué à la communauté, et présentent une résilience faible aux évènements de perturbation.Ce travail montre que les interactions entre plantes peuvent être cartographiées in situ à partir des associations spatiales. Il ouvre la voie vers une meilleure compréhension et capacité d'anticipation de la réponse des communautés écologiques face aux perturbations.
... Defining the spatial scale of interactions is critical (Génin et al., 2018) for defining density and this in return can have direct effects on positive or negative density effects. Often the 'local' interactions or fine scale is measured as e.g. ...
Article
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Spatio-temporal data are more ubiquitous and richer than even before and the availability of such data poses great challenges in data analytics. Ecological facilitation, the positive effect of density of individuals on the individual’s survival across a stress gradient, is a complex phenomenon. A large number of tree individuals coupled with soil moisture, temperature, and water stress data across a long temporal period were followed. Data-driven analysis in the absence of hypothesis was performed. Information theoretic analysis of multiple statistical models was employed in order to quantify the best data-driven index of vegetation density and spatial scale of interactions. Sequentially, tree survival was quantified as a function of the size of the individual, vegetation density, and time at the optimal spatial interaction scale. Land surface temperature and soil moisture were also statistically explained by tree size, density, and time. Results indicated that in space both facilitation and competition co-exist in the same ecosystem and the sign and magnitude of this depend on the spatial scale. Overall, within the optimal data-driven spatial scale, tree survival was best explained by the interaction between density and year, sifting overall from facilitation to competition through time. However, small sized trees were always facilitated by increased densities, while large sized trees had either negative or no density effects. Tree size was more important predictor than density in survival and this has implications for nature-based solutions: maintaining large tree individuals or planting species that can become large-sized can safeguard against tree-less areas by promoting survival at long time periods through harsh environmental conditions. Large trees had also a significant effect in moderating land surface temperature and this effect was higher than the one of vegetation density on temperature.
... Anisotropic effects can also emerge from differences in insolation driven by landscape aspect which greatly influences vegetation patterns, runoff generation and erosion processes [44]. These types of anisotropy or those arising from heterogeneity in external perturbations (i.e., grazing, logging [31,45]) can be easily incorporated into the analysis of landscape hydrological connectivity [14]. ...
Article
Anthropogenic activities and climate change are imposing an unprecedented pressure on drylands, increasing their vulnerability to desertification. The spatial organization of the sparse vegetation cover is fundamental for the healthy function of the system, and disturbances can trigger cascading feedbacks leading to catastrophic system collapse. Here we discuss some of the latest research aiming at understanding abrupt landscape transitions and possible non-reversible changes, as well as emerging research on the identification of early warning indicators of abrupt transitions to desert states. Robust indicators should take into account temporal system dynamics characteristics, vegetation organization/patch size distribution, functional connectivity measures and human intervention effects.
Chapter
This chapter aims to (1) provide background for conceptual mathematical models of spontaneous pattern formation, in the context of dryland vegetation patterns, and (2) review observational studies of the phenomenon. The chapter also highlights challenges and opportunities associated with the development of the models in light of increasing availability of remote sensing data. This includes both satellite imagery of the patterns and elevation data of the topography. The vast scales, in time and space, associated with the key processes further suggest avenues for improved mathematical modeling paradigms.
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We compared two biogeomorphic models that postulate how vegetation is intertwined in the response and recovery of barrier island dunes. Each model was developed in a separate coastal region using different methods. Both relied on simple elevational representations of topography. By comparing topographies among more islands of these two regions and by linking multiple representations of topographic pattern to resistance and resilience, we provide a synthesis that shows the validity of both models and the consequences of reifying one over the other. Using airborne LiDAR, topographic metrics based on point, patch, and gradient representations of topography were derived for fifty-two sites across eleven islands along the Georgia Bight and Virginia. These seventeen metrics were categorized in terms of resistance and resilience to disturbance from storm-forced high water levels and overwash. Resistance refers to intrinsic properties that directly counter expressions of power from disturbance. Resilience refers to the degrees of freedom to adjust and adapt to disturbance. Using a cross-scale data modeling approach, these data were visualized as topographic state space using multidimensional scaling. In this state space, similarity in topography as well as resistance and resilience were inferred through a site's position along low-dimension axes representing geomorphic resistance and high-dimension axes representing the spatial landscape properties of biogeomorphic resilience. The two models overlap in how they account for barrier dune resistance and resilience along the U.S. south Atlantic coast. Islands of the Georgia Bight have a propensity for higher resistance and resilience. The Virginia islands have lower resistance and resilience. Key Words: barrier islands, biogeomorphology, cross-scale structure, dunes, resilience.
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1.In diverse ecosystems, organisms cluster together in such a manner that the frequency distribution of cluster‐sizes is a power‐law function. Spatially‐explicit computational models of ecosystems suggest that a loss of such power‐law clustering may indicate a loss of ecosystem resilience; the empirical evidence in support for this hypothesis has been mixed. On the other hand, a well‐known dynamical feature of systems with reduced resilience is the slower recovery from perturbations, a phenomenon known as critical slowing down (CSD). Here, we examine the relationship between spatial clustering and CSD to better understand the use of cluster‐size distributions as indicators of ecosystem resilience. 2.Local positive feedback is an important driver of spatial clustering, while also affecting the dynamics of the ecosystem: Studies have demonstrated that positive feedback promotes abrupt regime shifts. Here, we analyse a spatial model of ecosystem transitions that enables us to disentangle the roles of local positive feedback and environmental stress on spatial patterns and ecosystem resilience. 3.We demonstrate that, depending on the strength of positive feedback, powerlaw clustering can occur at any distance from the critical threshold of ecosystem collapse. In fact, we find that for systems with strong positive feedback, which are more likely to exhibit abrupt transitions, there may be no loss of power‐law clustering prior to critical thresholds. 4.Our analyses show that cluster‐size distributions are unrelated to the phenomenon of CSD and that loss of power‐law clustering is not a generic indicator of ecosystemresilience. Further, due to CSD, a power‐lawfeature does occur near critical thresholds but in a different quantity; specifically, a power‐law decay of spatial covariance of ecosystem state. Our work highlights the importance of links between local positive feedback, emergent spatial properties and how they may be used to interpret ecosystem resilience.
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Prediction of ecosystem responses to global environmental change is a pressing scientific challenge of major societal relevance. Many ecosystems display nonlinear responses to environmental change, and may even undergo practically irreversible ‘regime shifts’ that initiate ecosystem collapse. Recently, early warning signals based on spatiotemporal metrics have been proposed for the identification of impending regime shifts. The rapidly increasing availability of remotely‐sensed data provides excellent opportunities to apply such model‐based spatial early warning signals in the real world, to assess ecosystem resilience and identify impending regime shifts induced by global change. Such information would allow land‐managers and policy makers to interfere and avoid catastrophic shifts, but also to induce regime shifts that move ecosystems to a desired state. Here, we show that the application of spatial early warning signals in real‐world landscapes presents unique and unexpected challenges, and may result in misleading conclusions when employed without careful consideration of the spatial data and processes at hand. We identify key practical and theoretical issues and provide guidelines for applying spatial early warning signals in heterogeneous, real‐world landscapes based on literature review and examples from real‐world data. Major identified issues include (1) spatial heterogeneity in real‐world landscapes may enhance reversibility of regime shifts and boost landscape‐level resilience to environmental change (2) ecosystem states are often difficult to define, while these definitions have great impact on spatial early warning signals, and (3) spatial environmental variability and socio‐economic factors may affect spatial patterns, spatial early warning signals, and associated regime shift predictions. We propose a novel framework shifting from an ecosystem perspective towards a landscape approach. The framework can be used to identify conditions under which resilience assessment with spatial remotely‐sensed data may be successful, to support well‐informed application of spatial early warning signals, and to improve predictions of ecosystem responses to global environmental change. This article is protected by copyright. All rights reserved.
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The response of drylands to environmental gradients can be abrupt rather than gradual. These shifts largely occur unannounced and are difficult to reverse once they happen; their prompt detection is of crucial importance. The distribution of vegetation patch sizes may indicate the proximity to these shifts, but the use of this metric is hampered by a lack of large-scale studies relating these distributions to the provision of multiple ecosystem functions (multifunctionality) and comparing them to other ecosystem attributes, such as total plant cover. Here we sampled 115 dryland ecosystems across the globe and related their vegetation attributes (cover and patch size distributions) to multifunctionality. Multifunctionality followed a bimodal distribution across our sites, suggesting alternative states in the functioning of drylands. Although plant cover was the strongest predictor of multifunctionality when linear analyses were used, only patch size distributions reflected the bimodal distribution of multifunctionality observed. Differences in the coupling between nutrient cycles and in the importance of self-organizing biotic processes characterized the two multifunctionality states observed. Our findings support the use of vegetation patterns as indicators of ecosystem functioning in drylands and pave the way for developing effective strategies to monitor desertification processes.
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Complex systems inspired analysis suggests a hypothesis that financial meltdowns are abrupt critical transitions that occur when the system reaches a tipping point. Theoretical and empirical studies on climatic and ecological dynamical systems have shown that approach to tipping points is preceded by a generic phenomenon called critical slowing down, i.e. an increasingly slow response of the system to perturbations. Therefore, it has been suggested that critical slowing down may be used as an early warning signal of imminent critical transitions. Whether financial markets exhibit critical slowing down prior to meltdowns remains unclear. Here, our analysis reveals that three major US (Dow Jones Index, S&P 500 and NASDAQ) and two European markets (DAX and FTSE) did not exhibit critical slowing down prior to major financial crashes over the last century. However, all markets showed strong trends of rising variability, quantified by time series variance and spectral function at low frequencies, prior to crashes. These results suggest that financial crashes are not critical transitions that occur in the vicinity of a tipping point. Using a simple model, we argue that financial crashes are likely to be stochastic transitions which can occur even when the system is far away from the tipping point. Specifically, we show that a gradually increasing strength of stochastic perturbations may have caused to abrupt transitions in the financial markets. Broadly, our results highlight the importance of stochastically driven abrupt transitions in real world scenarios. Our study offers rising variability as a precursor of financial meltdowns albeit with a limitation that they may signal false alarms.
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Ecosystems may exhibit catastrophic shifts, i.e. abrupt and irreversible responses of ecosystem functions and services to continuous changes in external conditions. The search for early warning signs of approaching shifts has so far mainly been conducted on theoretical models assuming spatially-homogeneous external pressures (e.g. climatic). Here, we investigate how a spatially explicit pressure may affect ecosystems’ risk of catastrophic shifts and the associated spatial early-warning signs. As a case study, we studied a dryland vegetation model assuming ‘associational resistance’, i.e. the mutual reduction of local grazing impact by neighboring plants sharing the investment in defensive traits. Consequently, grazing pressure depends on the local density of plants and is thus spatially-explicit. We focus on the distribution of vegetation patch sizes, which can be assessed using remote sensing and are candidate early warning signs of catastrophic shifts in drylands. We found that spatially explicit grazing affected both the resilience and the spatial patterns of the landscape. Grazing impact became self-enhancing in more fragmented landscapes, disrupted patch growth and put apparently ‘healthy’ drylands under high risks of catastrophic shifts. Our study highlights that a spatially explicit pressure may affect the nature of the spatial pattern observed and thereby change the interpretation of the early warning signs. This may generalize to other ecosystems exhibiting self-organized spatial patterns, where a spatially-explicit pressure may interfere with pattern formation.
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Ecological resilience is the ability of a system to persist in the face of perturbations. Although resilience has been a highly influential concept, its interpretation has remained largely qualitative. Here we describe an emerging family of methods for quantifying resilience on the basis of observations. A first set of methods is based on the phenomenon of critical slowing down, which implies that recovery upon small perturbations becomes slower as a system approaches a tipping point. Such slowing down can be measured experimentally but may also be indirectly inferred from changes in natural fluctuations and spatial patterns. A second group of methods aims to characterize the resilience of alternative states in probabilistic terms based on large numbers of observations as in long time series or satellite images. These generic approaches to measuring resilience complement the system-specific knowledge needed to infer the effects of environmental change on the resilience of complex systems.
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Facilitation is a major force shaping the structure and diversity of plant communities in terrestrial ecosystems. Detecting positive plant-plant interactions relies on the combination of field experimentation and the demonstration of spatial association between neighboring plants. This has often restricted the study of facilitation to particular sites, limiting the development of systematic assessments of facilitation over regional and global scales. Here we explore whether the frequency of plant spatial associations detected from high-resolution remotely sensed images can be used to infer plant facilitation at the community level in drylands around the globe. We correlated the information from remotely sensed images freely available through Google Earth with detailed field assessments, and used a simple individualbased model to generate patch-size distributions using different assumptions about the type and strength of plant-plant interactions. Most of the patterns found from the remotely sensed images were more right skewed than the patterns from the null model simulating a random distribution. This suggests that the plants in the studied drylands show stronger spatial clustering than expected by chance. We found that positive plant co-occurrence, as measured in the field, was significantly related to the skewness of vegetation patch-size distribution measured using Google Earth images. Our findings suggest that the relative frequency of facilitation may be inferred from spatial pattern signals measured from remotely sensed images, since facilitation often determines positive co-occurrence among neighboring plants. They pave the road for a systematic global assessment of the role of facilitation in terrestrial ecosystems.
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Critical transitions between alternate ecosystem states are often preceded by increased variance and autocorrelation in time series of ecosystem properties. Analogous changes may occur in spatial statistics as ecosystems approach thresholds for critical transitions. Changes in spatial statistics near thresholds have been described using models, laboratory experiments, and remotely sensed data, but there have been no tests using deliberate manipulations of whole ecosystems in the field. We previously documented a whole-lake manipulation resulting in a transition to predator dominance, a type of critical transition. The food web of an experimental lake was forced via cascading trophic interactions from a stable state characterized by abundant prey fish, small zooplankton, and high chlorophyll concentrations to an alternative state dominated by predatory fish, large zooplankton and low chlorophyll concentrations. Time series of zooplankton and chlorophyll concentrations provided early warning of the regime shift. Here we test if similar early warning signals were present in space by applying spatial variance and the discrete Fourier transform to spatially distributed prey fish catch data from this regime shift. Prey fish spatial distributions were monitored daily using minnow traps deployed around the lake perimeter. Added predators reduced prey fish populations and altered their spatial distributions. Increases in spatial variance and shifts to low frequency spatial variance were observed up to a year in advance of the shift. There was no response in an adjacent reference lake. Our results demonstrate that spatial signals of approaching thresholds can be detected at the ecosystem scale.
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With ongoing climate change, the search for indicators of imminent ecosystem shifts is attracting increasing attention (e.g., Scheffer et al. 2009). Recently, the spatial organization of ecosystems has been suggested as a good candidate for such an indicator in spatially structured ecosystems (Rietkerk et al. 2004, Ke´fi et al. 2007a, Guttal and Jayaprakash 2009).
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Complex dynamical systems, ranging from ecosystems to financial markets and the climate, can have tipping points at which a sudden shift to a contrasting dynamical regime may occur. Although predicting such critical points before they are reached is extremely difficult, work in different scientific fields is now suggesting the existence of generic early-warning signals that may indicate for a wide class of systems if a critical threshold is approaching.
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Local interactions, biotic and abiotic, can have a strong influence on the large-scale properties of ecosystems. However, ecological models often explore the influence of local biotic interactions where physical disturbance is included as a large-scale and imposed source of variability but is not allowed to interact with biotic processes at the local scale. In marine intertidal communities dominated by mussels, wave disturbances create gaps in the mussel bed that recover through a successional sequence. We present a lattice model of mussel disturbance dynamics that allows local interactions between wave disturbance and mussel recolonization, in which each cell of the lattice can be empty, occupied by a mussel bed element, or disturbed (which corresponds to a newly disturbed cell that has unstable edges). As in natural ecosystems, wave disturbance can also spread from disturbed to adjacent occupied cells, and recolonization can also spread from occupied to adjacent empty cells. We first validate the local rules from artificial gap experiments and from natural gap monitoring along the Oregon coast. We analyze the properties of the model system as a function of different oceanographic forcings of productivity and disturbance. We show that the mussel bed can go through phase transitions characterized by a large sensitivity of mussel cover and patterns to oceanographic forcings but also that criticality (scale invariance) is observed over wide ranges of parameters, which suggests self-organization. We also show that spatial patterns in the intertidal can provide a robust signature of local processes and can inform about oceanographic regimes. We do so by comparing the large-scale patterns of the simulation (scaling exponents) with field data, which suggest that some experimental sites are close to criticality. Our results suggest that regional patterns in disturbed populations can be explained by local biotic and abiotic processes submitted to oceanographic forcing.
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Unexpected sudden catastrophic shifts may occur in ecosystems, with concomitant losses or gains of ecological and economic resources. Such shifts have been theoretically attributed to positive feedback and bistability of ecosystem states. However, verifications and predictive power with respect to catastrophic responses to a changing environment are lacking for spatially extensive ecosystems. This situation impedes management and recovery strategies for such ecosystems. Here, we review recent studies on various ecosystems that link self-organized patchiness to catastrophic shifts between ecosystem states.
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Theory suggests that ecological systems exhibit a pronounced slow down in their dynamics, known as 'critical slowing down' (CSD), before they undergo regime shifts or critical transitions. As a result of CSD, ecosystems exhibit characteristic temporal and spatial changes which can be used as early warning signals of imminent regime shifts. For temporal data, statistical methods to detect these generic indicators of ecosystem resilience are well developed. However, for spatial data, despite a well developed theoretical framework, statistical methods such as data pre-processing and null models to detect EWS are relatively poorly developed. In this manuscript, we investigate the case of a common type of ecological spatial dataset which consists of binary values at each location (e.g. occupied/unoccupied, tree/grass or coralline/bleached). We employ a cellular-automaton based spatially-explicit model which generates data that mimics remotely sensed or field collected high-resolution spatial data with a binary classification of the state variables at each location. We demonstrate that trends in two spatial metrics, spatial variance and spatial skewness, of such binary spatial data lead to false, failed or misleading signals of transitions. We find that, two other indicators, spatial autocorrelation at lag-1 and spectral density ratio, accurately reflect CSD even with binary spatial data. To overcome the problems associated with detection of EWS using spatial variance and skewness, we investigate a data pre-processing method called 'coarse-graining' which is inspired from the physics literature on phase transitions. Coarse-graining reduces the spatial resolution of data by averaging state variables over small scales. Yet, it enables detection of CSD-based spatial indicators of impending critical transitions. In summary, our study provides a theoretical basis, and rigorous evaluation, of coarse-graining as a pre-processing step to analyse spatial datasets with discrete state classifications.
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Aim Theory suggests that as ecological systems approach regime shifts, they become increasingly slow in recovering from perturbations. This phenomenon, known as critical slowing down [CSD], leads to spatial and temporal signatures in ecological state variables, thus potentially offering early indicators of regime shifts. Indicators using temporal dynamics have been empirically validated in laboratory microcosms and other well‐mixed systems, but tests of spatial indicators of regime shifts at large spatial scales in the field are rare due to the relative absence of high‐resolution data and difficulties in experimental manipulations. Here, we test theoretical predictions of CSD‐based spatial indicators using large‐scale field data from the Serengeti–Mara grassland–woodland system. Location Serengeti–Mara ecosystem, Tanzania and Kenya. Time period Year 2000 Major taxa studied Vegetation Method We used a space‐for‐time substitution method to empirically test the validity of CSD‐based spatial indicators, i.e., we computed indicators along a spatial [in lieu of temporal] gradient of ecological states. First we used a model of vegetation dynamics to determine if our space‐for‐time substitution method was appropriate. Then we tested for CSD‐based spatial indicators using high‐resolution spatial vegetation [30 m] and rainfall [2.5 km] data from the Serengeti–Mara ecosystem. Results Our model predicts that CSD‐based indicators increase along a spatial gradient of alternative vegetation states. Empirical analyses suggest that grasslands and woodlands occur as alternative stable states in the Serengeti–Mara ecosystem with rainfall as one of the potential drivers of transitions between these states. We found that four indices of CSD showed the theoretically expected increasing trends along spatial gradients of grasslands to woodlands: spatial variance, spatial skewness, spatial correlation at lag‐1 and spatial spectra at low frequencies. Main conclusions Our results suggest that CSD‐based spatial indicators can offer early warning signals of critical transitions in large‐scale ecosystems.
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Foreseeing population collapse is an on-going target in ecology, and this has led to the development of early warning signals based on expected changes in leading indicators before a bifurcation. Such signals have been sought for in abundance time-series data on a population of interest, with varying degrees of success. Here we move beyond these established methods by including parallel time-series data of abundance and fitness-related trait dynamics. Using data from a microcosm experiment, we show that including information on the dynamics of phenotypic traits such as body size into composite early warning indices can produce more accurate inferences of whether a population is approaching a critical transition than using abundance time-series alone. By including fitness-related trait information alongside traditional abundance-based early warning signals in a single metric of risk, our generalizable approach provides a powerful new way to assess what populations may be on the verge of collapse.
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To derive the consequence of heritable traits of individual organisms upon the feature of their populations, the lattice Lotka-Volterra model is studied which is defined as a Markov process of the state of the lattice space. A lattice site is either vacant or occupied by an individual of a certain type or species. Transition rates of the process are given in terms of parameters representing the traits of an individual such as intrinsic birth and death and migration rate of each type. Density is a variable defined as a probability that a site is occupied by a certain type. Under a given state of a site the conditional probability of its nearest neighbor site being occupied by a certain type is termed environs density of the site. Mutual exclusion of individuals is already taken into account by the basic assumption of the lattice model. Other interaction between individuals can be taken into account by assuming that the actual birth and death and migration rates are dependent on the environs densities. Extending the notion of ordinary Malthusian parameters, we define Malthusians as dynamical variables specifying the time development of the densities. Conditions for the positive stationary densities and for the evolutional stability (ES) against the invasion of mutant types is given in terms of Malthusians. Using the pair approximation (PA), a simplest decoupling approximation to take account of spatial correlation, we obtain analytical results for stationary densities, and critical parameters for ES in the case of two types. Assuming that the death rate is dependent on the environs density, we derive conditions for the evolution of altruism. Comparing with computer simulation, we discuss the validity of PA and its improvement.
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Ecosystems may shift abruptly between alternative states in response to environmental perturbations [1-3]. Early warning indicators have been proposed to anticipate such regime shifts, but experimental field tests of their validity are rare [4-6]. We exposed rocky intertidal algal canopies to a gradient of press perturbations and recorded the response of associated assemblages over 7 years. Reduced cover and biomass of algal canopies promoted the invasion of algal turfs, driving understory assemblages toward collapse upon total canopy removal. A dynamic model indicated the existence of a critical threshold separating the canopy- and turf-dominated states. We evaluated common indicators of regime shift as the system approached the threshold, including autocorrelation, SD, and skewness [7]. These indicators captured changes in understory cover due to colonization of algal turfs. All indicators increased significantly as the system approached the critical threshold, in agreement with theoretical predictions [2, 8, 9]. The performance of indicators changed when we superimposed a pulse disturbance on the press perturbation that amplified environmental noise. This treatment caused several experimental units to switch repeatedly between the canopy- and the turf-dominated state, resulting in a significant increase in overall variance of understory cover, a negligible effect on skewness and no effect on autocorrelation. Power analysis indicated that autocorrelation and SD were better suited at anticipating a regime shift under mild and strong fluctuations of the state variable, respectively. Our results suggest that regime shifts may be anticipated under a broad range of fluctuating conditions using the appropriate indicator. Copyright © 2015 Elsevier Ltd. All rights reserved.
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In ecosystems with spatial dispersion of nutrients, organic matter, or organisms, dispersal may dampen variability that could provide an early warning of regime shifts. The discrete Fourier transform (DFT) of spatial pattern rescales system dynamics into spatial frequencies where early warnings may be sharpened. We analyzed four spatial ecological models with the DFT. Different models represented single species in discrete and continuous time, and two different prey-harvester systems. In all four systems, the DFT of transient data exhibited substantial increases prior to the critical transitions. The DFT adds to the arsenal of early warning indicators for spatially structured ecosystems. In addition it provides information about the spatial frequencies where destabilization first begins.
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Recent studies on forest dynamics in diverse forested ecosystems suggest that forest stands are disturbed more frequently if they are next to existing gaps, and that gaps once formed tend to expand their area in subsequent years. We examine total gap area and the size distribution of gaps at equilibrium in a lattice-structured forest model. Each site undergoes transition between two states (gaps and non-gaps), and the disturbance rate (transition from non-gap to gap) increases with the number of gap sites in the neighborhood. Dynamics based on a mean-field approximation (i.e. neglecting of spatial structure) failed to predict total gap area and the gap size distribution in the equilibrium forest. Pair approximation, which considers a closed dynamical system of average and local gap density (the conditional gap density among neighbors of a randomly chosen gap site), can predict the total gap area, the correlation between neighbors, and the gap size distribution fairly accurately. If the recruitment rate increases in proportion to non-gap area in the forest, the model may show bistability. We analyse data on forest spatial dynamics in the light of the model. We conclude that gap size distribution can often be described using two statistics (global and local gap densities) and that these in turn can be predicted by the dynamics of gap formation, gap expansion, regeneration, and gap closure.
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Avoidance and tolerance are the two means by which plants cope with herbivores. Avoidances internal to the plant, such as morphology, chemical repellants, thorns, etc., have received considerable attention in the plant-herbivore literature, but relatively little consideration has been given to avoidances external to the plant. We develop a conceptual framework of external plant avoidances of herbivory based on foraging selection impedances (associational avoidances), behavioral impedances (indirect avoidances), and physical impedances (refuges) organized along axes of efficiency, degree of protection, and necessity of tolerance characteristics. Associational avoidances are uncommon for terrestrial mammalian herbivores compared to plant-insect or marine situations. Indirect avoidances mediated through herbivore territoriality, predator avoidance, and other behaviors independent of foraging decisions are probably common in nature, but few have been formally documented. Biotic and geologic refuges providing a physical impedance are the only avoidances shown to have implications for plant biodiversity. This is particularly true for geologic refuges, where there is not a tradeoff between competition and the refuge effect. Small geologic refuges (rock outcrops, cliffs, etc.) are more likely to also positively or negatively alter associated plant microenvironments than large geologic refuges (mesas, islands, etc.). In a survey, 86% of small refuge studies reported positive effects on plant diversity compared to 50% for large refuges. Geologic refuges in more productive environments were more important in protecting diversity than refuges in less productive, semiarid environments, and the effects of protection were greater in communities with short compared to long evolutionary histories of grazing. Other characteristics of refuges such as extent across the landscape and the manner they alter or ameliorate the environment, as well as characteristics of the herbivore such as small or large, generalist or specialist may also determine the effectiveness of refuges, but there are too few studies to assess these factors.
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Regime shifts are substantial, long-lasting reorganizations of complex systems, such as ecosystems. Large ecosystem changes such as eutrophication, shifts among vegetation types, degradation of coral reefs and regional climate change often come as surprises because we lack leading indicators for regime shifts. Increases in variability of ecosystems have been suggested to foreshadow ecological regime shifts. However, it may be difficult to discern variability due to impending regime shift from that of exogenous drivers that affect the ecosystem. We addressed this problem using a model of lake eutrophication. Lakes are subject to fluctuations in recycling associated with regime shifts, as well as fluctuating nutrient inputs. Despite the complications of noisy inputs, increasing variability of lake-water phosphorus was discernible prior to the shift to eutrophic conditions. Simulations show that rising standard deviation (SD) could signal impending shifts about a decade in advance. The rising SD was detected by studying variability around predictions of a simple time-series model, and did not depend on detailed knowledge of the actual ecosystem dynamics.
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Dramatic changes at thresholds in multiple stable ecosystems may be irreversible if caused by man. The characteristic return time to an equilibrium increases when a threshold is approached. A universal law for this increase is found, which may be used to forecast the position of a threshold by extrapolation of empirical data. Harvesting experiments on populations are proposed that can be used to verify the method. Preliminary harvesting experiments on rotifer populations display a good agreement with the theory.
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Proper assessment and early detection of land degradation and desertification is extremely important in arid and semi-arid ecosystems. Recent research has proposed to use the characteristics of spatial vegetation patterns, such as parameters derived from power-law modeling of vegetation patches, for detecting the early signs of desertification. However, contradictory results have been reported regarding the suitability of those proposed indicators. We used an experiment with multiple grazing intensities as an analog of a desertification gradient and evaluated the performance of two predictors of desertification: percent plant cover and a transition from a patch-area distribution characterized by a power law to another portrayed by a truncated power law, in a desert steppe in Inner Mongolia, China. We found that spatial metrics, such as the largest patch index and coefficient of variation of mean patch area had negative linear relationships with grazing intensity, suggesting that vegetation patches became more fragmented and homogeneous under higher grazing pressure. Using a binning-based method to analyze our dataset, we found that the patch-area relationship deviated from a power-law to a truncated power-law model with increasing grazing pressure, while the truncated power law was a better fit than the power law for all plots when binning was not used. These results suggest that the selection of methodology is crucial in using power-law models to detect changes in vegetation patterns. Plant cover was significantly correlated with stocking rate and all spatial metrics evaluated; however, the relationship between cover and vegetation spatial pattern still deserves a thorough examination, especially in other types of ecosystems, before using cover as a universal early sign of desertification. Our results highlight a strong connection between the vegetation spatial pattern and the desertification associated with heavy grazing and suggest that future studies should incorporate information about vegetation spatial pattern in monitoring desertification processes. KeywordsGrassland-Dryland-Herbivory-Sheep grazing-Overgrazing-Spatial homogeneity-Habitat fragmentation-Heterogeneity-Small-scale
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Spatial vegetation patterns are recognized as sources of valuable information that can be used to infer the state and functionality of semiarid ecosystems, particularly in the context of both climate and land use change. Recent studies have suggested that the patch-size distribution of vegetation in drylands can be described using power-law metrics, and that these scale-free distributions deviate from power-law linearity with characteristic scale lengths under the effects of increasing aridity or human disturbance, providing an early sign of desertification. These findings have been questioned by several modeling approaches, which have identified the presence of characteristic scale lengths on the patch-size distribution of semiarid periodic landscapes. We analyze the relationship between fragmentation of vegetation patterns and their patch-size distributions in semiarid landscapes showing different degree of periodicity (i.e., banding). Our assessment is based on the study of vegetation patterns derived from remote sensing in a series of semiarid Australian Mulga shrublands subjected to different disturbance levels. We use the patch-size probability density and cumulative probability distribution functions from both nondirectional and downslope analyses of the vegetation patterns. Our results indicate that the shape of the patch-size distribution of vegetation changes with the methodology of analysis applied and specific landscape traits, breaking the universal applicability of the power-law metrics. Characteristic scale lengths are detected in (quasi) periodic banded ecosystems when the methodology of analysis accounts for critical landscape anisotropies, using downslope transects in the direction of flow paths. In addition, a common signal of fragmentation is observed: the largest vegetation patches become increasingly less abundant under the effects of disturbance. This effect also explains deviations from power-law behavior in disturbed vegetation which originally showed scale-free patterns. Overall, our results emphasize the complexity of structure assessment in dryland ecosystems, while recognizing the usefulness of the patch-size distribution of vegetation for monitoring semiarid ecosystems, especially through the cumulative probability distributions, which showed high sensitivity to fragmentation of the vegetation patterns. We suggest that preserving large vegetation patches is a critical task for the maintenance of the ecosystem structure and functionality.
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Ecology Letters (2011) 14: 29–35 Robust critical systems are characterized by power laws which occur over a broad range of conditions. Their robust behaviour has been explained by local interactions. While such systems could be widespread in nature, their properties are not well understood. Here, we study three robust critical ecosystem models and a null model that lacks spatial interactions. In all these models, individuals aggregate in patches whose size distributions follow power laws which melt down under increasing external stress. We propose that this power-law decay associated with the connectivity of the system can be used to evaluate the level of stress exerted on the ecosystem. We identify several indicators along the transition to extinction. These indicators give us a relative measure of the distance to extinction, and have therefore potential application to conservation biology, especially for ecosystems with self-organization and critical transitions.
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Ecology Letters (2010) 13: 464–472 Predicting regime shifts – drastic changes in dynamic behaviour – is a key challenge in ecology and other fields. Here we show that the class of ecological systems that will exhibit leading indicators of regime shifts is limited, and that there is a set of ecological models and, therefore, also likely to be a class of natural systems for which there will be no forewarning of a regime change. We first describe how nonlinearities in combination with environmental variability lead to model descriptions that will not have smooth potentials, concluding that many ecological systems are described by systems without smooth potentials and thus will not show typical leading indicators of regime shifts. We then illustrate the impact of these general arguments by numerically examining the dynamics of several model ecological systems under slowly changing conditions. Our results offer a cautionary note about the generality of forecasting sudden changes in ecosystems.
Article
The ubiquity of scale-free patterns in ecological systems has raised the possibility that these systems operate near criticality. Critical phenomena (CP) require the tuning of parameters and typically exhibit a narrow scaling region in which power laws hold. Here we show that an individual-based predator-prey model exhibits scaling properties similar to CP, generated by a percolation-like transition but with a broader scaling region. There are no drastic changes in ecological quantities across this critical point and species coexist broadly in parameter space. The implications of these findings for the stability of ecological systems “near” criticality is discussed. © 2003 Wiley Periodicals, Inc. Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/35202/1/10096_ftp.pdf
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The monitoring of desertification processes, and particularly the development of "early-warning" systems, is an increasingly important development in the management of drylands. It has been shown that the patch size distribution of dryland vegetation can be described using power laws and that deviations from such patterns may be used as an early-warning signal for the onset of desertification. We tested this idea using data from 29 semiarid steppes located along a latitudinal gradient in Spain. A truncated power law (TPL) fitted the patch size distribution of perennial vegetation better than a power law in all the evaluated sites. Variations in this distribution, as measured with the scaling exponent (gamma) of the TPL, were not related to total perennial cover, but a negative logarithmic relationship was found between gamma and soil variables related to desertification processes (total nitrogen, total phosphorus, and organic carbon). A positive and stronger linear relationship was found between total perennial cover and the same soil variables. Our results suggest that deviations from a patch size distribution characterized by a power law are not directly related to desertification. They also indicate that plant cover can be used to effectively monitor key variables linked to desertification processes.
Article
Transitions in ecological systems often occur without apparent warning, and may represent shifts between alternative persistent states. Decreasing ecological resilience (the size of the basin of attraction around a stable state) can signal an impending transition, but this effect is difficult to measure in practice. Recent research has suggested that a decreasing rate of recovery from small perturbations (critical slowing down) is a good indicator of ecological resilience. Here we use analytical techniques to draw general conclusions about the conditions under which critical slowing down provides an early indicator of transitions in two-species predator-prey and competition models. The models exhibit three types of transition: the predator-prey model has a Hopf bifurcation and a transcritical bifurcation, and the competition model has two saddle-node bifurcations (in which case the system exhibits hysteresis) or two transcritical bifurcations, depending on the parameterisation. We find that critical slowing down is an earlier indicator of the Hopf bifurcation in predator-prey models in which prey are regulated by predation rather than by intrinsic density-dependent effects and an earlier indicator of transitions in competition models in which the dynamics of the rare species operate on slower timescales than the dynamics of the common species. These results lead directly to predictions for more complex multi-species systems, which can be tested using simulation models or real ecosystems.
Article
Arid ecosystems are liable to undergo sudden discontinuous transitions from a vegetated to a desert state as a result of human pressure and climate change. A predictive framework about the conditions under which such transitions occur is lacking. Here, we derive and analyze a general model describing the spatial dynamics of vegetation in arid ecosystems considering local facilitation as an essential process. We investigate the conditions under which continuous or discontinuous transitions from a vegetated to a desert state are likely to occur. We focus on arid ecosystems but our approach is sufficiently general to be applied to other ecosystems with severe environmental conditions. The model exhibits bistability and vegetation patchiness. High local facilitation decreases the risk of discontinuous transitions. Moreover, for arid ecosystems where local facilitation is a driving process, vegetation patchiness indicates proximity to a transition point, but does not allow distinguishing between continuous and discontinuous transitions.
Article
The size of the basin of attraction in ecosystems with alternative stable states is often referred to as "ecological resilience." Ecosystems with a low ecological resilience may easily be tipped into an alternative basin of attraction by a stochastic event. Unfortunately, it is very difficult to measure ecological resilience in practice. Here we show that the rate of recovery from small perturbations (sometimes called "engineering resilience") is a remarkably good indicator of ecological resilience. Such recovery rates decrease as a catastrophic regime shift is approached, a phenomenon known in physics as "critical slowing down." We demonstrate the robust occurrence of critical slowing down in six ecological models and outline a possible experimental approach to quantify differences in recovery rates. In all the models we analyzed, critical slowing down becomes apparent quite far from a threshold point, suggesting that it may indeed be of practical use as an early warning signal. Despite the fact that critical slowing down could also indicate other critical transitions, such as a stable system becoming oscillatory, the robustness of the phenomenon makes it a promising indicator of loss of resilience and the risk of upcoming regime shifts in a system.
Article
Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution -- the part of the distribution representing large but rare events -- and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov-Smirnov statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data while in others the power law is ruled out. Comment: 43 pages, 11 figures, 7 tables, 4 appendices; code available at http://www.santafe.edu/~aaronc/powerlaws/