# Spatial variance and spatial skewness: leading indicators of regime shifts in spatial ecological systems

- 1k Downloads
- 90 Citations

## Abstract

Ecosystems can undergo large-scale changes in their states, known as catastrophic regime shifts, leading to substantial losses to services they provide to humans. These shifts occur rapidly and are difficult to predict. Several early warning signals of such transitions have recently been developed using simple models. These studies typically ignore spatial interactions, and the signal provided by these indicators may be ambiguous. We employ a simple model of collapse of vegetation in one and two spatial dimensions and show, using analytic and numerical studies, that increases in spatial variance and changes in spatial skewness occur as one approaches the threshold of vegetation collapse. We identify a novel feature, an increasing spatial variance in conjunction with a peaking of spatial skewness, as an unambiguous indicator of an impending regime shift. Once a signal has been detected, we show that a quick management action reducing the grazing activity is needed to prevent the collapse of vegetated state. Our results show that the difficulties in obtaining the accurate estimates of indicators arising due to lack of long temporal data can be alleviated when high-resolution spatially extended data are available. These results are shown to hold true independent of various details of model or different spatial dispersal kernels such as Gaussian or heavily fat tailed. This study suggests that spatial data and monitoring multiple indicators of regime shifts can play a key role in making reliable predictions on ecosystem stability and resilience.

## Keywords

Catastrophic regime shifts Stability Resilience Early warning signals Indicators Spatial variance Spatial skewness## Introduction

It has long been suggested that multiple stable states leading to sudden flips and hysteresis are essential features of ecological systems (Lewontin 1969; Holling 1973; May 1977). Empirical studies show that lakes (Scheffer et al. 2001; Scheffer and Jeppesen 2007), marine ecosystems (Hare and Mantua 2000), vegetation in semiarid regions (Peters et al. 2004; Rietkerk et al. 2004; D’Odorico and Porporato 2004), rangelands (Anderies et al. 2002; Bestelmeyer 2006), food webs (Daskalov et al. 2007), and climatic systems undergo abrupt transitions leading to severe and widespread changes in ecological properties, thus lending strong credence to the hypothesis of the existence of alternative stable states (Scheffer et al. 2001; Schroder et al. 2005). Such abrupt changes are often referred to as “catastrophic regime shifts.” They can result in the degradation of ecosystems and the services they provide to human societies. The possibility of alternative stable states has important consequences in the spread of an invasive species under Allee effects, which in turn can threaten the structure of communities, ecosystems, and biodiversity (Kot et al. 1996; Taylor and Hastings 2005; Huggett 2005).

Theoretical and empirical studies have led to significant advances in identifying the underlying mechanisms for the occurrence of multiple stable states, as well as causes for switches between them (Scheffer et al. 2001; Scheffer and Carpenter 2003). However, a thorough study of mechanisms and parameterizations of models across a wide spectrum of terrestrial and aquatic ecological systems is, in practice, an impossible task and, hence, regime shifts are difficult to predict (Clark et al. 2001; Folke et al. 2005; Groffman et al. 2006). Therefore, an important task in ecology is to develop easily computable predictive measures of proximity to a tipping point that do not depend on the detailed understanding of ecological processes. Such indicators can be employed as early warning signals of impending catastrophic transitions with potential applications to conservation and management of ecosystems.

The role of spatial scales is central to ecology (Wiens 1989; Levin 1992), and recent studies have underlined the importance of space in the context of catastrophic regime shifts (Peters et al. 2004; Groffman et al. 2006). However, the theoretical work in this direction has received limited attention (Rietkerk et al. 2004; van Nes and Scheffer 2005; Buenau et al. 2007). Models of vegetation in semiarid ecosystems incorporating spatial scales and water-vegetation positive feedback mechanisms suggest that regular patterns of spots or deviations from power law patchy distributions can be indicators of regime shifts in those systems (von Hardenberg et al. 2001; Rietkerk et al. 2004; Kéfi et al. 2007). In a model of population where the individuals grow logistically and the dispersal is nearest neighbor (i.e., local), a potential indicator of extinctions is the spatial variance of the population, which increases according to scaling laws (Obórny et al. 2005). However, the generality of these measures remains an open question. The aim of this study is to present new leading indicators of regime shifts in spatially extended ecological systems that can be widely applicable.

In this paper, we study a spatially extended model of a population that exhibits bistable dynamics under the influence of grazing pressure. We propose that increases in the variance and changes in the skewness of the spatial probability distribution, i.e., of a *spatial snapshot* of the relevant ecological variable can be useful leading indicators of catastrophic regime shifts. We obtain a striking result, using an analytical approximation and confirmed by numerical simulations, that our results are independent of the choice of dispersal kernels, such as Gaussian or fat-tailed, and holds in a reaction-diffusion system as well. A key drawback of the individual nonspatial indicators is that they need long time series for reliable evaluation. In contrast, we have focussed on the instantaneous distribution of data obtained at different spatial points, thus alleviating this problem. Furthermore, we show that the inclusion of spatially explicit dynamic processes can open up new ways of devising indicators. As an example, our numerical simulations and semianalytic calculations show a novel feature that an observation of an increasing spatial variance, together with a peak in the spatial skewness, is an unambiguous measure of proximity to an impending catastrophic regime shift. This warning may provide a clear signal to initiate conservation strategies. Finally, we discuss the limitations of our work and possibilities of false alarms and make suggestions for future work.

## Model and methods

### Mean field model

*V*is the logistically growing population with growth rate

*r*and carrying capacity

*V*

_{ c }under a mean grazing rate, which can reach the maximum value of

*c*. This is a popular model in the ecology literature exhibiting bistable dynamics and has been applied to study variety of systems such as exploitation of fish communities (Steele and Henderson 1984), vegetation in a semiarid ecosystem (Noy-Meir 1975), and spruce budworm dynamics (May 1977), but typically in a nonspatial context. In this paper, we view the model as that of a vegetation under the influence of grazing in a semiarid ecosystem (Noy-Meir 1975).

### Spatially explicit model with stochasticity

We now include stochasticity and spatial interactions in the model. We first discuss the model in one dimension (Eq. 2) with a general dispersion kernel *k*(*x*,*y*) to test the robustness of our results with respect to different kernels. We present the two-dimensional model with a diffusive spread of the seeds (Eq. 3) to address the issue of detecting and preventing a regime shift.

**Generic seed dispersal model:**The spatially explicit model in one dimension in which the seeds are dispersed according to a kernel

*k*(

*x*,

*y*) is given by a stochastic partial integro-differential equation:

*c*, and it is modeled as a Gaussian white noise process with mean zero and variance \(\sigma_c^2\), i.e., \(\langle \eta_c(x,t) = 0 \rangle\) and \(\langle \eta_c(x,t) \eta_c(x',t') \rangle = \sigma_c^2 \delta(t-t')\delta(x-x')\). The angular brackets in these expressions indicate an average performed over space and time. Formally, white noise

*η*

_{ c }is a space-time derivative of the Weiner process

*W*(

*x*,

*t*) (Gardiner 2003). Biologically, these properties of grazing rates can be interpreted as the total number of grazers remaining approximately constant in the large spatial area at any given time (when the mean grazing rate

*c*is constant). However, the number of grazers in a given “spatial patch” varies stochastically with time. In other words, this process can interpreted as arising from (random) change in the number of grazers in a given spatial area in response to changing conditions of vegetation or other sources of variability in the environment not accounted for in the model.

The spatial interactions determine how the vegetation interacts with the surroundings and spreads in space. We assume that the spatial habitat is homogeneous and the interactions are isotropic and depend only on the relative distance between the plant (at *x*) and the spatial region of interest (at *y*), i.e., *k*(*x*,*y*) = *k*(|*y* − *x*|). We assume that the kernel is always nonnegative and is normalized, \(\int_\Omega k(y) dy = 1\), where Ω is the spatial region over which the dispersal occurs: this normalization condition implies that the seed from the parent plant always falls within the spatial region Ω. With these characteristics of the kernel, the spatial interaction term in Eq. 2 leads to an increase in the growth rate when the local vegetation density is lower than the neighbors or a decrease in the growth rate if the local density is higher. In other words, the spatial kernels of the form assumed tend to make the vegetation density uniform.

Dispersal kernels and their properties

Name | k(x) | Moments | \(\tilde{k}(s)\) (generating function) |
---|---|---|---|

Gaussian | \(\frac{1}{\sqrt{2 \pi w^2}}e^{-\frac{x^2}{2w^2}} \) | \(k_2=w^2\), \(k_4=3w^4\) | \(e^{\frac{w^2 s^2}{2}}\) |

Fat tailed | \(\frac{1}{4\alpha}e^{-\sqrt{|x|/\alpha}}\) | \(k_2=5!\alpha^2\), \(k_4=9!\alpha^4\) | Does not exist |

Heavily fat-tailed | \(\frac{1}{\pi}\frac{\beta}{\beta^2+x^2}\) | Does not exist | Does not exist |

**Diffusive seed dispersal model:**By assuming near-neighbor dispersal only, we can reduce the integro-differential equation to a reaction diffusion equation:

*D*=

*w*

^{2}/2. Here, the symbol

*w*

^{2}represents the second central moment of the Gaussian dispersal kernel (see Table 1). This model is studied in two spatial dimensions (see Section Detection and prevention of an impending regime shift). We emphasize that we have investigated the indicators for models represented by Eqs. 2 and 3 in which the noise-averaged system is spatially homogeneous and has no spatial patterns. However, we focus on individual snapshots of the system that has spatial variations.

### Numerical methods

We discretize the vegetation system on a spatial lattice to reduce the continuous stochastic partial differential equations to a finite number of coupled stochastic differential equations. Specifically, the spatially extended system of Eq. 2 under different dispersal kernels is solved numerically on a one-dimensional lattice of size *N* = 16,384 with periodic boundary conditions. To solve the temporal part of the equation, we employ Ito calculus and discretize the equation using an Euler forward time scheme, which is first-order accurate in time when averaged over many realizations or \(\sqrt{dt}\) accurate for a single simulation. To numerically solve the reaction diffusion equation (Eq. 3), we use the fully explicit forward-time, centered-space differencing scheme on a two-dimensional spatial grid of 128 × 128 squares with periodic boundary conditions, using MATLAB and C++. We provide explicit expressions evaluated to solve these equations numerically in Appendix A (Electronic Supplementary Material).

We chose the discretization units of *dx* = 0.1 and *dt* = 0.01 for solving Eq. 2 and *dt* = 0.001 for Eq. 3. We provide below some representative values for various scales and parameters. Each square grid of the two-dimensional lattice can be taken to have an area \(100~\text{m}^2\) and one time unit to be 1 year. In these units, the time step of integration corresponds to approximately 3 days for Eq. 2 and one third of a day for Eq. 3. The values of diffusion constants used in this study, which range from *D* = 0.001 to 0.1 units, correspond approximately to 0.03 to \(3~\text{m}^2~\text{day}^{-1}\), which covers the typical values of plant dispersal coefficients used in previous studies (HilleRisLambers et al. 2001). However, we assume coarse graining of the system on scales such that the spatial patterns observed in semiarid regions are not important. Unless stated otherwise, the initial conditions for all of our simulations is a spatially homogeneous high-density stable vegetated state corresponding to the choice of grazing rate *c* at *t* = 0. We then let the system evolve according to Eqs. 2 or 3.

*V*(

*x*,

*t*) as a function of space and time, its spatial variance \(\bar{\sigma}_V^2\) and spatial skewness \(\bar{\gamma}_V\) at a time

*t*are obtained by averaging over space:

*V*at time

*t*when the spatial snapshot is taken and

*L*is the length (or the area in two dimensions) of the spatial region of interest, Ω.

## Results

The deterministic nonspatial model (Eq. 1) exhibits a bistable region with high-density and low-density vegetated states when the mean grazing rate *c* lies in the interval (18,26). When the grazing rate exceeds *c* ^{*} = 26, the system collapses from a densely vegetated state to a low-density state (see Fig. 1e). The key feature underlying our approach is the probability density (histogram) of a snapshot of the spatial pattern of the state variable, vegetation biomass density, changing qualitatively as we move closer to the threshold. This is seen in Fig. 1c, showing that the distribution has a narrow width and is nearly symmetric for *c* = 18.0, which is far from the threshold of transition. At *c* = 24.0, which is close to the threshold of the transition, we observe that the distribution is broadened and, in addition, it develops an asymmetric tail (see Fig. 1d). These features of the instantaneous spatial probability distribution function form the basis for the quantitative spatial indicators suggested and investigated in this paper. We emphasize that the probability distribution under consideration is evaluated for the spatial data instantaneously at any given time, and not from time series data studied in nonspatial models (Carpenter and Brock 2006; Guttal and Jayaprakash 2008).

### Spatial variance and spatial skewness: leading indicators of regime shift

We address the interesting issue of the dependence of our results on the form of the spatial kernel used in the evolution equation. To this end, we study the model under the influence of different spatial kernels in one spatial dimension as given by Eq. 2. We have quantified the width of the spatial probability distribution by its variance and the asymmetry by its skewness. The results for different kernels are shown as a function of grazing rate in Fig. 1f–g. Both the spatial variance and the spatial skewness increase as the grazing rate approaches the threshold at *c* ^{*} = 26 and, hence, can serve as early warning signals of an impending regime shift. The results are clearly independent of the details of the kernel.

An intuitive understanding for the behavior of indicators based on the features of potential landscape (Fig. 1a–b) and an estimate of spatial indicators can be obtained by a mean-field approximation, described in detail in Appendix B (Electronic Supplementary Material) (Van den Broeck et al. 1994). A striking prediction of this analysis, consistent with the results of numerical simulations, is that the spatial indicators thus obtained are independent of the spatial kernel, a result that holds true independent of dimension. Similar calculations on a two spatial dimensional lattice for the diffusion model of Eq. 3 show that the results of spatial variance and spatial skewness as early warning signals continue to hold true. Furthermore, our analytical calculations show that these results do not depend on the details of the specific model of this paper as long as we consider models where there is a regime shift between spatially homogeneous states. More research is needed to study the applicability of these indicators in models where inclusion of spatial interactions leads to patterns, for example, as found in Rietkerk et al. (2004). See the Discussion section and Appendix C (Electronic Supplementary Material) for detailed comments and calculations on the generality of these results.

### Detection and prevention of an impending regime shift

In this section, we establish that spatial indicators can be used to detect and prevent an impending vegetation collapse by simulating a scenario that is likely to occur in the field. We do this by employing the spatial model with diffusion in two dimensions (Eq. 3) that makes the study numerically feasible. We show that a joint monitoring of spatial variance and skewness as a function of time can lead to a novel and an unambiguous measure of an impending transition. Furthermore, the warning may occur sufficiently in advance so that management practices can be initiated to prevent the undesirable collapse of vegetation.

**Simulating effects of increasing grazing pressure:**We consider a scenario in which the grazing rate

*c*is increasing with time, a widely observed consequence of enhanced human influence on ecosystems (Millenium-Ecosystem-Assessment 2005; Kéfi et al. 2007). We begin with a high-density vegetated state with the mean grazing rate

*c*= 21, which undergoes a discrete increment of 0.1 units (approximately 0.5%) every year. The evolution of spatial patterns is depicted in Fig. 2a. The ecosystem continues in the high vegetation state, albeit slowly declining in its density until year 48. Within the next 4 years, the vegetation undergoes a dramatic collapse to a low-density state. During this interval of abrupt collapse, a number of tiny low-density patches emerge, which are clearly identifiable if the complete spatial data is available in year 48. These patches grow bigger in size with time and coalesce with nearby patches, leading to a rapid decline in vegetated areas.

The abrupt nature of collapse is readily captured in the plot of spatial mean, \(\bar{V}\) in Fig. 2b(i) and the spatial indicators in Fig. 2b(ii–iii), which exhibit substantial changes during the period of transition. The key issue is the ability to predict proximity to a transition, and hence, the question is whether the spatial indicators show observable changes prior to the obvious collapse seen either by the drop in the spatial mean \(\bar{V}\) or the visual observation of large patches. In order to answer this question, we have plotted a magnified version of the spatial mean, variance, and skewness from year 40 to 56. The spatial mean is decreasing slightly with time up to year 46, and a small number of tiny patches are visible. On the other hand, the spatial variance has increased by nearly 300% and the skewness has changed from zero to one by this time, therefore offering us two potential early warning signals of an impending transition when the spatial mean does not display a clear sign of the collapse.

**Peaking spatial skewness with increasing variance—an unambiguous indicator:** Next, we argue that a joint monitoring of spatial variance and skewness can be used to deduce an impending transition unambiguously. Results in Fig. 2b(iv–vi) shows that spatial skewness peaks at year 47 even as the spatial variance continues to increase and the spatial mean shows quick decline beginning year 48. We propose that a peak in skewness when observed with a continued increase in variance can constitute a clear signature of an imminent catastrophic vegetation collapse. We have observed this feature of skewness en route to a regime shift in all the spatial simulations we have performed. In Appendix D (Electronic Supplementary Material), we employ a generic bistable model and argue semianalytically that this feature is not dependent on the details of the choice of model in this paper.

**Prevention of regime shift:** We performed additional simulations to check if this sign of regime shift occurs sufficiently early to be utilized for the prevention of the regime shift by beginning to reduce the grazing activity from year 48, which is a year after the peak occurred in skewness, modeling a scenario of time taken for the managers to detect the peak. The results for different conservation strategies are shown in Fig. 2c(i–ii). These sample simulations suggest the plausibility of a maintaining ecosystem in the highly vegetated state if the reduction rate in the grazing activity is beyond a certain threshold. We emphasize that we have employed representative numbers, for example, for the time scale, for these simulations and a more accurate parameterizations of ecosystems is required for predictive purposes.

In order to mimic such a scenario where resolution of spatial data is limited, we utilized only a part of the data available from the simulation and evaluated the indicators based on sparse data. More specifically, we looked at data collected at grid points separated by eight sites (hence, a total of 16 × 16 data points), 16 sites (8 × 8), and 32 sites (4 × 4; see Fig. 2b(iv–vi)). It is striking that the temporal trends of indicators largely remain unaffected even for as small a data set as 8 × 8, but at further lower resolutions, they may not be statistically significant, limiting their practical utility.

## Discussion

Our analysis of a spatially explicit model of vegetation exhibiting alternative stable states has shown that an increase in the spatial variance and the changes in spatial skewness obtained from the instantaneous spatial snapshot of the ecological system can serve as leading indicators of impending regime shifts. This study shows that the availability of spatial data not only leads to quantitative improvements in providing early warning signals, but can lead to qualitatively different and novel ways of predicting proximity to a threshold. We demonstrated this by suggesting a new indicator based on the relative dynamical behavior of two independent spatial indicators: an increasing spatial variance together with a peak in the magnitude of the spatial skewness is an unambiguous measure of proximity to a catastrophic regime shift. Studying individual indicators poses an important limitation because it is not clear as to how much of a change in a specific individual indicator is sufficient to be signalled as an early warning signal. We have shown this joint measure of peaking spatial skewness with increasing spatial variance can point to a resolution of that problem. In addition, our calculations showed that quick action (i.e., reduction in grazing activity) is needed soon after the early warning is detected to prevent the vegetation collapse. We comment on various features of spatial indicators along with their limitations below.

Catastrophic transitions occur in a wide array of aquatic, terrestrial, climatic, and complex socioeconomic systems (Scheffer et al. 2001; Brock and Carpenter 2006). It is obvious that the scale of interactions can vary significantly between, and even within any of these complex systems: from the diffusive spread of nutrients to the complex hydrodynamics of storms in lakes. Empirical data on animals and seeds indicate that their dispersal can be strongly leptokurtic, i.e., the concentration of dispersal at the source and the tails is larger than what a Gaussian dispersal with comparable mean and variance would predict. Theoretical studies show that even infrequent long-range dispersal can disproportionately influence the structure of the population, community, and ecosystems (Kot et al. 1996; Levin et al. 2003). For the purpose of detecting an impending regime shift, we have shown in our calculations that spatial indicators proposed in this paper are insensitive to the details of the specific dispersal kernel, as long as they are positive definite, and the choice of the model. Due to these features, our results can be potentially applicable to a wide variety of real ecosystem data.

Clearly, moments such as spatial variance are important in understanding ecological patterns (Horne and Schneider 1995) and have been measured in contexts such as populations and abundance of species (Logerwell et al. 1998; He and Gaston 2003). Increasing availability of spatially explicit data makes our results relevant: e.g., remotely sensed data makes semiarid vegetation systems an excellent candidate ecosystem to test the applicability of indicators suggested in this work (Barbier et al. 2006; Scanlon et al. 2007; Kéfi et al. 2007). A practically important point is that nonspatial moments, calculated as moving averages, reflect the state of the system over the interval used to calculate the moving average, potentially underestimating the actual proximity to a threshold. In contrast, spatial indicators obtained from instantaneous measurements provide a more accurate measure of the state of the system at that time. Furthermore, we find that the spatial moments provide early warning signals even when nonspatial indicators cannot.

Next, we discuss the possibility of false alarms. Changes in variance and skewness as suggested in this paper can occur near other bifurcations with no abrupt transitions (eg. a bifurcation leading to cycles); they can also occur when the potential landscape corresponding to only a single stable state shows flattening and asymmetric properties. In such cases, they do not indicate abrupt switch to an alternative state. Unfortunately, this drawback exists in all of the indicators developed so far and we suggest that they need to be addressed in a context dependent way. For instance, in many cases, one is often studying alarms in ecosystems where there is a perceived threat of a regime shift based on historical data and/or model studies. Another way in which false alarms can occur is due to external noise that itself shows changing variance and skewness with time, in contrast to the fixed variance Gaussian noise we have used. Techniques such as dynamical linear modeling of time series that can filter such sources of noise can be potentially useful for such cases if the data of the external noise is also available (Carpenter and Brock 2006). For the model system and the spatial interactions studied in this paper, we do not find a false positive whenever sufficiently fine resolution data are available. We believe this is a consequence of the instantaneous nature of the (large) data available in spatially extended systems in contrast to time series indicators wherein a false positive can occur even under ideal conditions of data availability. This, however, does not rule out the possibility of false alarms while studying indicators in real ecosystems but merely as pointing to the greater reliability on signals provided by spatial indicators.

We note there are several sources of errors associated with ecological variables, as well as measurement of ecological data. An important constraint on this approach is the limited availability of spatial data that can render the evaluations of indicators inaccurate; for example, a conclusive evidence of peaking skewness may occur too late to be practically useful. While we have established (in Appendix D (Electronic Supplementary Material)) that the peaking of the magnitude of skewness before variance is a general phenomenon, the time interval between these will depend on the specific ecosystem; it might very well be too short to take corrective action in specific cases. We have also ignored the possibility of spatial heterogeneity and correlated noise in grazing rates that may be important in regime shifts (Scheffer and Carpenter 2003; van Nes and Scheffer 2005). We expect that weak heterogeneity will not affect the main conclusions of the paper. Empirical studies in semi-arid ecosystems suggest that observed patterns of vegetation are a result of self-organization with the spatial heterogeneity playing a minimal role (Barbier et al. 2006). Next, we note that an effective potential function does not exist in many complex ecological models with more than one variable (Gandhi et al. 1998; Durrett and Levin 1994). Even in these cases, as long as there is a regime shift between two homogeneous states in the absence of noise, we believe that our approach should be fruitful. For the model without spatial degrees of freedom, the similarities between models with and without a potential function have been shown (Guttal and Jayaprakash 2008). In addition, a number of ecological models that show self-organizing behavior and spatial patterns prior to regime shifts have been proposed (von Hardenberg et al. 2001; Rietkerk et al. 2004; Kéfi et al. 2007). In our model, both stable states are spatially homogeneous in the absence of noise. Whether the spatial indicators developed in this study can be applied to regime shifts occurring through the route of spatial self-organized patters such as regular spots or power law distribution of patches or in the absence of potential functions needs further study.

In summary, our findings based on semianalytic and numerical calculations of spatially explicit models show that studying the variance and skewness of the probability distribution obtained from instantaneous spatial data can provide early warning signals of regime shifts in ecosystems. With the empirical evidence for catastrophic transitions growing enormous across different ecosystems such as lakes, forests, and regional climates, the need of the hour is to apply these in field situations to validate the theoretical results. Our study suggests that high-resolution spatial data and monitoring multiple indicators of regime shifts can play a key role in making reliable predictions.

## Notes

### Acknowledgements

This work is supported by National Science Foundation grant DEB-0410336. VG was also supported by Presidential Fellowship at The Ohio State University and a travel fellowship from Institute for Complex Adaptive Matter through National Science Foundation grant DMR-0456669.

## Supplementary material

## References

- Anderies J, Janssen M, Walker B (2002) Grazing management, resilience, and the dynamics of a fire-driven rangeland system. Ecosystems 5:23–44CrossRefGoogle Scholar
- Barbier N, Couteron P, Lejoly J, Deblauwe V, Lejeune O (2006) Self-organized vegetation patterning as a fingerprint of climate and human impact on semi-arid ecosystems. J Ecol 94:537–547CrossRefGoogle Scholar
- Bestelmeyer B (2006) Threshold concepts and their use in rangeland management and restoration: the good, the bad, and the insidious. Restor Ecol 14:325–329CrossRefGoogle Scholar
- Brock WA, Carpenter SR (2006) Variance as a leading indicator of regime shift in ecosystem services. Ecol Soc 11:9Google Scholar
- Buenau KE, Rassweiler A, Nisbet RM (2007) The effects of landscape structure on space competition and alternative stable states. Ecology 88:3022–3031PubMedCrossRefGoogle Scholar
- Carpenter SR, Brock WA (2006) Rising variance: a leading indicator of ecological transition. Ecol Lett 9:308–315Google Scholar
- Clark J, Carpenter S, Barber M, Collins S, Dobson A, Foley J, Lodge D, Pascual M, Pielke R Jr, Pizer W et al (2001) Ecological forecasts: an emerging imperative. Science 293:657–660PubMedCrossRefGoogle Scholar
- Daskalov GM, Grishin AN, Rodionov S, Mihneva V (2007) Trophic cascades triggered by overfishing reveal possible mechanisms of ecosystem regime shifts. Proc Natl Acad Sci U S A 104:10518–10523PubMedCrossRefGoogle Scholar
- D’Odorico P, Porporato A (2004) Preferential states in soil moisture and climate dynamics. Proc Natl Acad Sci U S A 101:8848–8851PubMedCrossRefGoogle Scholar
- Durrett R, Levin SA (1994) The importance of being discrete (and spatial). Theor Popul Biol 46:363–394CrossRefGoogle Scholar
- Folke C, Carpenter S, Walker B, Scheffer M, Elmqvist T, Gunderson L, Holling CS (2005) Regime shifts, resilience, and biodiversity in ecosystem management. Ann Rev Ecol Evol Syst 35:557–581CrossRefGoogle Scholar
- Gandhi A, Levin S, Orszag S (1998) “Critical slowing down” in time-to-extinction: an example of critical phenomena in ecology. J Theor Biol 192:363–376PubMedCrossRefGoogle Scholar
- Gardiner CW (2003) Handbook of stochastic methods for physics, chemistry and the natural sciences, 3rd edn. Springler, HeidelbergGoogle Scholar
- Groffman P, Baron J, Blett T, Gold A, Goodman I, Gunderson L, Levinson B, Palmer M, Paerl H, Peterson G et al (2006) Ecological thresholds: the key to successful environmental management or an important concept with no practical application? Ecosystems 9:1–13CrossRefGoogle Scholar
- Guttal V, Jayaprakash C (2007) Impact of noise on bistable ecological systems. Ecol Model 201:420–428CrossRefGoogle Scholar
- Guttal V, Jayaprakash C (2008) Changing skewness: an early warning signal of regime shifts in ecological systems. Ecol Lett 11:450–460PubMedCrossRefGoogle Scholar
- Hare S, Mantua N (2000) Empirical evidence for North Pacific regime shifts in 1977 and 1989. Prog Oceanogr 47:103–145CrossRefGoogle Scholar
- Hastings A (2004) Transients: the key to long-term ecological understanding? Trends Ecol Evol 19:39–45PubMedCrossRefGoogle Scholar
- He F, Gaston KJ (2003) Scale-dependent spatial variance patterns and correlations of seabirds and prey in the southeastern bering sea as revealed by spectral analysis. Am Nat 162:366–375PubMedCrossRefGoogle Scholar
- Held H, Kleinen T (2004) Detection of climate system bifurcations by degenerate fingerprinting. Geophys Res Lett 31:L020972CrossRefGoogle Scholar
- HilleRisLambers R, Rietkerk M, van den Bosch F, Prins HHT, de Kroon H (2001) Vegetation pattern formation in semi-arid grazing systems. Ecology 82:50–61CrossRefGoogle Scholar
- Holling CS (1973) Resilience and stability of ecological systems. Ann Rev Ecolog Syst 4:1–23CrossRefGoogle Scholar
- Horne J, Schneider D (1995) Spatial variance in ecology. Oikos 74:18–26CrossRefGoogle Scholar
- Horsthemke W, Lefever R (1984) Noise-induced transitions. Springer, New YorkGoogle Scholar
- Huggett A (2005) The concept and utility of ‘ecological thresholds’ in biodiversity conservation. Biol Conserv 124:301–310CrossRefGoogle Scholar
- Kéfi S, Rietkerk M, Alados CL, Pueyo Y, Papanastasis VP, ElAich A, de Ruiter PC (2007) Spatial vegetation patterns and imminent desertification in mediterranean arid ecosystems. Nature 449:213–217PubMedCrossRefGoogle Scholar
- Kleinen T, Held H, Petschel-Held G (2003) The potential role of spectral properties in detecting thresholds in the Earth system: application to the thermohaline circulation. Ocean Dyn 53:53–63CrossRefGoogle Scholar
- Kot M, Lewis M, van den Driessche P (1996) Dispersal data and the spread of invading organisms. Ecology 77:2027–2042CrossRefGoogle Scholar
- Levin S (1992) The problem of pattern and scale in ecology: the Robert H. MacArthur award lecture. Ecology 73:1943–1967CrossRefGoogle Scholar
- Levin S, Muller-Landau H, Nathan R, Chave J (2003) The ecology and evolution of seed dispersal: a theoretical perspective. Ann Rev Ecol Evol Syst 34:575–604CrossRefGoogle Scholar
- Lewontin R (1969) The meaning of stability. Brookhaven Syps Biol 22:13–24Google Scholar
- Logerwell E, Hewitt R, Demer D (1998) Scale-dependent spatial variance patterns and correlations of seabirds and prey in the Southeastern bering sea as revealed by spectral analysis. Ecography 21:212–223CrossRefGoogle Scholar
- May RM (1977) Thresholds and breakpoints in ecosystems with a multiplicity of stable states. Nature 269:471–477CrossRefGoogle Scholar
- Millenium-Ecosystem-Assessment (2005) Ecosystems and human well-being: desertification synthesis. Island, Washington, DCGoogle Scholar
- Noy-Meir I (1975) Stabilty of grazing systems: an application of predator–prey graphs. J Ecol 63:459–482CrossRefGoogle Scholar
- Obórny B, Meszéna G, Szabó G (2005) Dynamics of populations on the verge of extinction. Oikos 109:291–296CrossRefGoogle Scholar
- Peters D, Pielke Sr R, Bestelmeyer B, Allen C, Munson-McGee S, Havstad K (2004) Cross-scale interactions, nonlinearities, and forecasting catastrophic events. Proc Natl Acad Sci U S A 101:15130PubMedCrossRefGoogle Scholar
- Rietkerk M, Dekker SC, de Ruiter PC, van de Koppel J (2004) Self-organized patchiness and catastrophic regime shifts in ecosystems. Science 305:1926–1929PubMedCrossRefGoogle Scholar
- Scanlon T, Caylor K, Levin S, Rodriguez-Iturbe I (2007) Positive feedbacks promote power-law clustering of Kalahari vegetation. Nature 449:209–12PubMedCrossRefGoogle Scholar
- Scheffer M, Carpenter SR (2003) Catastrophic regime shifts in ecosystems: linking theory and observation. Trends Ecol Evol 18:648–656CrossRefGoogle Scholar
- Scheffer M, Carpenter SR, Foley JA, Folke C, Walker B (2001) Catastrophic shifts in ecosystems. Nature 413:591–596PubMedCrossRefGoogle Scholar
- Scheffer M, Jeppesen E (2007) Regime shifts in shallow lakes. Ecosystems 10:1–3CrossRefGoogle Scholar
- Schroder A, Persson L, Roos AMD (2005) Direct experimental evidence for alternative stable states: a review. Oikos 110:3–19CrossRefGoogle Scholar
- Steele JH, Henderson EW (1984) Modelling long-term fluctuations in fish stocks. Science 224:985–987PubMedCrossRefGoogle Scholar
- Taylor C, Hastings A (2005) Allee effects in biological invasions. Ecol Lett 8:895–908CrossRefGoogle Scholar
- Van den Broeck C, Parrondo JMR, Toral R (1994) Noise-induced nonequilibrium phase transition. Phys Rev Lett 73:3395–3398PubMedCrossRefGoogle Scholar
- van Nes E, Scheffer M (2005) Implications of spatial heterogeneity for catastrophic regime shifts in ecosystems. Ecology 86:1797–1807CrossRefGoogle Scholar
- van Nes EH, Scheffer M (2007) Slow recovery from perurbations as a generic indicator of a nearby catastrophic regime shift. Am Nat 169:738–747PubMedCrossRefGoogle Scholar
- von Hardenberg J, Meron E, Shachak M, Zarmi Y (2001) Diversity of vegetation patterns and desertification. Phys Rev Lett 2001:198101:1–4Google Scholar
- Wiens JA (1989) Spatial scaling in ecology. Funct Ecol 3:385–397CrossRefGoogle Scholar
- Wissel C (1984) A universal law of the characteristic return time near thresholds. Oecologia 65:101–107CrossRefGoogle Scholar