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Ecological Thresholds and Regime Shifts: from theory to operation

Tom Andersen

Department of Biology, University of Oslo,

P.O. box 1066, Blindern, N0316 Oslo, Norway

Jacob Carstensen

National Environmental Research Institute, University of Aarhus

P.O. Box 358, DK-4000 Roskilde, Denmark

Emilio Hernández-García

IFISC (CSIC–UIB), Instituto de Física Interdisciplinar y Sistemas Complejos

Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain

Carlos M. Duarte

IMEDEA (CSIC-UIB), Instituto Mediterráneo de Estudios Avanzados,

C/Miquel Marqués 21, 07190 Esporles (Islas Baleares), Spain

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Abstract 1

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There is an apparent gap between the prominence of present theoretical frameworks involving

ecological thresholds and regime shifts, and the paucity of efforts to conduct simple tests and

quantitative inferences on the actual appearance of such phenomena in ecological data. There

is a wide range of statistical methods and analytical techniques now available that render

these questions tractable, some of them even dating half a century back. Yet, their application

has been sparse and confined within a narrow subset of cases of ecological regime shifts. Our

objective is to raise awareness on the range of techniques available, and to their principles and

limitations, in order to promote a more operational approach to the analysis of ecological

thresholds and regime shifts.

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Regime Shifts in Ecology 1

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The observation that managed ecosystems often fail to respond smoothly to changing

pressures has generated perplexity and eventually lead researchers to draw parallels between

the behaviour of ecological systems and other complex systems with non-linear dynamics,

such as the global climate, the human immune system, and the world economy (cf. [1] for a

popular account). Initial reports of kelp forest disturbance and recovery [2], freshwater

ecosystem shifts engineered by beavers [3], and vegetation shifts affected by fire [4] have lead

on to an ever-growing research effort on ECOLOGICAL THRESHOLDS and REGIME SHIFTS (see

Glossary), whose underlying theoretical framework [5, 6] (Box 1) has been shown to be

applicable to a broad range of ecosystems from coral reefs to forests and lakes [7,8]. These

concepts are now also making their way into the minds and discussions of policy makers and

might soon be translated into legislative frameworks [9].

Ecological regime shifts can be defined as abrupt changes on several trophic levels [10],

leading to rapid ecosystem reconfiguration between alternative states. These shifts are

generally thought to be driven by external perturbations (e.g. climatic fluctuations,

overexploitation, eutrophication, and invasive species), but the exact mechanism is often

unclear. The subject has become a fast growing scientific discipline, manifested by a 12-fold

increase in publications between 1991 and 2006, twice as fast as the growth rate of research

effort in ecology as a whole (7.7 % year-1, ISI Web of Science). Most of the reported cases of

ecological regime shifts are inferred from time series of monitoring data, while direct

evidence by controlled experiments of the existence of alternative states is difficult to find

[11]. Surprisingly, the general techniques available to test for regime shifts and thresholds

have only to a limited extent been applied to these data sets. As formal tests of regime shifts

have a long history in the context of climate change research (e.g. [12]) it is not surprising

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that formal statistical tests for ecological regime shifts have mostly been restricted to the

effects of climate change on marine communities [13]. These observations suggest that there

is a need to increase the awareness of ecologists on the availability and diversity of

approaches allowing inferential analyses of ecological regime shifts and thresholds, helping

this important research field to move to a more operational phase.

Here, after exploring research efforts in several fields we provide a review of methods for

regime shift and threshold detection relevant to ecosystems, including both informal

EXPLORATORY DATA ANALYSIS and formal HYPOTHESIS TESTING approaches, with the aim of

encouraging a more quantitative approach to the study of these phenomena. Finally, we

provide an operational summary of available software that can be useful for investigating

abrupt changes in ecological data sets. As some of the terms are used differently among

different research traditions, a glossary is provided.

Detecting thresholds and regime shifts in ecological data

Figure 1 shows that there are at least three ways by which an ecological system might exhibit

abrupt changes over time; two which are reversible in response to changes in environmental

drivers, while a third (Figure 1C) and most undesirable one is not [14]. Thus the existence of

an abrupt CHANGE-POINT is a necessary but not sufficient condition for demonstrating

BISTABILITY and HYSTERESIS (Box 1), as it might actually derive from sudden changes in the

main drivers of the systems. It should also be kept in mind that while most ecological regime

shifts are inferred from abrupt changes over time, time itself is never the actual underlying

driver. Identification of the environmental driver(s) is complicated by the general

interrelatedness of different social and environmental factors, and often also by the lack of

data. Identification of a change-point in time is therefore the natural first step towards

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identifying a potential driver, which again is the first step towards identifying a regime shift

mechanism that might eventually be relevant for policy-making.

There is an abundance of methods for identifying abrupt changes in time series, most of them

developed in scientific fields other than ecology. The basic change-point problem, i.e.

detecting a step change in the mean value in a sequence of random variables, has a long

history in statistical inference (Box 2). The general scientific literature contains a bewildering

diversity of methods that in a widest sense correspond to change-point detection, either in

time or space (Box 3). In this review we contend that terms like regime shift [10,14], abrupt

change [15], break- or change-point [16], STRUCTURAL CHANGE [17], ecological threshold

[18], tipping point [19], and observational inhomogeneity [20], basically address the same

problem and that methods developed for their analysis should have general relevance for the

study of ecological regime shifts. The rapid growth of this literature already makes it hard to

maintain an overview, and increases risk of unnecessary reinvention. For example, one of the

threshold detection methods proposed in Ref. [21] is basically a rediscovery of the basic

change-point problem presented a half a century earlier in Ref. [22] (Box 2).

Exploratory data analysis

A substantial part of the literature on ecological thresholds and regime shifts follows an

explorative approach where data are pre-processed in various ways that render the presence of

thresholds or jumps more evident to heuristic inspection, but usually without any statistical

significance tests. The AVERAGE STANDARD DEVIATES (ASD) compositing method [23] is a

rather popular representative based on simple heuristics rather than an underlying statistical

model. The ASD was for example used to propose the occurrence of regime shifts in the

North-Pacific in 1977 and 1989 [24] (Box 4). It has however been demonstrated that the ASD

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method is prone to false positives for AUTOCORRELATED time series, that is, to detect a regime

shift when in fact there is none [25]. We recommend therefore that ASD, despite its

popularity, is replaced by methods presented below for inference on regime shifts in ecology.

PRINCIPAL COMPONENT ANALYSIS (PCA) and related techniques are known under a variety of

names (EMPIRICAL ORTHOGONAL FUNCTIONS (EOF), SINGULAR SPECTRUM ANALYSIS (SSA),

etc.). PCA is used to compress, by linear combinations, a large number of correlated time

series into a small number of uncorrelated ones that contain as much as possible of the

original total variance [26, 27]. The presence of threshold phenomena in the reduced set could

become more evident to visual inspection, but further processing and statistical testing is

recommended. Applications to regime shift detection include the reduction of 100 climatic

and ecological time-series from the North-Pacific into just two variables [24] (Box 4) or the

combination of different climatic indices related to Pacific fisheries into a single one [28].

The conclusions drawn from a PCA can be strengthened by combining it with other

independent approaches to multivariate time series analysis, such as CHRONOLOGICAL

CLUSTERING [29,30] (Box 4). For example, PCA and chronological clustering were found to

yield comparable regime shift patterns in 78 time series from the North and Wadden Seas

[31].

PCA methods have well known limitations [27], such as the inability to capture relationships

that are not linear, and the possibility of the reduced variables being distorted by the

requirement of linear independence. Variants of non-linear dimensionality reduction [32] have

been developed independently and also partly based on very different underlying concepts, in

for example cognitive psychology [33] and oceanography [34]. The ARTIFICIAL NEURAL

NETWORK-based approach [34] is claimed to be able to reveal multimodality, which in

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principle would be very relevant for detecting regime shift mechanisms related to bistability

and hysteresis loops. Unfortunately, it has also been shown that this approach [34] is prone to

false positives, as it is reported to find multimodality even in data sets generated from a

multivariate normal distribution [35], which, by definition, cannot be MULTIMODAL. Our

opinion of the current state of this field is that non-linear dimensionality reduction methods

should primarily be used if simpler methods such as PCA and chronological clustering have

been documented to be incapable of capturing important variations in a data set. Conclusions

will also in this case be strengthened if there is a general agreement between several

independent methods.

Inferential statistics and hypothesis testing

In the search for ecological regime shifts there is always a risk for thresholds being detected

in what is actually just random fluctuation. Statistical hypothesis testing aims at limiting this

possibility to a predetermined fixed value, typically a significance level of 5%. If the time of

the threshold event is known (e.g. introduction of an invasive species, change in management

practice, deforestation event) the significance probability of the regime shift under a null

hypothesis of no change can be analyzed using intervention methods from standard statistical

textbooks [36]. While originally aimed at testing for a shift in a time series following a

particular action, INTERVENTION ANALYSIS has also been applied to data where the change-

point was not known a priori, but hypothesized following exploratory data analysis [37].

Classical intervention analysis cannot be used for situations with the change-point occurring

at an a priori unknown time. This calls for sequential tests where the existence of a regime

shift is tested for at every point in time, and which must be characterized by higher critical

values of the test statistic than in classical statistical methods (cf. Box 2) due to the so-called

type I error (false positive) inflation in multiple tests. The underlying principle of the

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sequential methods is to compare a test statistic with its distribution under the null hypothesis.

Critical values at different significance levels are tabularized for regularly observed data

points, typically time series [38], whereas critical values for irregularly spaced observations

must be calculated case-by-case and therefore can be computationally costly, but the

continuous increase in computing power has greatly alleviated this constraint. Sequential test

methods have mainly been developed for univariate time series, particularly within

econometrics [17, 39] (Box 3) and climate research [40, 41] (Box 3).

The most commonly investigated regime shift hypothesis is a step change in mean level using

parametric [40, 42, 43] or non-parametric [44] methods. Regime shift detection methods

involving changing variance, shift in the frequencies of fluctuations, or even simultaneous

interrelated shifts in several ecosystem components at a particular point have also been

proposed [45], but their application to practical data analysis has so far been limited. The

computational burden increases exponentially with the number of change-points in the data

set [17]. While methods intended for identifying only single thresholds can also be employed

to the individual subsets separated by a significant change-point in a hierarchical fashion [46],

this will normally be less efficient than a DYNAMIC PROGRAMMING approach [39]. As the

goodness of fit will generally increase with the number of change-points, model selection

procedures involving penalties for the number of model parameters are to be recommended

[47].

One problem associated with the classical statistical framework for investigating regime shifts

against a null hypothesis with no regime shift is the lack of STATISTICAL TEST POWER for

robust inferences. Ecological time series displaying regime shifts are generally much shorter

(typically 20-40 time steps, usually years) than the typical time series that has driven the

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development of threshold detection methods in econometrics and climate research (often >

100 time steps). As change-points occurring at the extremes of a time series do not lend much

power to hypothesis testing, it is only those change-points located near the middle of the time

series that can be detected with confidence. Moreover, since ecological data are typically also

noisier than climatic or economic data, a null hypothesis of no change-point is unlikely to be

rejected within a classical statistical testing framework.

Testing the existence of hysteresis poses a statistical challenge even greater than threshold

identification, because modelling hysteresis requires a memory effect to be included into the

model formulation such that the present regime becomes dependent on previous states.

Statistical inference must therefore be based on comparing the observations with the output of

a dynamical model. In THRESHOLD AUTOREGRESSIVE (TAR) models the dynamics can switch

between different linear autoregressive models depending on a linear function of the previous

state relative to a threshold value [48]. The classical Canadian lynx population data could be

modelled with a TAR model having two regimes, representing the increasing and declining

phases of the lynx population cycle [49]. Otherwise, quantitative statistical studies of regime

shifts with hysteresis in ecology are remarkably few. Needless to say, data requirements are

high, as several transitions are needed to identify hysteresis effects (e.g. typically >10

transitions in the Canadian lynx data sets [49]). Additional complications caused by to

missing values, measurement errors, and non-stationarity could also contribute to the paucity

of applications of these analyses.

Although most of the threshold testing procedures described in the literature are univariate,

they can naturally be expanded to include multiple variables. The advantage of multivariate

analyses is that the power of testing increases provided that all variables show similar trends

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and have interactions that can be accounted for in the analysis. However, if only a subset of

the variables shows a threshold response, the power of the test decreases and the outcome can

become less clear. Simultaneous estimation of changes in the community interaction matrix

(i.e. the density-dependent effects of a population both on itself and on other populations) has

been suggested [50], but this approach will usually inflate the number of parameters such that

the data requirements will be beyond what is realistic for ecological time series.

Consequently, parsimonious consideration of the variables to be included in a multivariate

test is recommended.

Available software for analysing regime shifts

Most of the statistical methods discussed in this review are implemented in available software

packages (Table 1). Table 1 is not an exhaustive list of relevant software, but rather a

selection of possible starting points for scientists interested exploring different approaches to

quantitative regime shift detection. The list contains both tools requiring little background

knowledge, such as standalone products or Excel add-ins, as well as toolboxes or packages for

some of the major statistical computing environments such as R, Matlab, and O-matrix. The

emphasis in Table 1 is on inferential tools for hypothesis testing, but some of the software

listed also implements exploratory analysis methods.

Conclusions and future perspectives

The remarkable paucity of inferential analyses of ecological regime shifts and thresholds in

the literature is at odds with the vigorous growth of this research direction, and could be

attributable to the perception that these techniques are so data-demanding that only

exceptionally few long-term ecological data sets would meet the requirements. However, the

impressive impetus to the development and implementation of observational platforms across

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a broad range of ecosystems over the past two decades (e.g. the US NSF Long term ecological

research (LTER) network and the EU Water framework directive (WFD) monitoring system)

has already and will continue to deliver a wealth of data sets that could meet the requirements

of even the more demanding of the techniques available. Lack of awareness on available

techniques or misperceived data requirements should not keep ecologists from applying

statistical techniques for threshold detection.

As human pressures on ecosystems continue to increase worldwide, the need for analytical

approaches allowing the detection of ecological thresholds and regime shifts becomes a

matter of urgency. Particularly, the impacts of climate change on biodiversity and ecosystems

are currently assumed to be smooth, involving a continuous increase in impacts and

extinctions as global temperature rises [51]. Well-documented fisheries statistics have shown

that even relatively smooth climatic changes might lead to strong regime shifts in ocean

ecosystems [13], increasing the likelihood of more prevalent and abrupt regime shifts as the

planet warms beyond ecological thresholds for a growing fraction of species and ecosystems.

Ecologists should increasingly contribute quantitative evidence of ecological thresholds for

the future environmental policy making.

This review has documented a diversity of approaches, differing in complexity, power and

requirements, which we hope will stimulate the transition from a phenomenological

assessment of ecological regime shifts and thresholds to an operational one. All of the

exploratory and inferential techniques covered here require that the threshold has to be

crossed in order to be detected, which means that they cannot directly be used to prevent

abrupt changes in ecosystems [1]. However, the accumulation of a broad empirical basis on

the presence of ecological threshold and regime shifts in response to key pressures, such as

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increased nutrient inputs, ecosystem fragmentation or climate change, will certainly help

develop a predictive framework to be used to anticipate and avoid changes associated with

loss of essential ecosystem functionality. The accumulating knowledge base of ecological

thresholds across different ecosystems should be accompanied by developing quantitative

methods that allow confident extrapolation of thresholds for ecosystems that have not yet

experienced, in particular irreversible, regime shifts.

Acknowledgements

This is a contribution to the THRESHOLDS integrated project funded by the 6th Framework

Program of the European Union (www.thresholds-eu.org, contract # 003933-2). 10

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Glossary 1

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Artificial neural network: a mathematical model where input signals are processed through

one or more layers of interconnected computational nodes resembling biological neurons

Autocorrelation: the smoothness of a time series expressed as the correlation between its

successive values

Autoregressive model: a linear regression model that uses past values to predict the present

value of a time series variable

Average standard deviates (ASD): a regime shift detection method for multiple time series

where the individual variables are forced to have the same sign on the same side of a

change-point; known to have unacceptable false positive rate on autocorrelated data

Bifurcation: a qualitative change in the behaviour of a dynamical system resulting from a

small change in a system parameter

Bistability: the existence of more than one locally stable stationary state in a dynamical

system

Brownian bridge: a Brownian motion (random walk) where both ends are clamped to zero

Change-point: a step change in the mean value, or more generally, the distribution of a time

series variable

Chronological clustering: a hierarchical grouping of successive steps in a multivariate time

series according to a dissimilarity measure, also called constrained or stratigraphic

clustering

CUSUM: cumulative sum of scaled deviations from a target value, such as the mean of a time

series

Dissimilarity measure: a single numerical value expressing a distance between two

multivariate objects, such that identical objects have 0 dissimilarity

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Dynamic programming: a computationally efficient method for solving sequential decision

problems by recursion

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Ecological regime shift: a sudden shift in ecosystem status caused by passing a threshold

where core ecosystem functions, structures, and processes of are fundamentally changed

Ecological threshold: the critical value of an environmental driver for which small changes

can produce an ecological regime shift

Ecotone: a transitional area between two adjacent ecological communities

Empirical orthogonal function (EOF): a principal component decomposition of a

multivariate time series

Exploratory data analysis: the analysis of data for the purpose of formulating hypotheses

worth testing, thus complementing the conventional statistical tools for hypothesis testing

Hypothesis testing: making statistical decisions about data by asking a hypothetical question

formulated as a null hypothesis

Hysteresis: a property of systems that can follow different paths when increasing and when

relaxing a perturbation

Intervention analysis: a test of the hypothesis that an event at a known time caused a change

in an autoregressive time series model

Likelihood ratio: the relative probabilities of an observed data set under two alternative

hypotheses

Matrix decomposition: expressing a matrix as a product of (usually) simpler matrices

Multimodal distribution: a probability distribution with more than one peak

Principal component analysis (PCA): an orthogonal MATRIX DECOMPOSITION of the

covariance or correlation matrix of a multivariate data set to reduce the dimensionality of

interrelated variables

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Recursive processing: a computational method for processing new data incrementally as

they arrive, instead of processing them all in a single batch

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Singular spectrum analysis (SSA): a technique for estimating the frequency components of

a time series through a principal component decomposition of its autocorrelation matrix

Statistical test power: the probability that a statistical test will reject a false null hypothesis;

the sensitivity of the test

Structural change: a change-point with a step change in the parameters of the generating

model for a time series

Threshold autoregressive model: a time series model that can change structurally depending

on the past values of the series (rather than at a particular change-point in time)

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