Regime Shifts in a Socio-Ecological System: Working Paper

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Regime shifts in a social-ecological system Steven J. Lade, Alessandro Tavoni, Simon A. Levin, Maja Schlüter February 2013 Centre for Climate Change Economics and Policy Working Paper No. 125 Grantham Research Institute on Climate Change and the Environment Working Paper No. 105

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Regime shifts in a social-ecological system

Steven J. Lade · Alessandro Tavoni · Simon A. Levin · Maja Schl¨uter

Received: date / Accepted: date

Abstract Ecological regime shifts are rarely purely eco- warning signals for the social-ecological regime shifts logical. Not only is the regime shift frequently triggered we observe in the models. by human activity, but the responses of relevant actors Keywords Regime shifts · tipping points · early to ecological dynamics are often crucial to the devel- warning signals · bifurcation · generalised modelling · opment and even existence of the regime shift. Here, social-ecological system we show that the dynamics of human behaviour in response to ecological changes can be crucial in deter- mining the overall dynamics of the system. We find a 1 Introduction social-ecological regime shift in a model of harvesters of a common-pool resource who avoid over-exploitation Many ecological systems can undergo large, sudden and of the resource by social ostracism of non-complying long-lasting changes in structure and function (Scheffer harvesters. The regime shift, which can be triggered by et al 2001). Such changes, often called regime shifts1 several different drivers individually or also in combina- or critical transitions (Scheffer et al 2009), have been tion, consists of a breakdown of the social norm, sudden found in a range of ecological systems, including eu- collapse of co-operation and an over-exploitation of the trophication of freshwater lakes, soil salinisation, degra- resource. We use the approach of generalised modelling dation of coral reefs, collapse of fisheries and encroach- to study the robustness of the regime shift to uncer- ment of bushland (Biggs et al 2012a). tainty over the specific forms of model components such Most ecological systems, and especially those sys- as the ostracism norm and the resource dynamics. Im- tems that are at risk of sudden nonlinear changes such portantly, the regime shift in our model does not occur as regime shifts, are subject to influence by humans if the dynamics of harvester behaviour are not included (Millennium Ecosystem Assessment 2005). Furthermore, in the model. Finally, we sketch some possible early humans not only influence the ecological system but also adapt their behaviour in response to ecological Steven J. Lade · Maja Schlüter changes (Folke et al 2010). However, many traditional Stockholm Resilience Centre, Stockholm University, ecological modelling approaches reduce the social sub- Kräftriket 2B, 114 19 Stockholm, Sweden E-mail: [email protected] system to a simple driver, such as fishing pressure or resource extraction rate (Schlüter et al 2012a). Mirror- Steven J. Lade NORDITA, KTH Royal Institute of Technology and ing this problem, in bioeconomic models of the optimal Stockholm University, Roslagstullsbacken 23, management of renewable resources the descriptions of 106 91 Stockholm, Sweden resource dynamics are often very simple, with neither Alessandro Tavoni 1 Grantham Research Institute, London School of Economics, In this article, we use the term ‘regime shift’ in a social London WC2A 2AZ, United Kingdom as well as an ecological context. We intend the term ‘regime shift’ to be understood in such a social context as not (neces- Simon A. Levin sarily) a political regime change but rather any recognisably Department of Ecology and Evolutionary Biology, Princeton sudden, large and persistent change in the behaviour of rele- University, Princeton, NJ 08544, USA vant actors. 2 Steven J. Lade et al. resource nor management strategy capable of regime ings for regime shifts in social-ecological systems. We shifts (with exceptions including recent work from eco- use the conventional variance and autocorrelation in- logical economics such as Crépin and Lindahl 2009; Ho- dicators (Dakos et al 2012b) as well as the generalised ran et al 2011). modelling-based early warning signal (Lade and Gross Here, we show that modelling changes in the be- 2012). haviour of the humans who interact with the ecologi- Section 2 describes in greater detail the models and cal system is crucial to understanding regime shifts in the methods used to analyse them, including bifurca- the ecological populations. In particular, a central re- tion diagrams and generalised modelling for both bi- sult is that such systems, often referred to as social- furcation analysis and early warning signals as well as ecological systems (Carpenter et al 2009), can display conventional early warning signals. Section 3 presents regime shifts that are absent from the ecological subsys- the results of these bifurcation and early warning sig- tem in isolation. We find that even a non-linear linkage nal analyses, and in Section 4 we discuss their implica- between completely linear social and ecological subsys- tions for modelling studies and for governance of social- tems, which have no regime shifts of their own, can ecological systems. Concluding remarks are presented induce regime shifts in the coupled system. in Section 5. To develop these results, we analyse a social-ecological system that captures essential properties of a class of systems encountered frequently in natural resource man- 2 Methods agement: a common-pool resource, for which harvesters choose between a community-efficient co-operation or a We begin by introducing background theory and the self-interested defection that leads to over-harvesting, analytical tools we propose to use in our analysis, then together with a social mechanism that encourages but describe the model to which they will be applied. does not guarantee co-operation (Ostrom 1990, 2006). We represent this system using two types of models: a generalised model, where the details of the processes in 2.1 Bifurcation theory the system are left unspecified; and simulation models based on the model of Tavoni et al (2012) (hereafter re- Bifurcation theory describes sudden, qualitative changes ferred to as the TSL model) that have functional forms in the dynamical behaviour of systems. Specifically, a fully specified, as is necessary to perform time series bifurcation of a dynamical system occurs when a small, simulations. smooth change in a parameter of the dynamical system We search for possible regime shifts in this social- causes a qualitative change to the dynamics of the sys- ecological system by analysing the fold bifurcations of tem (Kuznetsov 2010). In this article, we analyse the the models. Fold bifurcations, one of a family of pre- bifurcations of our social-ecological models to under- cisely mathematically defined qualitative changes in the stand qualitative changes in their behaviour. dynamical behaviour of a system (Kuznetsov 2010), can Regime shifts are a type of qualitative change in lead to regime shifts (Scheffer et al 2001). Bifurcation behaviour that can be caused by bifurcations (Kuehn analysis of the generalised model therefore permits gen- 2011). One important type of regime shift-inducing bi- eral statements about the presence and robustness of furcation is the fold bifurcation (Scheffer et al 2001). regime shifts over a wide range of systems. This gener- In a fold bifurcation, a stable fixed point collides with ality is useful when analysing social-ecological systems, an unstable fixed point and both fixed points disappear in which the specific functional forms required by simu- (Fig. 1). A system initially on or near the stable fixed lation models can be difficult to determine. Bifurcation point will undergo a sudden and possibly large change analysis of simulation models allows us to test the gen- in state to a distant attractor (in Fig. 1 this is another eralised results in specific cases and to predict the pres- stable fixed point). This change is also persistent, as the ence and effects of regime shifts with respect to specific parameter must often be changed to a significantly dif- parameters of the simulation model. ferent value in order for the system to transition back Since regime shifts are sudden, persistent and often to the first fixed point. Mathematically, fold bifurca- have significant consequences, early warning signals for tions occur (in the ordinary differential equations we an impending regime shift would be highly desirable. use here) when a real eigenvalue of the Jacobian ma- Early warning signals have recently been developed for trix at the fixed point passes through zero. The Jaco- regime shifts in ecological and physical systems (Schef- bian matrix is a matrix of derivatives that characterises fer et al 2009, 2012). We perform preliminary inves- the dynamics of the system in the vicinity of the fixed tigations on the possibility of using these early warn- point. Regime shifts in a social-ecological system 3

obtained from the model are of a general nature or are specific to the functional forms chosen. The generalised modelling approach (Gross and Feudel 2006; Kuehn et al 2013) permits precise mathematical statements about bifurcations of a dynamical system to be made in the presence of uncertainties about its precise form. Here, we use generalised modelling to make two types of inferences about a dynamical system. The first, and more traditional, use of generalised modelling will be to calculate the types of bifurcations that a generalised model can and is likely to undergo. Further below, we use generalised models together with time series data to generate early warning signals for regime shifts. A generalised modelling bifurcation analysis pro- Fig. 1 Bifurcations and regime shifts. Fold bifurcations ceeds as follows: occur when a stable (solid lines) and an unstable (dashed line) fixed point collide, and lead to regime shifts. In the pair of 1. Write down a generalised model structure for the fold bifurcations sketched here (manually, without computer state variables and processes present in the system. simulation), a system on stable fixed point 1 will, upon the 2. Symbolically calculate the Jacobian matrix of a fixed driver passing fold bifurcation 1, undergo a sudden and large point in this generalised model. shift to stable fixed point 2 (trajectory 1). This change is persistent, because the parameter must be returned to a value 3. Parameterise the Jacobian matrix directly using the below fold bifurcation 2 (if this is possible) in order for the so-called generalised parameters. Assign likely ranges system to return to its original state, in another regime shift of the generalised parameters based on knowledge of (trajectory 2). the system. 4. Calculate, using appropriate methods, the likely bi- Another important type of bifurcation that can also furcations to which these values of generalised pa- be detected with a generalised modelling analysis is the rameters can lead. Hopf bifurcation, which often (in the case of supercriti- This procedure will be described in further detail below cal Hopf bifurcations) lead to a transition to oscillatory with the aid of a simple example. We also note that the dynamics. A Hopf bifurcation occurs when the Jacobian procedure described here above skips the normalisation matrix at a fixed point has a complex conjugate pair of step often used in previous generalised modelling stud- eigenvalues that pass through the imaginary axis (that ies (Gross and Feudel 2006; Kuehn et al 2013). This is, have zero real part) (Kuznetsov 2010). leads to the presence of an additional parameter in the For simulation models of dynamical systems, where Jacobian, which we call the ‘steady-state ratio’ below, the forms of all interactions and the values of all pa- but which has no effect on the bifurcation analysis. rameters are known, numerical continuation software are often used to locate and track bifurcations. Here, Generalised modelling shares its mathematical ba- we used the software XPPAUT (Ermentrout 2011) to sis, the bifurcations of dynamical systems, with both plot bifurcation diagrams of the form of Fig. 1, where the qualitative theory of differential equations (Kelley the values of a parameter or ‘driver’ are plotted on the and Peterson 2010) and catastrophe theory (Zeeman horizontal axis, and the stable and unstable fixed points 1977). Indeed, generalised modelling could be consid- of the system corresponding to each value of the driver ered a systematic way of parameterising a model be- on the vertical axis. fore analysing its bifurcations with the tools of qualitative differential equation theory. Two key points of distinction of the generalised modelling are that: (i) it 2.2 Generalised modelling for bifurcation analysis parameterises the Jacobian matrix directly, rather than the original functional forms; and (ii) it provides a sys- Bifurcation diagrams give great insight into possible tematic framework for connecting a generalised model qualitative behaviours of the system. Frequently, how- structure to properties of real systems, through the gen- ever, the forms of interactions and the values of param- eralised parameters (see below) or time series obser- eters in the model on which the bifurcation analysis vations (in the case of a generalised modelling-based was performed cannot be accurately known. In a related early warning signal). Generalised modelling therefore problem, if functional forms are chosen in a model, there allows for the investigation of how processes, even in- can be significant uncertainty over whether conclusions completely characterised processes, lead to bifurcations 4 Steven J. Lade et al. of the system. Frequent failure to investigate how system- To better relate the derivatives in the model to prop- specific processes could lead to the abstract, general erties of a real-world system, we re-write Eq. (2) in a geometries of catastrophes was a major contributor to different form. We introduce: the controversy over some strands of catastrophe theory – the scale parameter α = G∗/X∗, which, with di- (Guckenheimer 1978). mensions of inverse time, is a measure of the char- Conceptually, generalised modelling is also similar acteristic time scale of X. to systems dynamics approaches (Sterman 2000), struc- – the dimensionless elasticity parameters tural equation modelling (Kline 2011) and flexible func- ∗ ∗ tional forms (Chambers 1988). A generalised model could X∗ dG X∗ dL G = and L = . be considered a causal loop diagram or stock and flow X G∗ dX X L∗ dX diagram from system dynamics in mathematical form, which we then further manipulate to obtain general The elasticity parameters give an indication of the mathematical results without resorting to simulating non-linearity of the process near the fixed point. For specific systems. Like generalised modelling, structural example, a linear function f(x)= ax has elasticity 1 equation modelling (SEM) explores the consequences of for all x. A constant function f(x)= a has elasticity linkages within a network of interacting variables; un- 0 for all x. like generalised modelling, SEM can statistically test The scale and elasticity parameters, and in more com- for the presence and strength of linkages, but does not plicated examples another type of parameter called the commonly explore nonlinear dynamics and transitions ratio parameter, are collectively referred to as gener- of the variables. In economics, flexible functional forms alised parameters, as they are used to represent the dy- are a general way to determine functional relationships namics of a general class of models rather than parame- directly from data, but they generally are not forms terising a particular model. Using these generalised pa- that are convenient for a subsequent bifurcation analy- rameters, Eq. (2) becomes sis. To formulate a generalised model, the important λ = α (GX − LX ) , state variables in the system and the processes through ∗ ∗ which they interact must first be identified. As a sim- where we have also used the fact that G = L . ple example, consider a single population X (which may Next, we identify what conditions on the generalised be of animals, of people, of people holding a particular parameters can give rise to different types of bifurca- opinion) that can increase due to a gain process G(X) tions. As described in the previous section, a zero eigen- and decrease due to a loss process L(X), both of which value of J indicates a fold bifurcation, while a pair of may depend on the current population X. A generalised imaginary conjugate eigenvalues indicates a Hopf bifur- model, in differential equation form, for the population cation. In this simple example, we can conclude that no X is then Hopf bifurcations are possible (since there is only one eigenvalue, which is real) and, provided α > 0, a fold dX = G(X) − L(X). (1) bifurcation can only occur when GX = LX . dt The final step of the generalised modelling process To investigate the bifurcations of this generalised is to make use of contextual knowledge about the pro- model, we assume that the system has a fixed point, cesses to evaluate the likelihood of the identified bi- that is, some value X∗ where if X(0) = X∗ then X(t)= ∗ furcations. In this example, suppose we know that the X for all t > 0. Throughout this article we use the loss process L(x) is approximately linear (elasticity 1), asterisk to denote a quantity evaluated at the fixed while the gain process G(X) is linear at X = 0 (elastic- point. We then calculate the Jacobian matrix of the ity 1) but saturates at high X (elasticity 0). Therefore generalised model at the fixed point, and calculate the LX ≈ 1 and 0 < GX < 1 for any non-zero population so eigenvalues of the Jacobian, from which we can estab- a fold bifurcation is unlikely. If, however, G(X) took a lish stability and bifurcations as described in Sec. 2.1. sigmoidal-type shape (with a quadratic or higher-order For Eq. (1), the Jacobian matrix J consists of a single shape near X = 0) then the elasticity GX could reach element which is also the eigenvalue, λ, and exceed 1 and a fold bifurcation may be possible. ∗ ∗ dG dL The generalised model that we analyse below is two- J = λ = − . (2) dX dX dimensional. For this system, we search for bifurcations

by identifying combinations of generalised parameters Therefore to determine the bifurcations of the system, that satisfy we need to determine the possible ranges of these two derivatives. det J = 0 (3) Regime shifts in a social-ecological system 5 for fold bifurcations and model and on the available data; we will derive an algorithm appropriate to the model studied here. tr J =0 and det J > 0 (4) for Hopf bifurcations, where tr denotes the trace of the 2.4 Social ostracism and resource model matrix.2 We also test whether det J (for fold bifurca- J tions) or tr (for Hopf bifurcations) changes sign at the We use the tools described above to analyze regime candidate bifurcation points, as is necessary for a true shifts in a stylised social-ecological model. Based on the bifurcation. TSL model, we consider a resource, of resource level R, that is being harvested by a community of users. A proportion fc of the harvesters co-operate to harvest at 2.3 Early warning signals a socially optimal level while the remaining harvesters ‘defect’ and harvest at a higher level out of self-interest. Consider a dynamical system that is undergoing noisy We write the resource dynamics in generalised form fluctuations due to a fast noise forcing. As the dynam- as ical system approaches a fold bifurcation the standard deviation of these fluctuations, under a linear approx- dR = c − D(R) − Q(E(f ),R), (5) imation, diverges to infinity and their autocorrelation dt c increases towards one (Scheffer et al 2009; Kuehn 2013). Although the linear approximation will not hold arbi- where c is the resource inflow or growth rate (and is trarily close to a fold bifurcation, increasing variance independent of the current resource level), D(R) is the and autocorrelation are two well-known early warning natural resource outflow rate or mortality, and Q(E,R) signals that have both been shown to precede regime is the resource extraction. Here E(fc) is the total effort shifts in simulation studies, in laboratory and field ex- exerted by the harvesters, which decreases with increas- periments and in data of past regime shifts (Lenton ing proportion of co-operators. In practice, the resource 2012; Scheffer et al 2012). could be fish in a fishery, water in an irrigation system, We examine whether variance and autocorrelation an unpolluted atmosphere, etc. early warning signals also precede the regime shifts we We, like TSL, use the replicator dynamics of evo- explore in our social-ecological system. To calculate the lutionary game theory to model the dynamics of the early warning signals we use the R package ‘earlywarnings’ fraction of co-operators fc. In its most general form, (Dakos et al 2012a) developed by Dakos et al (2012b). the replicator equation for two strategies is dfc/ dt = − − A bandwidth of 20 for the detrending of the time series fc(1 fc)(Uc Ud), where Uc and Ud are the utilities was used. for a co-operator and a defector, respectively. In our We also explore a third early warning signal, the re- model, the utilities of defectors and co-operators can cently developed generalised modelling-based early warn- differ in three ways. ing signal (Lade and Gross 2012). Unlike the variance First, the income received by defectors and co-operators and autocorrelation approaches, which rely purely on differs due to their harvesting activities, by an amount time series data of the system, this approach also makes we denote by F (E,R) > 0 that depends on the cur- use of structural knowledge about the system. As the rent resource level R and the total effort E. Second, name suggests, the approach begins by constructing a the costs incurred by defectors and co-operators differ generalised model and then formally calculating its Ja- due to their harvesting activities by an amount W > 0. cobian matrix as described in Sec. 2.2. Instead of as- We expect both income and costs to be higher for defec- signing ranges to the elements of the Jacobian matrix tors than for co-operators, since defectors invest more through generalised parameters, time series observa- effort. tions of the state variables and processes are used to The income difference minus the cost difference con- directly estimate the derivatives in the Jacobian ma- stitutes the payoff difference between defectors and co- trix. The particular computations used to estimate the operators. Like TSL, we include a third difference be- derivatives depend on the structure of the generalised tween co-operator and defector utility, that the defectors can be socially ostracised by the co-operators, to 2 These conditions can be easily derived by noting that reduce the utility of the larger payoff that they would the eigenvalues of a two-dimensional Jacobian matrix are otherwise obtain. We denote by ω(fc) > 0 the degree 1 ± 1 2 − λ = 2 tr J 2 ptr J 4 det J. We emphasise that Eq. (4) is to which ostracism reduces a defector’s utility, with the only valid in two dimensions; approaches that also work for higher dimensions include the Routh-Hurwitz criteria and the property that the size of this reduction is expected to in- method of resultants (Gross and Feudel 2004). crease with increasing fc. Ostracism can occur with the 6 Steven J. Lade et al. presence of individuals in the community with other- Outflow D(R) regarding preferences (Fehr and Fischbacher 2002) and Inflow c Resource level when fear of community disapproval leads to pressure to Extraction Q(E,R) conform with the social norm (Cialdini and Goldstein R 2004). We note that the social ostracism modelled by TSL, which we follow in generalised form here, is non-costly, Ecological/Physical Effort E(f ) in that it does not cost the co-operators to impose this Social c punishment. Non-costly ostracism may occur when the community builds upon available social capital (Bowles and Gintis 2002) to deny defectors important services, such as refusing to loan machinery or refusing trans- Defection due to portation to market (Tarui et al 2008; Tavoni et al relative income F(E,R) Fraction of 2012). Indeed, sanctioning can even provide benefits to defectors co-operators the enforcer (Ostrom 1990). Other models in which co- Co-operation due to f relative costs W c operation is encouraged in a non-costly manner include those of Osés-Eraso and Viladrich-Grau (2007), Iwasa Co-operation due to social ostracism ω(f ) et al (2007) and Tarui et al (2008). c These three differences between co-operator and de- Fig. 2 Schematic of the generalised model Eqs. (5-6). fector utility result in the following equation for changes Employing some of the systems dynamics conventions (Ster- in the fraction of co-operators, man 2000), we represent flows by double-line arrows, influ- ences by single-line arrows, state variables by rectangles, in- df termediate quantities by ovals, and sources and sinks of flows c = f (1 − f ) −F (E(f ),R)+ W + ω(f ) . (6) dt c c c c with explosion symbols. As described in the Supporting Text (Section 1), some of our definitions differ from those originally used by (Traulsen et al 2005). During the simulation, which TSL. These changes were made in order to reduce Eq. 6 lasted from t = 0 to 500, we varied the TSL parame- to a number of unknown functions and parameters that ters c and w (see Supporting Text) according to c(t)= − is as small as possible while still clearly representing the 40 + 0.024t and w(t) = 18 0.08t. The results of the processes at work in the social-ecological system. simulation were sampled at intervals of ∆t = 1 time Eqs. (5-6) constitute our generalised model of social unit. ostracism and resource dynamics, hereafter referred to as ‘our social-ecological model’. We denote by the eco- 3 Results logical or social subsystem the relevant part of the system with any feedback from the other part set to a 3.1 Generalised modelling bifurcation analysis constant value. It is clear that there are two key processes that link the ecological and social subsystems: Following the generalised modelling procedure, let there extraction of the resource Q, and the income (differ- ∗ ∗ be a fixed point (R,fc) = (R ,fc ) where dR/dt = ence) F gained by harvesters due to extraction (Fig. 2). dfc/dt = 0. Furthermore, we concentrate on mixed- For both these processes, the effect of the social subsys- strategy fixed points, 0

Table 1 Deﬁnitions and interpretations of the generalised parameters used in the bifurcation analysis. Values of the generalised parameters that correspond to the TSL model are indicated, as are other values used in the bifurcation analysis. For deﬁnitions of TSL’s k, ed, ec, a, b, see the Supporting Text.

Type Symbol Definition Interpretation TSL value Other values ∗ Scale αR c/R Fractional rate of replenishment of 0.5 to 2 the resource from the inflow c − − ∗ ∗ αf (1 fc )F Fractional rate of defector recruit- 0 to 0.5 ment due to income difference F ∗ ∗ ∗ Ratio βR D /(D + Q ) Relative rate of resource loss from 0.1 to 0.2 outflow compared to harvesting βf w/(w + ωc) Relative rate of co-operator recruit- 0.6 to 1 ment from effort cost compared to ostracism Elasticity DR Of the form Non-linearity, see Sec. 2.2 k =2 1,0to6 QR XY ≡ 1 ∗ ∗ QE Y ∂X 1 X∗ ∂Y − − a Efc 1 ed/ec = 2.8 to 0 FE a − 1= −0.4 FR b =0.2 b ωfc 0 to 8 1 R ∗ ∗ State vari- δfc R /fc Ratio of state variable values able ratioc a The effort elasticity is negative because an increase in co-operators decreases the effort. The elasticity is zero under full defection (fc = 0), because a large fractional change in the number of co-operators leads to a small fractional change in the total effort. In the TSL model, the elasticity takes its largest negative value at full co-operation (fc = 1). b The particular ostracism function chosen by TSL causes the elasticity ωfc to reach extremely high values at values of fc where the magnitude of the function ω(fc) itself is very small. Another function that reproduces TSL’s ω(fc) very 8 8 8 closely is ω(fc)=0.34fc /(0.55 + fc ), which has the more reasonable range of elasticities indicated above. c R The state variable ratio δfc is not required in any of the following calculations.

Using J and the condition in Eq. (3), we find that for generalised parameters corresponding to the TSL 0 model (Table 1), fold bifurcations are widespread (Fig. 3). Specifically, for any combination of ωfc and DR in −1 the range plotted, there is an Efc that leads to a fold fc bifurcation. This matches the bifurcations observed by E −2 TSL both in their previous work, as indicated by the −3 appearance and disappearance of their mixed equilib- 0 rium states, as well as in an upcoming study (Schlüter 2 et al in prep.). 0 2 Although the Jacobian J does permit Hopf bifur- 4 4 6 cations for some extreme parameter combinations, the D 6 8 ω Hopf conditions [Eq. (4)] are not simultaneously satis- R fc fied anywhere in the generalised parameter space plot- Fig. 3 Generalised modelling analysis. Surface of fold bi- ted in Fig. 3. We conclude that Hopf bifurcations, and furcations for ranges of generalised parameters matching the consequently oscillatory states, are unlikely to be ob- TSL model. served for models similar to the TSL model. If we ignore the social dynamics in this system and that by ignoring the social dynamics it is impossible to set the total effort E to a constant, the eigenvalue at a appropriately model the regime shift that can occur in fixed point of R in the ecological subsystem is our social-ecological system.

λecol = −αRβRDR − αR(1 − βR)QR. To obtain a regime shift in the ecological subsystem alone, DR or QR would have to be sufficiently negative Provided that DR and QR are always positive (as they for λecol to reach zero. In traditional models of ecologi- are in the TSL model), this eigenvalue is always nega- cal regime shifts, this is achieved because D(R) (or in- tive. Therefore, in this model, no bifurcation can occur flow rate, which here is a constant c) is sufficiently non- and in particular no regime shift can occur. It is clear linear. There are of course many ecological and physical 8 Steven J. Lade et al. systems with such a regime shift-inducing non-linearity increase in the defector income than in co-operator in- (Scheffer et al 2001). Our purpose here is to show that come, due to allocation of net production according to regime shifts of ecological states can occur even if the effort. Increased defection then decreased the effective- ecological subsystem alone does not have a regime shift. ness of social ostracism and also increased extraction of Returning to Fig. 3 we note, in addition to the ubiq- the resource, culminating in a collapse in co-operation uity of fold bifurcations, that (i) fold bifurcations are and in resource levels. possible for a large range of ωfc , including values near The bifurcation remained (Fig. 4b) when we used a 1, and (ii) that the presence of fold bifurcations is not variant of the TSL model with ω(fc) ∝ fc and D(R) ∝ strongly affected by the value of DR. This indicates that R (Supporting Text, section 1). The size of the regime close to linear ostracism and resource outflow functions shift was not as large as in the case of strongly non- ω(fc) and D(R) may be sufficient to produce a fold linear ω(fc) (Fig. 4a), however. We conclude that, as bifurcation. We confirm this prediction below using a predicted by the generalised modelling analysis, the pres- simulation model. ence of the regime shift is robust to the functional form Furthermore, setting ω(fc) ∝ fc and D(R) ∝ R of the ostracism process ω(fc), and also to the func- removes all non-linearity from both the purely social tional form of the resource outflow D(R). As predicted and purely ecological components of the dynamics. The by the generalised modelling analysis, a simulation model only remaining non-linearities are contained in the link- of the ecological subsystem alone, however, did not have age between the social and ecological subsystems. This a bifurcation (Fig. 4h). linkage is comprised by the processes Q(E,R), which Although the ecological subsystem alone does not specifies the amount of resource extracted by the har- display a regime shift, the consequences of the social- vesters, and the income difference F (E,R), which spec- ecological regime shift can be just as serious for the ifies the effect of resource extraction on the fraction of resource levels as a purely ecological shift. The regime 3 co-operators. Thus as well as arising from non-linearities shift associated with increasing resource inflow (Fig. 4a) in the ecological or social dynamics, regime shifts can led to a significant drop in resource levels (Fig. 4c). The also arise from non-linearities in the linkages between total payoff (income minus costs) that the community them. received also collapsed (Fig. 4d). In this state, social ostracism is largely ineffective due to the small population of co-operators. Re-establishing the ostracism 3.2 Bifurcations of simulation models norm and the associated high resource state would in this model require a large drop in resource inflow, or We next tested the general predictions of the gener- may even be impossible in the absence other mecha- alised modelling analysis above with simulations of the nisms to re-establish the norm. TSL model, and variants thereof. Beginning with the parameter set used by TSL (see Supporting Text, Section 1), changing the resource inflow readily triggered a fold bifurcation (Fig. 4a). In 3.3 Early warning signals fact, changes in any of many different drivers, including effort cost (Fig. 4e), the strength of ostracism (Fig. A good standard deviation or autocorrelation warning 4f), or even multiple drivers changing simultaneously signal should display a clear upwards trend well in ad- (resource inflow and effort cost, Fig. 4g), could trig- vance of the critical transition. In the early warning lit- ger the fold bifurcation. We conclude that, as predicted erature, these trends in indicators are often quantified by the generalised modelling analysis, fold bifurcations with the Kendall-τ statistic (Dakos et al 2012b). We and therefore regime shifts are easy triggered in our applied the standard early warning suite to time series social-ecological model. of both R and fc leading up to the transition (Fig. 5). Somewhat counter-intuitively, the regime shift from We observed, for the simulation dataset used (Fig. 5, a high co-operation state that led to breakdown of the first row), weak trends in autocorrelation and stronger social norm and collapse of the resource was triggered trends in standard deviation (largest Kendall τ statistic by an increasing resource inflow, countering the com- 0.86 for standard deviation of residuals of fc). mon understanding that it is scarcity that leads to con- We next constructed a generalised modelling-based flict. In this model, the regime shift occurred due to an early warning signal. We assumed that the following initially increasing resource level that led to a greater quantities could be measured:

3 In the TSL model, these linkages have the following non- – The resource level, R a−1 b linearities: Q(E,R) ∝ ER and F (E,R) ∝ E R . – The fraction of co-operators, fc Regime shifts in a social-ecological system 9

(a) 1 (b) 1 * * c c

f 0.5 f 0.5

0 0 0 20 40 60 0 20 40 60 80 Resource inflow, c Resource inflow, c (c) 60 (d) 10

40 5 R* 20 0 Total payoff

0 −5 0 20 40 60 0 20 40 60 Resource inflow, c Resource inflow, c (e) 1 (f) 1 * * c c

f 0.5 f 0.5

0 0 10 15 20 0 0.5 1 Effort cost, w Ostracism strength, h (g) 1 (h) 200

150 * c f

0.5 R* 100

0 0 40 45 50 0 5 10 Resource inflow, c Harvesting pressure, qE 18 17 16 15 14 Effort cost, w

Fig. 4 Bifurcation diagrams of simulation models. The fraction of co-operators fc are plotted for the fixed points of the TSL model (Supporting Text, Section 1) with respect to changes in: (a) the resource inflow; (b) the resource inflow with the functions ωfc and D(R) set to linear forms (see Supporting Text, Section 1); (e) the cost of harvesting effort; (f) the strength parameter of the ostracism function; (g) both resource inflow and effort cost at the same time. (h) Fixed points R of the isolated ecological subsystem (Supporting Text, Section 1). In (c) the resource levels and in (d) the total community payoff ∗ − ∗ ∗ ∗ ∗ − n[fc ec + (1 fc )ed][f(E ,R )/E w] (see Supporting Text, Section 1 for definitions of symbols) corresponding to the fixed points in (a) are shown. Solid lines denote stable fixed points, dotted lines denote unstable fixed points. 10 Steven J. Lade et al.

R fc 60 55 0.8 50 0.6 45 40 0.4 35 0.2 30 2 0.020.0

0.01 1 0.00 0 −0.01

−1 −0.02 residuals residuals −2 −0.03 0.78 autocorrelation autocorrelation 0.88 0.76 0.86 0.74 0.84 0.72 Kendall tau= 0.599 Kendall tau= 0.462 0.70 0.82 0.68 0.80 0.66 0.78 0.0090 standard deviation standard deviation 0.65 0.0085 0.0080 0.0075 0.60 Kendall tau= 0.684 0.0070 Kendall tau= 0.864 0.0065 0.55 0.0060 0.0055

0 100 200 300 400 0 100 200 300 400 time time

−0.08 eigenvalue −0.09 −0.10 Kendall tau= 0.760 −0.11 −0.12 −0.13

0 100 200 300 400 time

Fig. 5 Early warning signals. (top) Time series with filtered fit, detrended fluctuations, and autocorrelation and standard deviation of the detrended fluctuations for R and fc, respectively, in the lead-up to the regime shift. (bottom) Generalised modelling-based early warning signal preceding the regime shift. Regime shifts in a social-ecological system 11

– The resource outflow, D (for example, natural fish The early warning signal approaches described here mortality or natural water losses due to evaporation also have differing demands on the amount and type or outflow). Could be replaced by observations of of data and knowledge required. The autocorrelation resource inflow c if more easily measured. and standard deviation approaches require only high- – The total resource extraction, Q (for example, total frequency observations of a single quantity. The gener- fish caught or total water used for irrigation) alised modelling-based warning signal, in contrast, re- – The income difference between a defector and a co- quires knowledge of the structure of the social-ecological operator, F . system as well as regular observations of all state vari- – The cost difference between a defector and a co- ables and several of the processes by which they inter- operator, W . act. It is hoped that, for some regime shifts, such additional, system-specific information will improve the We assumed that the regime shift was being triggered reliability of the warning signal, as well as decreasing by changes in resource inflow c and/or effort costs W . the frequency at which time series need to be sampled To complete the generalised modelling analysis we also (Lade and Gross 2012; Boettiger and Hastings 2013). required the assumptions that: resource extraction is The generalised modelling approach for early warn- linear in the resource level, Q ∝ R; and income differing signals is also itself in an early stage of develop- ence is sub-linear in resource level with known elasticity ment. Future improvements could include statistical ap- b, F ∝ Rb. We found however that the results of the proaches: to incorporate partial knowledge about deriva- generalised modelling early warning signal are not sen- tives in the Jacobian matrix; similar to the approach of sitive to the value of b used. In the following, we used Boettiger and Hastings (2012), to test the fit of alter- b = 0.5, significantly different to the elasticity actually native generalised models; and to calculate the level of used in the simulation (b = 0.2). confidence in an early warning trend. Following the approach outlined by Lade and Gross (2012), we derived an algorithm to calculate the eigen- 4 Discussion and implications for management values of the generalised model from the quantities in the above list (Supporting Text, Section 2). Key out- In the social-ecological system studied here, the ecologi- puts of this algorithm included the derivatives of the in- cal subsystem could not by itself undergo a regime shift come difference F , resource extraction Q and ostracism at all, whereas regime shifts in the social-ecological sys- ω with respect to the fraction of co-operators fc. tem were common. The results of the social-ecological From the TSL model with changing parameters de- regime shift were as dramatic as purely ecological regime scribed at the end of Section 2.4, we generated time shifts that occur when the human impact acts as a sim- series for the list of quantities above. We then applied ple driver: there was a rapid, large and persistent col- our algorithm, which yielded time series of two eigenval- lapse of the resource, along with an associated collapse ues. A clear warning signal would be a negative (stable) of the social norm and community payoffs. We conclude eigenvalue increasing consistently towards the stability that failing to model a natural resource as a social- boundary of zero eigenvalue. One of the eigenvalues we ecological system, which requires including the dynam- calculated was always stable and far from the stabil- ics of human (and institutional) behaviour, can lead to ity boundary. The other eigenvalue, however, displayed severely underestimating the potential for regime shifts. a clear increasing trend (Fig. 5, Kendall tau statistic In a related result, undesirable regime shifts were previ- 0.760). ously shown to be avoidable if multiple feedbacks from On the basis of these results, we find standard devi- the state of a complex ecological system were incorpo- ation and generalised modelling eigenvalue to be good rated into management planning (Horan et al 2011). candidates for early warning signals for regime shifts The generalised modelling analysis showed the regime in this social-ecological system. However we empha- shift persists under variations to the shape of the os- sise that these are preliminary results. A more thor- tracism function and the natural resource outflows. A ough analysis would explore the sensitivity of the ob- regime shift even occurred when the social and eco- served trends to different algorithm parameters such as logical subsystems were completely linear, with non- smoothing or detrending constants and rolling window linearities arising only in the extraction and production size (Dakos et al 2012b), a consideration of false alarm processes that link the two subsystems. These findings and missed detection rates through receiver-operator have two implications for natural resource management. characteristics (Boettiger and Hastings 2012), as well First, careful attention should be paid to the links be- as repetitions over an ensemble of realisations of the tween natural resources and human actors, for example, noise. the way ecosystem services are used and contribute to 12 Steven J. Lade et al. human well-being. The link between natural resources co-operation and sudden decrease in the resource level and human well-being are in particular not well stud- and payoffs, due to the defectors gaining more from an ied. Second, social-ecological systems that under cur- increase in resource level than co-operators. We con- rent conditions are managed sustainably through high clude that sometimes not only can the regime shift it- levels of cooperation (such as the Maine lobster fishery, self be surprising, but the direction of change in a driver Acheson and Gardner 2011) could potentially easily be that triggers a regime shift can also surprise. destabilised by small changes in or increasing variability Given the widespread existence and sometimes sur- of important processes. This becomes particularly rel- prising nature of these social-ecological regime shifts, evant in the context of global change, where resource some early warning of an impending regime shift would dynamics are expected to become more variable or more be highly desirable, in order to avoid or at least miti- extreme, thus potentially pushing a successful system gate the effects of the regime shift. We tested the perfor- rapidly into an unsustainable state. mance of standard early warning signals for one of the Indeed, the ease with which a regime shift can be regime shifts produced by the TSL model. The autocor- triggered lends support to a precautionary-like approach relation warning signal showed only a weak indication to managing social-ecological systems (Raffensperger of the transition, with standard deviation (particularly and Tickner 1999): assume the system can undergo of fc) and the generalised modelling-based signal show- regime shifts, unless there is evidence otherwise. Such a ing stronger signals. In practice, the effectiveness of an precautionary approach may imply, for example: hold- early warning signal can depend on a number of factors, ing stocks at higher levels (Polasky et al 2011); using including: the magnitude of the noise (Contamin and generic principles for increasing resilience (Biggs et al Ellison 2009; Perretti and Munch 2012); appropriate 2012b), such as engaging in a process of adaptive gover- choice of variable(s) to observe; whether a potential as- nance; or preparing to mitigate the effects of the regime sociated with the dynamics exists and is smooth (Hast- shift if it is unlikely to be avoided (Crépin et al 2012). ings and Wysham 2010); the rate at which the driver Examining regime shifts demands viewing the social- is changing relative to the inherent time scales of the ecological system as a complex adaptive system (Levin systems, such as life spans (Bestelmeyer et al 2011); the et al 2012). As well as the possibility of non-equilibrium observation rate compared to these inherent time scales behaviour (such as regime shifts), an important com- (Bestelmeyer et al 2011); non-stationary noise statistics plex adaptive property of the system we studied here is (Dakos et al 2012c); and indeed whether the regime the ability of human actors to switch their harvesting shift is driven at all or is instead triggered by noisy strategies in response to changing resource levels. The fluctuations (Ditlevsen and Johnsen 2010). The auto- tool of generalised modelling that we have used com- correlation, standard deviation and generalised mod- bines a complex adaptive systems view of the social- elling early warning approaches also have different re- ecological system with a precise mathematical setting quirements for the amount and type of data and knowl- and the ability to obtain results in the presence of un- edge required (Sec. 3.3). We conclude that early warn- certainty about specific forms of interactions. Given ing signal approaches show potential for warning of the high degree of uncertainty often associated with social-ecological regime shifts, which could be valuable the detailed workings of social and ecological processes, in natural resource management to guide management we anticipate generalised modelling to be a useful tool responses to variable and changing resource levels or in future work on social-ecological systems. Although changes in resource users. However, investigation of spe- recently developed, generalised modelling (sometimes cific cases of social-ecological regime shifts is required to also called structural kinetic modelling) has already ascertain, first, the availability of the required data in yielded successes in ecology (Gross et al 2009; Aufder- those cases and, second, the robustness of the resulting heide et al 2012), physiology (Zumsande et al 2011) and early warning signals. molecular biology (Steuer et al 2006; Gehrmann and A third and very important criterion by which to Drossel 2010; Zumsande and Gross 2010), where the evaluate an early warning signal in a specific case study details of specific interactions can likewise be difficult is whether the signal can give sufficiently early warn- to determine. ing for the transition to be avoided. Successfully avert- We also studied regime shifts in our social-ecological ing a transition depends on a number of case-specific system using a simulation model. The regime shifts factors, including which drivers can be manipulated could be triggered by many different social and eco- (Biggs et al 2009), the rate at which this can be done logical drivers, and also a combination of drivers. We (Biggs et al 2009), how fast the system responds to a also obtained the result, on first glance counterintuitive, change in management (Contamin and Ellison 2009), that increasing the resource inflow led to a collapse of and, importantly, how fast the uncontrolled driver is Regime shifts in a social-ecological system 13 itself changing. In the limit of a very slowly changing FC, Rassweiler A, Schmitt RJ, Sharma S (2011) Analy- driver, for example, warning signals are likely to pro- sis of abrupt transitions in ecological systems. Ecosphere vide sufficient notice for action to be taken, while in 2:129 Biggs R, Carpenter SR, Brock WA (2009) Turning back from the limit of a very quickly changing driver, effectively the brink: Detecting an impending regime shift in time to an unpredictable shock, no warning signal could be suf- avert it. Proc Natl Acad Sci USA 106(3):826–831 ficiently fast. Including management responses in the Biggs R, Blenckner T, Folke C, Gordon L, Norström A, social-ecological system and evaluating whether warn- Nyström M, Peterson G (2012a) Regime shifts. In: Hast- ings A, Gross LJ (eds) Encyclopedia of Theoretical Ecol- ing signals can give sufficient notice for management ogy, University of California Press, pp 609–617 actions to avert a regime shift are beyond the scope Biggs R, Schlüter M, Biggs D, Bohensky EL, BurnSilver S, of the stylised models used here. 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Environ- mental and Resource Economics 44:29–46 Crépin AS, Biggs R, Polasky S, Troell M, de Zeeuw A (2012) Acknowledgements The authors would like to thank Ralf Regime shifts and management. Ecological Economics Eichhorn, Thilo Gross, Jamila Haider, Juan Carlos Rocha and 84:15–22 Nanda Wijermans for helpful comments on the manuscript. Dakos V, Carpenter S, Cline T, Lahti L (2012a) Early warn- The research leading to these results has received funding ing signals toolbox. Version 1.0.2, http://earlywarnings.r- from the European Research Council under the European forge.r-project.org Unions Seventh Framework Programme (FP/2007-2013)/ERC Dakos V, Carpenter SR, Brock WA, Ellison AM, Guttal V, grant agreement no. 283950 SES-LINK. 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