LETTER Regime shifts in ecological systems can occur with no warning Alan Hastings1* and Derin B. Wysham1,2 1 Department of Environmental Science and Policy, University of California, Davis, CA 95616, USA 2 Department of Computational and Systems Biology, John Innes Center, Norwich NR4 7UH, UK *Correspondence: E-mail: amhastings@ucdavis.edu Abstract Predicting regime shifts ��� drastic changes in dynamic behaviour ��� is a key challenge in ecology and other fields. Here we show that the class of ecological systems that will exhibit leading indicators of regime shifts is limited, and that there is a set of ecological models and, therefore, also likely to be a class of natural systems for which there will be no forewarning of a regime change. We first describe how nonlinearities in combination with environmental variability lead to model descriptions that will not have smooth potentials, concluding that many ecological systems are described by systems without smooth potentials and thus will not show typical leading indicators of regime shifts. We then illustrate the impact of these general arguments by numerically examining the dynamics of several model ecological systems under slowly changing conditions. Our results offer a cautionary note about the generality of forecasting sudden changes in ecosystems. Keywords Chaos, dynamics, potential, predator���prey, regime shifts, Ricker model. Ecology Letters (2010) 13: 464���472 INTRODUCTION A key problem in ecology and population biology is both to understand and to predict the sudden shifts that occur in a variety of ecological systems (Scheffer et al. 2001 Conners et al. 2002 Bond et al. 2003 Foley et al. 2003 Scheffer & Carpenter 2003 Scheffer & van Nes 2004 van Nes & Scheffer 2005 Brock & Carpenter 2006 Carpenter & Brock 2006 Guttal & Jayaprakash 2007, 2008 Carpenter et al. 2008 Biggs et al. 2009 Contamin & Ellison 2009 Chisholm & Filotas 2009 Takimoto 2009). Current research has focused on the possibility of determining leading indicators that would precede these regime shifts in both ecological and other more general settings (Scheffer et al. 2009). There is a wide variety of natural systems that can exhibit sudden changes in their state (Folke et al. 2004), with dramatic shifts possible in both the levels of different variables, such as vegetation, or shifts in the kinds of observed dynamics. Often the sudden changes lead from a more desirable ecosystem state to a less desirable one, such as eutrophi- cation in lakes (Carpenter 2005 Amemiya et al. 2007) or desertification (Guttal & Jayaprakash 2007). In most cases, it would be beneficial to predict when a change is going to occur before it does so, in order to allow for preventative management solutions or for ameliorative steps to lessen impacts. Studies (Brock & Carpenter 2006 Carpenter & Brock 2006 Carpenter et al. 2008) that have tried to determine likely leading indicators of regime shifts have typically begun with a description of an ecosystem using a dynamic model that includes both slowly changing underlying conditions (a slowly changing parameter) and some form of stochas- ticity. It has been argued that there is an increase in variance (Carpenter & Brock 2006) or skew (Guttal & Jayaprakash 2008, 2009) or slowing down in dynamics (Chisholm & Filotas 2009) prior to a regime shift. The change in variance would be an increase in the variance around the mean population size or some other measure, a change in skew would be a change in the third statistical moment, and a slowing in dynamics would be reflected by a shift in the power spectrum toward lower frequencies. In some instances the presence of leading indicators has been supported by the behaviour of entire distributions of solutions. We suggest that real systems only present one case history for study, so any successful leading indicator that would be used in practice must be able to accurately predict a regime shift from the results using only one initial condition in a single model simulation. Ecology Letters, (2010) 13: 464���472 doi: 10.1111/j.1461-0248.2010.01439.x �� 2010 Blackwell Publishing Ltd/CNRS

We take a different approach here and argue that for classes of ecological systems which exhibit several important features, including complex dynamics and multiple possible outcomes, the search for leading indicators of regime shifts will generally fail. Our argument first begins with the underlying mathematical reasons for the lack of leading indicators of regime shifts for this class of model systems. Clearly, our general approach does not rule out the presence of leading indicators of regime shifts in those cases where natural systems are well described by models that have smooth potentials, and our work therefore suggests the importance of a better understanding of which models of ecological systems can provide a reasonable description of natural dynamics. We then confirm and illustrate the predicted lack of leading indicators in a series of examples. In these models, we make two kinds of small changes in the parameters that produce regime shifts: we either change underlying para- meters by a small amount or we vary the size of the (stochastic) perturbations. Although we clearly only present simulation results for a selected set of models, we argue that these models are a fairly general representation of many ecological systems. Since essentially any non-spatial model with strong enough overcompensatory density dependence will show the period doubling cascade to chaos and any spatially coupled model will inherit this trait, the spatially explicit, overcompensatory (i.e. period doubling) models we choose here are in fact adequate, or at least plausible, characterizations of a large set of real ecological settings. In none of the cases we simulate are the proposed leading indicators of regime shifts found. Obviously preceding any regime shift there has to be some form of change in dynamics, but here these do not appear to be those that have been used to predict a regime shift. GENERAL ARGUMENT Suggestions that there are leading indicators of regime shifts have been largely based on the assumed properties of a potential (Graham & Tel 1984), which is essentially the mathematical abstraction of the physical notion of potential energy and acts as a guide for the type of expected dynamics. It is typically represented by ��ball and landscape�� diagrams such as found in Hastings (1997, p. 123). If the potential exists, depends smoothly on the underlying state of the ecosystem, and changes smoothly as underlying condi- tions change, then the prior arguments for the existence of leading indicators of regime shifts (summarized in Scheffer et al. 2009) are valid and in fact have been confirmed empirically. Simple ecological models based on Lotka- Volterra equations can satisfy the conditions for the existence of a smooth potential, and in these cases leading indicators are found. However, models for ecological systems can easily exhibit complex dynamics, and systems like this do not, in general, have smooth potentials (Graham & Tel 1984, 1986 Graham et al. 1991). In many cases the description of a system undergoing a regime shift in the terminology used by ecologists is the one which undergoes a crisis (Grebogi et al. 1983), which is defined to be a sudden change in (long-term) system behaviour as a parameter is varied. For most types of crises to occur, a system must be capable of producing complex dynamics and thus will not have a smooth potential. For example, if a chaotic attractor exists, then the potential has a non-smooth fractal structure (down to the level of the noise) similar to the fractal structure of the attractor (Graham et al. 1991). These are precisely the types of systems for which the presence of leading indicators have not been studied, as indicated in (Scheffer et al. 2009). The models that display chaotic or complex dynamics are only a subset of the systems for which the potential will not be smooth. If multiple attractors of any type are present, which is exactly the situation often modelled when looking to understand regime shifts, then the potential is likely to be non-smooth precisely in the transition area between two attractors. The physical meaning of the points where the potential has discontinuous derivatives is that the stochastic effects, which must be included in any realistic ecosystem model, causes there to be several most probable paths the system may take out of the bottom of any local potential well. The consequence of only piecewise smooth potentials for a system is that we would not in general expect that there would be obvious precursors to a regime shift (or crisis) as parameters are varied. The reason for the discrepancy between our results and those that have been previously reported is subtle. If one writes down a given model, derives a potential, concludes it is smooth, and then adds noise during simulation, then this invariably points to the probable existence of leading indicators. Whereas, if one writes down a model that includes a stochastic term, and then derives the (non- equilibrium) potential using the methodology as presented in (Graham & Tel 1984), then one will see that this potential is in fact generically non-smooth and that the proposed leading indicators will generally no longer apply. SPEC IFIC MODELS AND M ETHODS We numerically simulate several models that are likely to lack a smooth potential in order to illustrate the character- istics of systems prior to, or during, regime shifts. Our general argument shows there are broad classes of ecological systems that we would expect to lack a smooth potential and therefore we would expect to fail to have leading indicators of regime shifts, namely those that have either multiple attractors or have at least the possibility of chaotic Letter Regime shifts with no warning 465 �� 2010 Blackwell Publishing Ltd/CNRS