Definitions and basics Stepwise change in time series

Definitions and basics Stepwise change in time series Radical change of between alternative stable states (attractors) • • Resistance, resilience Reversibility, trajectory of change Recovery time Thresholds for shift and recovery

Patterns in RS (Scheffer, van Nes 2004) 1. Catastrophic transitions: Dramatic regime shifts occurring in response to a small change in an external condition (e. g. temperature or exploitation pressure). 2. Hysteresis: The phenomenon that catastrophic shifts are not reversible by an equally small reverse change in the external condition. 3. Changing resilience: The size of the attraction basin (resilience) changes with external conditions, implying changes in brittleness, i. e. the likelihood of a stochastic event triggering a regime shift changes.

Types of RS (Collie et al. 2004) 1. The ‘‘smooth’’ regime shift is characterized by a quasi-linear relationship between the forcing and response variables. This would be the case if the response variable directly tracks fluctuations in the forcing variable, for example, if production were directly related to wind stress. If fluctuations in the forcing variable approximate a step function, the coupled system could exhibit a biomodal frequency distribution, spending more time in the high and low regimes and less time in the intermediary states. 1. The ‘‘abrupt’’ regime shift has a nonlinear relationship between the forcing and response variables. As a result, gradual changes in the forcing variable will be amplified to abrupt shifts in population abundance. Even for strictly uncorrelated random variations (white noise) in the forcing variable, there would be a bimodal distribution of the response variable. However, the more interesting case in marine ecosystems is when persistent shifts in population abundance occur simultaneously with autocorrelated changes in the forcing variables 2. The ‘‘discontinuous’’, regime shift involves an abrupt response between alternative stable states. The discontinuity occurs when the forcing variable exceeds a threshold value and the response variable passes through the unstable equilibrium (broken line in Fig. 2(c)) to the lower stable equilibrium. When the forcing variable is decreased, the response variable will follow a different trajectory to the upper equilibrium, thus exhibiting hysteresis.

Ø 1. Is there a discrete step function or intervention in the time series? A simple test is whether the means during the two periods are significantly different, considering all years as potential break points. A significant step is a necessary condition for a regime shift. However, the type of regime shift cannot be inferred from time series alone. 2. Does the response state variable(s) have a bimodal (multimodal) distribution? Answering this question in the affirmative indicates the occurrence of a regime shift but not the type. 3. Is there a different functional relationship in different regimes? For example, is there a different stock-recruitment relationship for warm and cold regimes? Again, a positive answer to this question demonstrates a regime shift but not the type. 4. With different starting conditions, will the system go to different stable states? A positive answer here would indicate the existence of a discontinuous regime shift. However, it is notoriously difficult to conduct experiments in marine ecosystems that allow the use of different starting conditions. 5. Does the system switch to an alternative state when perturbed? Assuming that the forcing variable is known, the system should switch states when this variable changes. A positive answer to this question indicates a discontinuous regime shift. 6. Does the system have a different trajectory when the forcing variable increases, compared to when it decreases? Is this difference more than a time lag between the response and the forcing variable? If yes, this is evidence for hysteresis and the existence of a discontinuous regime shift. 7. Does the second derivative of the time series have peaks? If the system crosses a saddle point, it should accelerate as it ‘‘falls off the cliff or down the waterfall’’. This effect, if present, is likely to be subtle. It should be observed in mathematical models, but perhaps not in noisy data.

Discussion groups • Detecting regime shifts using timeseries and other data • Comparison of case studies and their relevance to theory • Management of aquatic systems toward recovery and sustainable use of ecosystem services