This week’s Nature has a review article in it I have been waiting for for some time, and which I suspect may become something of a classic:
Marten Scheffer et al, Early-warning signals for critical transitions Nature 461, 53-59 (3 September 2009) | doi:10.1038/nature08227. Here’s the abstract
Complex dynamical systems, ranging from ecosystems to financial markets and the climate, can have tipping points at which a sudden shift to a contrasting dynamical regime may occur. Although predicting such critical points before they are reached is extremely difficult, work in different scientific fields is now suggesting the existence of generic early-warning signals that may indicate for a wide class of systems if a critical threshold is approaching.
One of the problems with tipping points in complex systems is that straightforward analysis is incredibly unlikely to be precise and accurate about the tipping point’s threshold. A model may happily tell you that a system has a tipping point, bit it will not tell you where it is. In climate terms, you can be sure that there is a point at which the Greenland ice sheet will collapse, but you don’t know how far we are from it. This article reviews work which suggests a way round this. The system itself may tell you when it is getting close to a tipping point through subtle changes in the way its behaviour varies over time — in particular changes associated with “critical slowing”. What follows is my interpretation of the paper, which seems impressively approachable for a piece of mathematics, but which I may nevertheless be getting wrong; any real mathematicians in the audience should feel free to chip in in the comments.
A key symptom of critical slowing are that the system becomes lower to restore itself to its usual state after being perturbed. This means that if the system is fluctuating, its present state will become more closely determined by previous states — its “memory” will increase. In mathematical terms, this is an increase in autocorrelation. At the same time, and seemingly contradictorily, its variance may also increase because it becomes less able to recover from external shocks; that effect appears not to be as well grounded as the autocorrelation, but it does turn up in a lot of models.
As well as slowing down in this way the system will also start to get assymetric, because fluctuations that push it towards the tipping point and those that push it away will not be responded to in quite the same ways. It may also start to “flicker” as it moves back and forth across the boundary between two states before plumping firmly for one or the other. There are also ways of looking at differences over time, which I can’t really sum up: instead I’ll quote an example from the paper dealing with desertification:
Models of desert vegetation show that as a critical transition to a barren state is neared, the vegetation becomes characterized by regular patterns because of a symmetry-breaking instability. These patterns change in a predictable way as the critical transition to the barren state is approached, implying that this may be interpreted as early-warning signal for a catastrophic bifurcation [that being is one of the types of tipping point under discussion]
More on the desert stuff in this fascinating Science paper from 5 years back, one of the authors of which, Max Rietkerk, is also an author on the Nature paper.
How does this work in practice? Here’s an example using data from a 2008 PNAS paper by Dakos et al, on which many of the review’s authors worked.
The top plot is showing calcium carbonate percentages in a deep sea sediment core as a marker for the influence of the carbon cycle on the climate, as discussed in this 2005 Nature paper. The bottom plot is the measure of autocorrelation. As you can see, the time series goes from being not autocorrelated at all to being highly autocorrelated, and then bang. Similar autocorrelations can be seen in front of seven other abrupt climate shifts the PNAS authors looked at. The review looks at similar patterns in the onset of asthma attacks, ecological events, epileptic seizures and sudden stockmarket surges, and as the authors conclude
Flickering may occur before epileptic seizures, the end of a glacial period and in lakes before they shift to a turbid state; self-organized patterns can signal an imminent transition in desert vegetation and in asthma; increased autocorrelation may indicate critical slowing down before all kinds of climatic transitions and in ecosystems; and increased variance of fluctuation may be a leading indicator of an epileptic seizure or instability in an exploited fish stock.
So these processes really do seem to have a lot in common, much of which is related to the mathematical treatments reviewed in the paper. This is not to say everything is settled:
More work is needed to find out how robust these signals are in situations in which spatial complexity, chaos and stochastic perturbations govern the dynamics. Also, detection of the patterns in real data is challenging and may lead to false positive results as well as false negatives.
There are a lot of complexities here to do with how you filter the data, what data you choose, whether all the bifurcations that cause critical slowing are really catastrophic tipping points, and more. There are probably people who think it is all hogwash (and people should feel free to point me towards them). But I must say that after reading the review and feeling I have come a little way towards understanding what is going on, I look forward over the next years to people with climate models that show tipping-point behaviour getting stuck into this sort of analysis looking for precurssors. (Here’s a topical question: what does an autocorelation on the year-by-year arctic sea ice minimum show?) The idea that such mathematical work will ever reach a level where you would feel justified in saying “the Greenland ice tipping point is ten years away” may be far fetched. But you don’t know til you try.
Image copyright Nature Publishing Group
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