Numbers warn of looming collapses

Math tools predict when systems are on the brink

Brains have seizures, ecosystems collapse, economies crash — and it sure would be great if we could predict when. Despite the complexity of these seemingly disparate events, recent research suggests that tipping points are foreseeable.

A study published online February 2 in PLoS Computational Biology offers a way to discern when a complex system such as a fishery may be teetering towards collapse. The new work uses mathematical indicators to help researchers understand systems when there’s not enough data to build the kind of complex supercomputer simulations that are typically used to study things like climate change. And other recent studies have turned up even more mathematical red flags that a system is approaching a point of no return.

“At one end, there’s the brute force approach,” says study coauthor Steven Lade of the Max Planck Institute for the Physics of Complex Systems in Dresden, Germany. “You make a very detailed model of the system, try and add everything that’s going on, and run it into the future.”

But scientists often don’t know enough of the details to make such simulations accurate. “The more specific you can be, the better, but you shouldn’t be specific about the things you don’t know about,” says Marten Scheffer of Wageningen University in the Netherlands.

The latest work makes use of generic signals that a system might be going awry but also allows for any specific information that may be available. Lade and his Max Planck colleague Thilo Gross start with that known information, such as the year-end catch numbers for a particular fish and what that species eats. Then they link that information to general math that describes a system at near equilibrium. By adding to their simulation an outside perturbation that might push the system over the edge, such as habitat destruction or a new disease, the researchers can track the stability of the system through time. In the fisheries case the researchers used to test their model, the method revealed a glaring signal when collapse was imminent that became easier to distinguish as the collapse approached.

“It’s making the point that it’s better to use the information that you have,” Scheffer says.  “Anything you know about the system can sharpen your search image.”

Surprisingly, when even less data is available than the scant amount used in the new work, there are some simple mathematical clues that can provide a red alert that an abrupt shift is coming, be it in a tangle of brain cells or an entire ecosystem.

“In complex systems it’s really hard to predict anything,” says Scheffer. “But the fascinating thing is there are universal mathematical principles that should hold.” Much of this theory has been around for decades, Scheffer says, but no one knew whether the math would work for predicting major transitions in real systems. Now experimental data is starting to come in —mostly from the realm of ecology — and it suggests that predicting the future may soon be more about science than soothsaying.

In the Jan. 29 Nature, for example, a team led by Scheffer reported success using one mathematical test of an approaching tipping point. Theory says that when a shift is coming, a system exhibits what scientists call a critical slowing down. Normally, a really stable system quickly recovers after being perturbed. But when everything is about to come unglued, the recovery time from even a small perturbation becomes slower and slower.

Whether you’re looking at a timescale of years or centuries for a climate system, or at the scale of milliseconds in the brainwaves of someone about to have an epileptic seizure, this slowing down of recovery time appears to be an important signal that some serious action is about to go down.

Scheffer and his colleagues tested this idea by growing cyanobacteria in flasks. Under nice, stable conditions, these aquatic microbes provide shade for each other — shifting around so each gets enough light to grow, but not so much that they’re scorched to death. But when the light gets too intense, some of the cyanobacteria die. This leads to even less shade, so more of the microbes are killed and the system crashes. The researchers meddled with growing microbes, removing about 10 percent of the critters from the culture every four or five days. For a few weeks, the populations didn’t seem to mind. But the signature warning sign was there, Scheffer and his team reported. The critters were slower and slower to bounce back, and suddenly they crashed and literally burned.

In addition to watching for slowing systems like the bacteria, researchers can also look for signs that a system gets jittery when a shift is coming. By looking for variance — how much a set of measurements of a particular variable deviates from its average — researchers may see impending upheaval.

Researchers led by Stephen Carpenter of the University of Wisconsin–Madison recently experimentally tested the variance signal by adding more and more largemouth bass to a lake over a three-year period. The researchers took measurements of the light spectra of chlorophyll in the lake every five minutes (and in a control lake where they were not adding fish). Fifteen months before the food web of the whole lake shifted, the variance signal appeared in the chlorophyll measurements, Carpenter and his colleagues reported in Science last May.

“I had no idea if it was going to work,” Carpenter says. “It was pretty darn good.”

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