The first thing you notice about the campus of the Indian Institute of Science, Bangalore, is its serene, if casual, ambience. On the third floor of the spanking New Biological Sciences building in Dr Vishwesha Guttal’s office a few PhD students, project assistants, short-term visiting researchers peer into their computers for so long that their eyes seem glued to the screens, lounge on sofas fiddling with laptops, and bang away on the keys.

Shorts and uncombed hair—whatever is left of it—are common. They come and go as they please. On a recent morning, after overnight rain lashed the city, an earthy aroma drifts in through the windows. A dark grey pigeon skitters inside, threatening to knock down tea and coffee things, all because some bleary-eyed bloke left a window open the previous night. The feeling you get is what you might expect in a coffeehouse rather than a laboratory where serious work is being done on tipping points, regime shifts, critical transitions, systems going over the edge, and on collective group behaviour.

“We call them ‘regime shifts’ as in regime shift in Iraq from Saddam Hussein’s to the current one. The term comes from the social sciences. We also call them critical transitions, coming from a physics background,” Vishu, as he is called by students and colleagues alike, says.

Ecosystems can suddenly flip, and that flip can be so abrupt, an all-body curveball, a muscular sharp snap, a veritable arse-over-tip, that most of us wouldn’t recognise it even if it kicked us, and stuck us head-down-and-feet up on a vaulting pole. It’s the physics equivalent of water turning to gas exactly at 100°C, called a phase transition. These drastic changes occur in lakes, marine ecosystems, rangelands, food webs and climatic systems and even in our own bodies. The shift from a stable state to alternative stable state is not necessarily good, from a human point of view.

Various studies have shown that lakes undergo sudden shifts. In a lake, normal feedback loops keep it running and healthy at relatively low nutrient levels but enough to support a variety of aquatic life and maintain clear water. Now, due to run-off or human-induced intervention, the lake gets more nutrients, especially phosphorus and nitrates (eutrophication). The clear water may then suddenly turn turbid, and this smothers the submerged vegetation. The lake gets infested with algae. Marine life diversity goes for a toss. It is practically impossible to return the lake to the original state even after reducing the nutrient load.

In India, the Green Revolution seems to have had one unforeseen by-product. The increased use of fertilisers has affected our lakes, as the nutrient runoff may have crossed the threshold, leading to their degradation. One of the signs is the massive hyacinth blooms that we often see on water bodies.

It is reported that nearly 90 per cent of China’s lakes have eutrophic water due to pollution. Many lakes in developing countries suffer from eutrophication. The largest freshwater body in the world, Lake Baikal in Russia, 1.7 km deep, has increased concentrations of algae and decreased transparency.

But it’s just not lakes. Intensive fishing and changed land use—Marten Scheffer et al say in an article reviewing various “Catastrophic shifts in ecosystems”, published in Nature—have degraded Caribbean reefs and they are overgrown by fleshy macro-algae.

Although no data exist to predict a tipping point on land degradation in India, the Environment Ministry’s report submitted in April to the United Nations Convention to Combat Desertification raises several concerns. Over one-fourth of India’s geographical area is undergoing desertification, that is, loss of perennial vegetation in arid and semi-arid regions.

When vegetation falls below a certain threshold, water infiltration also goes down. Seeds cannot germinate and grow, which further reduces water infiltration, and the land turns into desert. Then even an improved monsoon cannot bring it back to the earlier state. Are we too far gone over fishing in our coastal waters that no amount of fishing holidays can replenish fishing stocks? It is anybody’s guess.

Scientists say the same thing happens in epileptic seizures: at critical thresholds, systems abruptly change from one state to an alternative stable state. In a paper titled “Epileptic seizures may begin hours in advance of clinical onset: a report of five patients”, and published in Neuron, Brain Litt et al observe that “epileptic seizures may begin as a cascade of electrophysiological events that evolve over hours and that quantitative measures of pre-seizure electrical activity could possibly be used to predict seizures far in advance of clinical onset.”

Stochastic events—highly unpredictable events such as extreme weather, pest or fire outbreaks—can cause such catastrophic changes that the system can shift to an alternative stable state with tremendous ramifications. Although large extreme events can drive catastrophic changes, what is more interesting is that it can also happen for small and gradual changes. This is what makes it scientifically interesting to study and also makes it notoriously difficult to predict.

According to some studies, recently increasing climate variability points to an impending shift. They say that every example in history of sudden changes in climate was preceded by oscillations from hot temperatures to cold temperatures, and cold to hot.

Although no data exist to predict a tipping point on land degradation in India, the Environment Ministry’s report submitted in April to the United Nations Convention to Combat Desertification raises several concerns. Over one-fourth of India’s geographical area is undergoing desertification, that is, loss of perennial vegetation in arid and semi-arid regions.

In a paper published in Nature and titled, “Early-warning signals for critical transitions,” Marten Scheffer, Stephen R Hunter et al reference that: “Interest in the possibility of critical transitions in the Earth system has been sparked by records of past climate dynamics revealing occasional sharp transitions from one regime to another. For instance, about 34 Myr [million years] ago Earth changed suddenly from the tropical state in which it had been for many millions of years to a colder state in which Antarctica was glaciated, a shift known as the greenhouse–icehouse transition. Also, glacial cycles tend to end with an abrupt warming.”


Mathematical ecology is using mathematical and computation models to study ecosystems. It is a science that requires a certain relish for the abstract and extreme complexity. Although most biologists steer clear of abstractions, mathematical ecologists revel in it. “The real world is complex, but our equations can only capture the tiniest of complexity. Nevertheless, such a simplistic view, if wisely adopted, offers us profound insights into ecology,” Vishu says.

Moreover, if you’re working with these equations, you feel they are alive. Feed the data and run the equation on computer, it’s alive, it bleeds, it’s vascular, and also visceral, especially if you see how a lake, a pristine lake, can come crashing down, or an infectious disease cruises in the population.

Vishu, assistant professor at the Centre for Ecological Sciences, is one of a handful of researchers in India working in the area. Today, he is wearing a black T-shirt, with “Brooklyn Boxing Club” emblazoned across the front, two menacing-looking boxers about to demolish each other, their muscles rippling off the T-shirt. But Vishu is no boxer. He is a stalk of a man, 32, his voice wispy, comfortable in office and with computational models, because, you know, the “real world is messy and complex”.

Studying ecosystems is not easy. External conditions such as nutrient input, “after all, it doesn’t remain constant throughout the day”, groundwater reduction, habitat fragmentation—all these external conditions have a tremendous effect on ecosystems. They may respond in a smooth way to these external changes, or, after remaining inactive over a certain range, respond in stronger ways when a critical level is reached. Such states never announce themselves, and predictions are notoriously difficult to predict.


Predictions in linear dynamical systems are easy to make. For example, in population dynamics, governed by linear dynamics, it’s possible to predict the trends—how the population is spatially distributed, and how that changes over time—given certain assumptions. In Newtonian deterministic models, given the initial conditions, we can predict everything. If you know the initial positions of the moon, sun and planets, their direction, speed, and the distance between them, you can calculate where they will be the next day. It’s a fairly predictable world.

Or is it?

A nonlinear world is entirely different. For example, the famous Butterfly Effect counters this notion of predictability. It says that, in complex nonlinear systems, small initial differences can amplify over time and can lead to dramatically different outcomes, even though we know it’s deterministic, meaning we know the trajectory precisely given a system’s initial state. This is called deterministic chaos.

It was Robert May who introduced and made substantial contributions to chaos theory and its application to ecology. He showed how systems are highly sensitive to initial conditions. Smallest differences in initial conditions can yield entirely different outcomes.

May was also one of the pioneers of the idea of threshold dynamics in ecology, which he introduced in the 1970s. In nonlinear dynamical systems, the system responds disproportionately. “For example, if you double the values of the drivers, the system may respond four times as strongly.”

Nonlinear equations govern ecology, gas dynamics, relativity, chemical reactions, fluid and plasma dynamics, biomechanics and others. Nonlinear iterative systems—iteration is repeated use of a function—show up in the spread of communicable diseases such as AIDS, in the growth and extinction of biological species, and in predator-prey interactions, climate change and in many other natural phenomena.

Predictions of drastic changes are very hard to make. “You simply don’t know when the system changes into an alternative stable state.” However researchers have found generic early warning signals that indicate the system is getting closer to the tipping point.

Vishu’s educational trajectory itself seems to have followed a kind of nonlinear trajectory, a fascinating phase transition: M.Sc. in Physics, then a year of fiddling with folding RNA, and then a PhD thesis on early warning signals of tipping points at the Ohio State University with condensed matter physicist C Jayaprakash, and then the work on collective animal behavior with Ian Couzin, a biologist at Princeton.

Some time in 2006, when he went to the annual meeting of the Ecological Society of America, in Memphis, Tennessee, he heard a researcher talking about “recovery time increasing as a system gets closure to a tipping point”. That led to “how we can think about the whole phenomena in terms of a ball rolling in a landscape with hills and valleys”.

Now, at IISc, in his office, in a conversation that involves lots of pacing to the whiteboard, plotting on X-and Y-axes, and drawing curves, he talks about how the lake gets closer to the tipping point and flips. The landscape picture he draws shows mountains and valleys.

Imagine a ball rolling to the bottom of the valley. The ball eventually comes to rest at the bottom. In this abstraction of ecosystems into valleys and hills, the system going close to the tipping point is equivalent to the valley getting flatter at the bottom. Therefore the ball wandering in such a landscape would fluctuate wildly, leading to increased recovery time to regain lost ground.

Also the hillsides are not equally steep in different directions. The ball wanders off more in one direction than the other, and that leads to asymmetry in fluctuations around the valley bottom. Vishu and his PhD adviser C Jayaprakash exploited these features of the dynamics of the ball in such landscapes to devise an early warning signal of tipping points.

It is the idea of changing skewness—a mathematical quantity that measures asymmetry—as a system approaches a tipping point that is their seminal contribution to the field of regime shifts in particular. In a paper titled “Changing skewness: as an early warning signal of regime shifts in ecosystems”, published in Ecology Letters (considered one of the top journals in Ecology), they “propose a novel early warning signal that exploits two ubiquitous features of ecological systems: nonlinearity and large external fluctuations”.

Do mathematical abstractions hold well in the real world?

“Yes,” Vishu says.

In an empirical study on Daphnia (small planktonic crustaceans of the order Cladocera) populations, “Early warning signals of extinction in deteriorating environments”, by John M. Drake1 & Blaine D. Griffen (Nature), the authors state that “Coefficient of variation, skewness, autocorrelation, and spatial correlation in population size are leading indicators of extinction.”


Squirreled away in a corner and looking intently into his computer is Jaideep, a Ph.D student. He has been here for a year, studying the evolution of cooperation in collectively moving animals.

Let’s take bacterial aggregation. This is how it works: When bacteria secrete certain chemicals, the chemicals help them and other bacteria around them to grow faster. Since secretion involves energy and other costs, the very bacteria that secrete the chemicals grow slower than those that don’t secrete chemicals. This is a paradox from the perspective of Darwinian evolution, because such behaviours apparently aren’t selfish and therefore, shouldn’t occur in nature.

Our hypothesis is that formation of groups, and fission and fusion of groups can facilitate the evolution of such non-selfish cooperative behaviours. We are building computation models to test the idea,” Vishu and Jaideep say.

The dynamics of collective behaviour is another field of research Vishu pursues, and which dovetails into the study of critical transitions. “It’s fascinating to look at how localised interactions, repeated over and over again for a length of time scale up to interesting patterns of stability, robustness, and also of fragility,” he says.

As a post-doctoral scholar at Princeton, Vishu wondered why collective behaviour—starlings, locusts, fish—patterns arise in nature. Combining the ideas of evolutionary game theory and collective animal behaviour, he tried to answer the question “How has the natural selection favored those sorts of local interactions between individuals that lead to these macroscopic patterns?”

“The group behaviour helps prey evade predation,” Vishu says. In an experiment conducted fusing real and simulated animal behaviour, the team of Ian Couzin, C. C. Ioannou, and Vishu found that “Prey with a tendency to be attracted toward, and to align direction of travel with, near neighbors tended to form mobile coordinated groups and were rarely attacked. These results demonstrate that collective motion could evolve as a response to predation, without prey being able to detect and respond to predators.” (Science, September 2012).

In the case of locusts, Vishu says, at high densities, collective behaviour reduces the risk of cannibalism among them. “It’s counterintuitive,” he continues, as he shows a simulation of locusts forming groups, “cannibalism should lead to running away from each other. It happens at low densities. What we find at high densities is the collective reduces the risk of cannibalism.”

In “Cannibalism can drive the evolution of behavioral phase polyphenism in locusts,” published in Ecology Letters, 2012, Vishu, Pawel Romanczuk et al write that “this behavioral strategy minimizes the risk associated with cannibalistic interactions and may account for the empirically observed persistence locust groups during outbreaks.”
“It’s fascinating,” he says. And the equation fed into his computer iterates.