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THE MIND OF THE SWARMMath explains how group behavior is more

than the sum of its partsBY ERICA KLARREICH

Few people can fail to marvel at a flock of birdsswooping through the evening sky, homing inwith certainty on its chosen resting place. Thenatural world abounds with other spectacularexamples of animals moving in concert: a school

of fish making a hairpin turn, an ant colony buildinggiant highways, or locusts marching across the plains.

Since ancient times, scientists and philosophers have ponderedhow animals coordinate their movements, often in the absence ofany leader. Coordinated groups can range in scale from just a fewindividuals to billions, and they can consist of an intelligent speciesor one whose members have barely enough brainpower to recog-nize each other.

Despite these differences, similarpatterns of motion appear again andagain throughout the animal king-dom. This congruence in behaviorhas led researchers to speculate forabout 70 years that a few simplerules might underpin many sophis-ticated group motions. However,establishing just what these rulesare is no easy matter.

“Imagine a space alien lookingat rush hour traffic on the L.A. free-way,” says Julia Parrish of the Uni-versity of Washington in Seattle,who studies fish schooling. “Itthinks the cars are organisms andwonders how they’re moving in apolarized way without collisions.The reason is that there’s a set of rules everyone knows.

“We’re the space aliens looking at fish, and we don’t have the dri-ver’s manual,” she says.

In recent years, mathematicians and biologists have started toget glimpses of just what may be in that manual. They have con-structed mathematical models of animal swarms and colonies thattake inspiration from decades of physics research. In physicists’studies of magnetism, for instance, they have elucidated how sim-ple local interactions give rise to complex, large-scale phenomena.Using a combination of computer simulations and experimentswith real animals, researchers are explicating how a trio of physicsand engineering principles —nonlinearity, positive feedback, andphase transitions—may be basic ingredients from which a widevariety of animal-swarming behaviors takes shape.

“This is a more and more exciting area in which to work,” saysIain Couzin, who studies collective animal behavior at the Uni-versity of Oxford in England and Princeton University. “We havethe mathematical foundations to investigate phenomena quicklyand effectively.”

POSITIVE FOOD BACK Anyone who has left crumbs on thekitchen counter knows the brutal efficiency with which ants cancapitalize on such a mistake. As soon as one ant discovers atempting morsel, thousands more create and follow a trailbetween the food source and their nest.

“Ants follow only local rules … but the resulting trail structureis built on a scale well beyond that of a single ant,” said DavidSumpter of the University of Oxford in England in an article onanimal groups in the January Philosophical Transactions of theRoyal Society B.

In 2001, using mathematical modeling and lab experiments,Sumpter and two colleagues studied how foraging pharaoh’s antsbuild trails. The researchers turned up a striking group behav-ior: Just as water abruptly turns to ice at the freezing point, for-aging behavior undergoes a “phase transition” at a certain criti-

cal colony size. If an ant colony is small, for-

agers wander about randomlyand, even if some of the ants dis-cover food, no trail persists. If acolony is large, the ants’ trailsbuild into a superhighway to thefood that they find. Somewhere inbetween—in the case of the exper-imental ants, at a colony size of700 members—the colony’sbehavior switches suddenly.

While this sharp transitionmight seem unexpected, theresearchers weren’t altogether sur-prised to find it because the math-ematical principles underlyingtheir model of foraging behaviormake such a transition likely.

When an ant discovers a food source, it deposits chemicals calledpheromones along its trail back to the nest. If another ant hap-pens to wander across the trail, it detects the pheromones andtends to follow that trail. Once it discovers the food, it will depositpheromones of its own along the trail, reinforcing it and mak-ing future ants that encounter it even more likely to follow it—exemplifying what engineers call a positive-feedback loop. How-ever, pheromones gradually evaporate, so if a trail is little used,it will eventually vanish.

If a colony is small, few ants wander around and they areunlikely to happen upon a trail before the pheromones evapo-rate. The colony collects only as much food as each ant, workingindependently, can find.

By contrast, in a large colony, many ants are likely to find agiven trail, and their combined deposits of pheromones have amultiplicative effect on the colony’s behavior. There’s a jump inefficiency that makes a large colony more than the sum of inde-pendently working ants, Sumpter says.

In mathematical terms, the ants’ behavior is nonlinear: If a

ON THE TRAIL — Ants’ foraging behavior may obey simple mathematical rules.

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colony, say, doubles in size, its trails more than double in strength.This happens because, at any moment, a trail’s growth reflectsthe product of how many ants have already found the trail andhow many ants are now likely to stumble upon it.

The result of this nonlinear growth is to eliminate the middleground. If a trail doesn’t evaporate, it will burgeon into a bustlingsuperhighway.

In each of these extremes, the individual ants are following thesame rules, points out Stephen Pratt, who studies collective ani-mal behavior at Arizona State University in Tempe. “In the olddays, the focus would have been onwhat has changed about the animalwhen it goes from one state toanother,” he says. “What’s new is tomove the question up a level and askhow changing a single environmen-tal variable, like density, can causethese dramatic changes in groupbehavior.”

Positive feedback and nonlinear-ity, which are ingredients in a widerange of animal interactions, enableanimal groups to generate behaviorsthat are more than the sum of theirparts, Sumpter says.

GLOBAL SWARMING Phasetransitions, far from being limitedto ant colonies, appear to be a ubiq-uitous feature of animal groups. In2002, for instance, Couzin and histeam showed that a few simple rulesfor fish interactions yielded phasetransitions between swarmingbehaviors.

Basing their work on a particle-interaction model from physics, theresearchers represented each fish asa single particle. They assumed threerules about how the particles inter-act: Each fish tries to avoid colliding with other fish, stays withthe group, and aligns its swimming direction with that of nearbyfish within some defined zone around itself.

Variations of these rules have been studied for decades, but onlyrecently has computer power grown to the point whereresearchers can simulate the movements of, say, 10,000 fish.

The researchers also assumed that fish can modify their sen-sitivity to their neighbors, that is, the size of their alignmentzones. The team found that as the individuals’ alignment zonesgrow, the school’s architecture undergoes two sharp transitions.

When the alignment zone around each fish is negligible insize, so that the fish barely pay attention to their neighbors’ direc-tions, each fish swims in a random direction within the group.At a certain critical size of the alignment zone, the fish suddenlystart following each other to produce a doughnut-shaped swarm.As the alignment zone continues to grow, the fish start swimmingin parallel, as in a migration.

“The model switches very dramatically and quickly betweenpatterns,” Couzin says.

As fish take into account more and more of their neighbors,the alignment between them grows nonlinearly, leading to thesharp transitions that the team observed. While biologists havenever, to Couzin’s knowledge, studied a fish school in the act ofchanging from one of the three swarming patterns to another,each of the patterns has been observed frequently in nature.

“When we first saw [the doughnut] pattern in the simulations,I thought ‘That’s really weird!’” Couzin recalls. “But then wefound in the literature that it really does appear in nature.”

The model shows that simple rules for how fish interact withtheir neighbors can give rise to complex, schoolwide patterns.Couzin reports, “There’s nothing in the individual rules that says,‘Go in a circle,’ but it happens spontaneously.”

Likewise, the model offers an explanation for how fish schoolschange their behaviors on the fly—for instance, if a predator sud-denly appears. “Very subtle adjustments in the rules allow youto create all these structures without complicated actions on thepart of the individual,” Couzin says.

Preliminary experimental evidence suggests that fish canindeed adjust the sizes of their align-ment zones. Parrish and DanielGrunbaum, also of the University ofWashington, have been filming fishschools in the lab, then using com-puterized sensing software to trackeach fish’s path. “Sometimes, theypay attention to a lot of neighbors;sometimes, to just one,” Parrish says.

However, Parrish and Grunbaumcaution, considerably more experi-mental data are needed beforeresearchers can say with confidencethat the alignment-zone model cap-tures what fish are doing. At pres-ent, tracking fish is so computation-ally intensive that Parrish andGrunbaum can follow only about 16fish at a time in the lab.

Parrish is optimistic, though, thattechnological advances will soonenable researchers both to track fishin their natural habitats and toquickly crunch the resulting data.

LOCUSTS OF CONTROL In themeantime, a team made up of Couzin,Sumpter, and several other research-ers has made progress in explainingone of the most dramatic of all ani-

mal swarms: a locust plague. Locust swarms—which often appearsuddenly and seemingly out of nowhere—can quickly grow to abillion insects that migrate together, eating every scrap of plantmatter in their path. Even in antiquity, philosophers marveledat the insects’ ability to coordinate their motion: “The locustshave no King, yet all of them march in rank,” the Bible com-ments.

Couzin, Sumpter, and their colleagues suspected that locustsfollow rules similar to those in the researchers’ fish model. In workpublished in the June 2 Science, they carried out simulations andlab experiments to test this hypothesis.

Their simulations suggested that as a locust population growsdenser, its swarming behavior changes from chaos to order. Whenthe researchers tracked locusts in a small area and graduallyincreased the insects’ number, the small group’s behavior mir-rored the model’s predictions. When there were just a few locusts,they wandered randomly, interacting only occasionally. Once thepopulation reached 10 locusts, the insects formed small bandsthat changed direction frequently. At 30 locusts, the insects sud-denly started marching as one.

“I think it’s surprising that it worked out that cleanly,” Grun-baum comments. “In the overwhelming majority of biological sit-uations, you have a nice, plausible theory, but the reality worksout differently.”

The model illuminates why it’s so hard to control locust swarmsonce they reach the density at which the insects start to marchin unison: The laws of physics overwhelm human efforts to resistthe migration.

RAPID RESPONSE — Mathematical modeling isexplaining how a school of fish can quickly changeshape in reaction to a predator.

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REACHING A CONSENSUS As with the pheromone modelof ant foraging, the positive feedback built into the alignment-zone model helps explain how an animal swarm achieves behav-ior that is more than the sum of its parts. In the Feb. 3, 2005Nature, Couzin and a group of coauthors demonstrated,through computer simulations, how a handful of informedindividuals can guide the rest of agroup along a migration route orto a food source, even if the groupmembers are incapable of recog-nizing which individuals haveexpert knowledge.

“We’re not assuming anythingabout what these animals know—they don’t know if anyone agreeswith them, and they can’t tell any-one, ‘Follow me,’” Couzin says.

The researchers assumed, sim-ply, that the experts’ choice ofdirection at any given moment isbalanced between their desire tomove in the correct direction andtheir desire to align with theirneighbors; by contrast, ignorantanimals simply do the latter.

These alignment rules create a positive-feedback effect: Themore animals are already turned in the correct direction, themore animals are likely to turn that way in the future. As longas the number of expert animals is big enough for the correctdirection to get a toehold, positive feedback amplifies the experts’influence.

The team found that the larger the group, the smaller the pro-portion of experts needed to get the group moving in the cor-rect direction. In the researchers’ simulations, for example, a

group of 30 ants needed four or five experts to get the groupmoving in the right direction, while a group of 200 could alsobe led accurately by just five of its members.

The researchers also studied what happens if the experts dis-agree. They found that the group will quickly reach a consensusand move in the direction preferred by a slight majority of the

experts—although no individualknows how the experts’ preferencesstack up or even who the expertsare. Once again, positive feedbackamplifies the majority’s tiny edgeinto a commanding lead.

“For humans, to reach consen-sus is very complicated—itrequires language and recognitioncapabilities,” Couzin says. “Butanimals can do it using very sim-ple behavioral rules.”

This simplicity has importantimplications. Couzin says, “Itmeans natural selection is muchmore likely to find this kind ofconsensus behavior” than it wouldif consensus building requiredfancy cognitive skills.

Couzin and his collaborators are now testing their model in awide range of systems, including fish and people. For instance,they’re training a few lab-kept fish in the location of a food sourceand then seeing whether they lead a group there.

One of the most important contributions that the mathemat-ical models can make, Couzin says, is to give biologists concrete,testable hypotheses to pursue. “It’s so difficult to do experiments,that starting with a good theoretical basis is important,” he says.“Theory can drive new ways of doing experiments.” ■

MIGRANT WORKERS — Positive feedback explains how afew experienced birds may lead the rest during a migration.

ZOOLOGY

Tough policingdeters cheating in insects

Coercion plays a big role in keeping work-ers in line in insect societies—in somespecies, as big a role as family ties do,according to a new study.

In many wasp and bee societies, work-ers are anatomically equipped to lay theirown eggs, but rarely do so while theirqueen’s alive. Instead, they raise thequeen’s offspring.

Several forces could drive such altru-istic babysitting, and a research teamcame up with a way to compare thestrength of two forces: family ties and

police work. The queen’s offspring are theworkers’ siblings and half-siblings, soraising them could be a worthwhilereproductive effort for the workers.Meanwhile, in the insect version of apolice crackdown, the queen or workerskill an egg that was laid illicitly by aworker (SN: 3/19/05, p.184).

Francis Ratnieks of the University ofSheffield in England suggests that espe-cially tough, thorough policing mightavoid the wasted effort that goes into pro-ducing an illicit egg.

To investigate, Ratnieks and TomWenseleers of the University of Leuven inBelgium collected police records from hon-eybees and nine Vespidae wasp species.The researchers found a link betweenpolicing and egg laying: The more thor-ough the policing was in a species, the lesslikely the workers were to lay illicit eggs.

In contrast, the closer the family tieswithin a species’ colonies, the more likelythe workers were to lay illicit eggs whilethe queen was alive. So, in insect specieswith policing, that force does more to

keep the crime rate down than family tiesdo, the researchers argue in the Nov. 2Nature. —S.M.

BIOLOGY

Jet lag might hastendeath in elderly

When old mice experienced artificial jet lag,their death rate increased, scientists report.

Gene Block of the University of Virginiain Charlottesville and his colleagues studyhow the body’s natural timekeeping, orcircadian rhythm, changes with age. Sev-eral years ago, the researchers noticed thata surprisingly large fraction of their elderlylab rats died soon after researcherschanged the daily cycle of light and darkin rooms containing the animals’ cages.

To examine this phenomenon in moredetail, Block’s team worked with middle-aged and elderly mice. Some of the ani-mals lived in cages where the researchersshifted daytime forward every week by

OFNOTE

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