NEW

S FE

ATU

RE

www.pnas.org/cgi/doi/10.1073/pnas.1301980110 PNAS Early Edition | 1 of 3

NEWS FEATURE

First, he tracked basking sharks—filter- feeding leviathans that look like super-sized great whites—in the coastal waters near Great Britain, and then Atlantic cod, leatherback turtles, Magellanic penguins, and bigeye tuna. He’s wrangled and tagged ocean sunfish, blue sharks, and shortfin mako sharks (and claims success by the fact that he still has all his fingers). Only after David Sims had exhausted many other spe-cies did he find his ultimate prey: the alba-tross.

Sims, a senior research fellow at the Ma-rine Biological Association of the United Kingdom, Plymouth, and a marine ecology professor at the University of Southampton, United Kingdom, has tracked many a spe-cies during his career, and it’s all to answer a fundamental ecology question: What’s the

best way for an animal to search for small and irregularly distributed pockets of food when it has limited information, and the whole ocean is open to it?

After investigating the feeding patterns of all these species, Sims has found that the optimal search often traces a mathemati-cal fractal called the Lévy flight, which is characterized by long segments followed by shorter hops in random directions. Sims proposes the pattern has evolved as a natu-rally selected strategy that gives animals an edge in the search for sparse prey.

This strategy may also guide Homo sa-piens: More recent studies suggest that the fractal also fits some human behaviors, like bidding in online auctions. Moving beyond behavior, it shows up in models of dripping faucets, variations in a healthy heartbeat,

the movement of light in certain materials, and astrophysical phenomena as well. The pattern isn’t just a mathematical curiosity that helps animals find food, it’s an optimi-zation strategy. From engineering to data analysis to financial decisions, optimization offers a way to make better forecasts when confronted with random data. That’s ap-pealing to business analysts who have a dif-ferent prey in mind: consumers.

Business entrepreneurs have been con-tacting Sims, eager to turn data on people’s decisions into profitable predictions of fu-ture behavior. “It does rather suggest that there might be a deeper mechanism un-derlying the behavior we’re seeing [in ani-mals],” Sims says.

Chasing the AlbatrossAfter seeing a Nature paper by Sims and his collaborators in 2010 that con-nects Lévy-like foraging paths of ma-rine predators to the distribution of available food in the water (1), French bird ecologist Henri Weimerskirch, Centre d’Études Biologiques de Chizé, Villiers en Bois, France, got in touch and said he’d been electronically tracking the giant albatrosses that lived on or near the Kerguelen Islands, a craggy and cold ar-chipelago in the South Indian Ocean (Cap-tain Cook christened them the “Desolation Islands”). Sims jumped at the chance to move his work beyond the water and study albatross wanderings. “It’s this incredible data set,” Sims says. The sensors recorded a bird’s position every second or 10 seconds, for days. “It gives you unprecedented in-sight into the twists and turns of an alba-tross.”

That collaboration led to a paper pub-lished in PNAS in May 2012, where Sims, Weimerskirch, and their collaborators showed that when food was scarce, like far from shore in the open ocean, the birds flew

Stephen OrnesScience Writer

Foraging flights

Discovering the feeding patterns of the albatross could lead to a broader understanding of optimization strategies in human behavior and business.

Black-browed albatross (Diomedea melanophris) flying near Saunders Island, Falkland Islands.

©iS

tock

ph

oto

.co

m/G

ento

oM

ult

imed

iaLi

mit

ed.

Dow

nloa

ded

by g

uest

on

Sep

tem

ber

10, 2

020

2 of 3 | www.pnas.org/cgi/doi/10.1073/pnas.1301980110 Ornes

long, uninterrupted stretches followed by staccato jumps in different directions in quick succession, the characteristic chicken scratch of Lévy flight (2). When food was plentiful and evenly distributed, like in the shallower waters near shore, the birds fed by flapping around randomly.

The hallmark characteristic of a fractal is scale invariance, which means the pattern appears whether one zooms in or out. The part looks like the whole, and the whole looks like the part. But showing invariance in real-world data can be tricky: In the real world, animals are too complex to fit a pure mathematical model. They don’t spend all their time foraging in sparse environments. They may switch behaviors in different hab-itats. Scientists like Sims have to figure out which data they can use, and how to decide whether or not that information matches the pattern. They also need to cleverly track the animals: In a recent study published in PNAS, Japanese researchers found hints of Lévy flight when they used video cameras and accelerometers to track foraging pen-guins (3).

By the time Sims came to the albatross, he had already found that sharks, cod, tur-tles, penguins, tuna, and the funny-looking ocean sunfish all follow Lévy-like flights as they search for food. Other studies have found the pattern in the foraging habits of jellyfish, some insects, and shearwaters.

Perhaps not surprisingly, the pattern also emerges deep in Sims’ own behavior. His description of how to find basking sharks

sounds like his strategy for finding lost keys and bears an uncanny resemblance to the fish he’s watching: “We would search an area quite intensively, and then jump to somewhere completely different, and search that area intensively,” he recalls.

The widespread study of Lévy flights has caught the interest of researchers from a wide variety of disciplines, but not every-one is convinced. The last few years have brought a growing dispute over whether Lévy flights are really there, or the product of statistical wishful thinking.

Finding the Right FlightThe history of Lévy flights goes back to the dawn of statistical physics in the 19th century. In 1827, British biologist Robert Brown observed pollen grains wandering aimlessly, in short steps, when suspended in water. This “Brownian motion” is now known as an example of what mathema-ticians call a random walk. An object un-dergoing random walk is often likened to a drunk on a sidewalk, equally likely to stag-ger in any direction.

In 1937, Parisian mathematician Paul-Pierre Lévy introduced a type of random walk in which step length is determined by a certain type of probability distribu-tion that introduces the possibility of tak-ing a giant step (4). The term “Lévy flight” appeared in mathematician, and student of Lévy, Benoit Mandelbrot’s 1982 landmark Fractal Geometry of Nature (5).

The extension to the animal kingdom ar-

rived in the 1990s, when G. M. Viswana-than, a physics graduate student at Boston University, studied data recorded by sen-sors attached to the legs of five wandering albatrosses that lived on Bird Island, South Georgia, a South Atlantic hotbed of bird research. His analysis of the flight of the albatrosses turned up the telltale soaring straightaways and shorter segments of Lévy flight. In the resulting 1996 Nature paper, Viswanathan and his collaborators made their case for the foraging fractal (6), but also acknowledged the study’s limitations: “Our findings represent a first attempt at studying Lévy flight search patterns in ani-mal behavior, and we expect the quality of such studies to improve with better data.” As a first attempt, it introduced a bold new idea. But the conclusions didn’t stick.

Trading Statistical ShotsEcological modeler Andrew Edwards, at Fisheries and Oceans Canada’s Pacific Bio-logical Station, Nanaimo, says he started out enchanted by Lévy flights. “It’s quite an appealing idea, and I quite liked it when I got into it,” Edwards says, “but it was a too-good-to-be-true kind of thing.”

In a 2007 paper in Nature, with Viswana-than and others, Edwards reexamined the albatross data and found no evidence of the fractal (7). In a 2008 paper in the Journal of Animal Ecology, he suggested that the emergence of Lévy flight in the analysis might have been a consequence of the bias of the statistical methods used (8).

GPS foraging track of a black-browed albatross (BBA46) off the Kerguelen Islands, Southern Indian Ocean, viewed across large (100 s km; Left) to small (10 s m; Right) scales showing similar patterns of trajectory complexity at all scales. Background color denotes bathymetry in meters. Each red square denotes area covered by panel adjacent right. Image and legend text reproduced from ref. 2.

Dow

nloa

ded

by g

uest

on

Sep

tem

ber

10, 2

020

www.pnas.org/cgi/doi/10.1073/pnas.1301980110 PNAS Early Edition | 3 of 3

NEW

S FE

ATU

RE

Sims wasn’t deterred: Those problems articulated by Edwards, he says, have long since been accounted for.

Since then, Sims and Edwards have been trading analyses in a volley in peer-reviewed journals, focusing on the appropriateness of their models and their choices of which data to include. Edwards has published a series of analytical papers criticizing stud-ies, including Sims’ study of marine preda-tors that claim Lévy flight; meanwhile, Sims maintains evidence of the pattern in an ever-growing list of species (11). “I think the whole idea is kind of getting watered down,” Edwards says. The original, prob-lematic albatross studies stuck closer to true Lévy’s distribution, he says, but now researchers like Sims are lowering the bar, content to look for looser, “Lévy-like” be-havior. “It’s hard to keep up with such mov-ing goalposts,” he says.

The albatross back-and-forth, says physi-cist Luis Amaral at Northwestern Univer-sity, Evanston, IL, who has studied Lévy flights in human behavior, arises from the natural discordance between real data and pure theory. “Reasonable people that are smart are disagreeing in some of these aspects because of the limitations in real data,” Amaral says.

CyberforagingScientists from other fields are finding Lévy flights on a different, abstract realm, and studies of human behavior suggest that we, too, obey fractal laws. Consider lowest-unique-bid online auctions: In these games, people pay an entry price to be able to sub-mit a ridiculously low bid on an expensive item. The auction has curious rules: A per-son can bid as low as a single cent on an item that’s valued at thousands of dollars. But the only way to win the auction is to be the sole bidder on a particular amount. (If

two people bid $0.01, then neither will win. If two people bid $0.01 and another bids $0.02, then $0.02 will win the auction.) A person can’t see anyone else’s bids, but they can know if they’re winning.

Andrea Baronchelli, a physicist at North-eastern University, Boston and collaborator Filippo Radicchi at Northwestern Univer-sity tracked bidding behavior in these auc-tions. They weren’t looking for Lévy flight, but they found it. In a paper published in PLOS ONE in January 2012, together with Amaral, they showed that an individual bidder will cluster bids close together on a single item, then jump—from $0.01 to $20, for example—to cluster more close bids (12).

The strategy appeared to be hardwired in players’ brains. “The behavior of the us-ers was almost universal in our study,” says Baronchelli, “Everybody does these Lévy flights.”

That gambling study resonates with the idea that Lévy flights are intrinsic, evolved behaviors that optimize the search for a gleaming prize in a seemingly empty sea. The scientists kept going: In a June 2012

paper in Physical Review E, Baronchelli and Radicchi explicitly addressed the idea of evolution (13). They showed that real-world Lévy flight data from their experiments on online bidding line up neatly with a com-puter model of evolved behavior. Winners gravitate toward the pattern, and the pat-tern maximizes a person’s financial return. Baronchelli thinks the online bidding be-havior shows that human beings map the same approach animals take to look for food onto an abstract, mental space.

Sims says the next step for ecologists is to tie scale-invariant behaviors, like Lévy flights, to specific environments where they appear. If the Lévy flight foraging hypoth-esis is correct, he says, then animals that live together and have evolved together should share the foraging pattern. In his latest study, he’s been looking at 15 such species—and has found the Lévy flight in abundance.

“I think we’re interested now in under-standing how these patterns arise,” Sims says, “Not just the where and when, but also the how and the why. Those are the impor-tant questions.”

1 Humphries NE, et al. (2010) Environmental context explains

Lévy and Brownian movement patterns of marine predators.

Nature 465(7301):1066–1069.

2 Humphries NE, et al. (2012) Foraging success of bio-

logical Lévy flights recorded in situ. Proc Natl Acad Sci USA

109(19):7169-7174.

3 Watanabe YY, Takahashi A (2013) Linking animal-borne

video to accelerometers reveals prey capture variability. Proc

Natl Acad Sci USA 110(6):2199–2204.

4 Lévy P (1937) Théorie de L’addition des Variables Aléatoires

(Jacques Gabay, Paris).

5 Mandelbrot BB (1982) The Fractal Geometry of Nature

(Macmillan, New York).

6 Viswanathan GM, et al. (1996) Lévy flight search patterns of

wandering albatrosses. Nature 381:413–415.

7 Edwards AM, et al. (2007) Revisiting Lévy flight search

patterns of wandering albatrosses, bumblebees and deer.

Nature 449(7165):1044–1048.

8 Edwards AM (2008) Using likelihood to test for Lévy flight

search patterns and for general power-law distributions in

nature. J Anim Ecol 77(6):1212–1222.

9 Edwards AM (2011) Overturning conclusions of Lévy flight

movement patterns by fishing boats and foraging animals.

Ecology 92:1247–1257.

10 Edwards AM, Freeman MP, Breed GA, Jonsen ID (2012)

Incorrect Likelihood Methods Were Used to Infer Scaling Laws

of Marine Predator Search Behaviour. PLoS One 7(10):e45174.

11 Sims DW, Humphries NE (2012) Levy flight search patterns

of marine predators not questioned: a reply to Edwards et al.

arXiv:1210.2288.

12 Radicchi F, Baronchelli A, Amaral LA (2012) Rationality,

irrationality and escalating behavior in lowest unique bid

auctions. PLoS One 7(1):e29910.

13 Radicchi F et al. (2012) Evolution of optimal Lévy-flight

strategies in human mental searches. Phys Rev E 85: 0611211-

0611218.

Dow

nloa

ded

by g

uest

on

Sep

tem

ber

10, 2

020