http://www.sciencenews.org/articles/20071215/mathtrek.asp
Science News Online
Week of Dec. 15, 2007; Vol. 172, No. 24
Flying without Fractals
A new study raises doubts about fractal patterns in animal behavior
Julie J. Rehmeyer
Scientific fa****ons can rise and fall, like the hems of each year's
skirts. Unlike haute couture, however, scientific fads can shed light on
the world.
Fractals have been in fa****on for a couple of decades or so, and
researchers have been finding fractal-based patterns in the ways that
many animals search for food. Albatrosses, deer, and bumblebees are
among the growing array of animals with fractal-based search patterns.
But a new study has found flaws in the methodology used in all such
animal studies, raising questions about whether fractal search patterns
really do occur in animal behavior.
f9116_1130.jpg
Albatrosses don't use fractal search patterns after all, a new study
shows.
iStockphoto
Wandering albatrosses were the first animal studied that seemed to use
fractal search techniques. In 1996, Gandhimohan M. Viswanathan of Boston
University and his colleagues clipped recording devices to the legs of
albatrosses to track when the birds were in the water and when they were
not. The researchers figured that the birds were resting or feeding when
wet and flying when dry.
Most of the time, the albatrosses flew short distances, as the
researchers had expected. But sometimes, they flew very far indeed. On
rare occasions, the data showed, the birds would fly for as long as 96
hours. The pattern formed by frequent short journeys, less frequent
longer journeys and rare very long journeys is known as a "Lévy flight."
These very long journeys are the critical element that distinguishes
Lévy flights from other, more common statistical patterns. By contrast,
a speck of dust would never leap to the other side of a room in its
random jostlings, so its motion wouldn't form a Lévy flight.
Lévy flights are the statistical fingerprint of a fractal. If the
albatross's travels form a Lévy flight, a graph of it would show
connected clumps, and these clumps would be made up of smaller clumps,
which in turn would be made up of still smaller clumps. If it were a
perfect fractal, this pattern would continue forever like an infinite
set of Russian dolls. In nature, it can only continue for a few steps.
f9116_244.jpg
The drawing on the left shows the path of a dot traversing a Lévy
flight. Most steps are small, but occasionally, the dot jumps a long
way. This forms a pattern of clumps made up of smaller clumps made up of
smaller clumps. The drawing on the right, by contrast, shows Brownian
motion. The random motions of a speck of dust are Brownian. The dust can
travel significant distances, but only by a series of steps. It doesn't
make huge jumps.
Wikipedia
At the time, the researchers weren't sure why the birds would fly in
that particular pattern. They thought perhaps food was distributed on
the ocean in similar clumps. But a few years later, Viswanathan and his
colleagues showed that Lévy flights provide the best strategy in
searching for objects at random locations. This suggested that the
bird's behavior has adaptive value. "If I lost my child in the woods,
the best way to find my child would be to do a search with Lévy flight
motion," says H. Eugene Stanley of Boston University, a co-author on all
of the studies. "A simple random walk retraces its same sites over and
over and over again."
Soon, Viswanathan's team and other researchers found similar behavior in
reindeer, spider monkeys, bumblebees, zooplankton, and jackals. The
research on jackals was especially im****tant because jackals can spread
rabies. If they travel in Lévy flights, that would mean they could
spread rabies quickly over long distances.
In the decade since the original albatross study, animal tracking
methods and statistical methods have both improved. So Andrew M. Edwards
of the British Antarctic Survey collaborated with Viswanathan and his
colleagues to revisit the albatross study and see if it still stands up.
They found that it doesn't.
The researchers started by analyzing recent data from tracking
albatrosses, which includes GPS data. From the new data, they found that
the albatrosses weren't taking the very long flights at all. So they
reexamined the old data to work out the discrepancy and discovered that
the albatrosses weren't necessarily flying whenever they weren't in the
water. They also spent time sitting on their nests. So, the animals'
longest dry periods don't correspond to very long journeys after all.
Next, the researchers turned to their studies of other animals. The data
on deer foraging had a problem similar to the issue with the albatross
data, and the bumblebee data didn't stand up to stricter statistical
scrutiny.
Stanley says the original study was valuable even though it turned out
to be wrong. "Niels Bohr got a Nobel Prize for the Bohr atom," he says,
"but his theory is absolute rubbish. The truth is much more complicated
and less poetic. But Bohr's atom theory was very im****tant nonetheless,"
because it was a step that led to the more sophisticated theories. The
development of the theory of Lévy flights in animal behavior is
following a similar trajectory, he says.
Frederic Bartumeus of Princeton University says he's not surprised to
find out that some of the old studies were flawed. "They're showing that
we have to be careful how we analyze data to say that we definitely have
these Lévy flight patterns, and that's good," he says. "But that doesn't
mean there are no Lévy flights in nature."
Indeed, many researchers are continuing to study a variety of animals,
from honeybees to humans. Bartumeus and others are optimistic that some
of these animals will be rigorously shown to exhibit Lévy flight
behavior. Viswanathan himself shares the optimism, but "the jury is
still out," he says.
If you would like to comment on this article, please see the blog version.
References:
Edwards, A.M. . . . H.E. Stanley, and G.M. Viswanathan. 2007. Revisiting
Lévy flight search patterns of wandering albatrosses, bumblebees and
deer. Nature 449(Oct. 25):1044-1048. Abstract available at
http://dx.doi.org/10.1038/nature06199.
Peterson, I. 1997. Fractal past, fractal future. Science News Online
(March 1). Available at
www.sciencenews.org/pages/sn_arc97/75th/ip_essay.htm.
Peterson, I. 1996. Trails of the wandering albatross. Science News
150(Aug. 17):104. Available at www.sciencenews.org/pages/pdfs/
data/1996/150-07/15007-11.pdf.
Viswanathan, G. M. . . . and H.E. Stanley. 1999. Optimizing the success
of random searches. Nature 401(Oct. 28):911–914. Abstract available at
http://dx.doi.org/10.1038/44831.
Viswanathan, G. M. . . . and H.E. Stanley. 1996. Lévy flight search
patterns of wandering albatrosses. Nature 381(May 30):413–415. Abstract
available at http://dx.doi.org/10.1038/381413a0.
For a tremendous amount of information about fractals, visit
http://cl*****.yale.edu/fractals.
http://www.sciencenews.org/articles/20071215/mathtrek.asp
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