From ’Sharks hunt via Lévy flights’, Physics World, June 11, 2010:

They were menacing enough before, but how would you feel if you knew sharks were employing advanced mathematical concepts in their hunt for the kill? Well, this is the case, according to new research, which has tracked the movement of these marine predators along with a number of other species as they foraged for prey in the Pacific and Atlantic oceans. The results showed that these animals hunt for food by alternating between Brownian motion and Lévy flights, depending on the scarcity of prey.

Animals don’t use advanced mathematical concepts. This statement encompasses many humans as well because it’s not a statement about intelligence but one about language and reality. You see a shark foraging in a particular pattern. You invent a language to efficiently describe such patterns. And in that language your name for the shark’s pattern is a Lévy flight. This doesn’t mean the shark is using a Lévy flight. The shark is simply doing what makes sense to it, but which we — in our own description of the world — call a Lévy flight.

The Lévy flight isn’t an advanced concept either. It’s a subset of a broader concept called the random walk. Say you’re on a square grid, like a chessboard. You’re standing on one square. You can move only one step at a time. You roll a four-sided die. Depending on the side it lands on, you step one square forwards, backwards, to the right or to the left. The path you trace over time is called a random walk because its shape is determined by the die roll, which is random.

There are different kinds of walks depending on the rule that determines the choice of your next step. A Lévy flight is a random walk that varies both the direction of the next step and the length of the step. In the random walk on the chessboard, you took steps of fixed lengths: to the adjacent squares. In a Lévy flight, the direction of the next step is random and the length is picked at random from a Lévy distribution. This is what the distribution looks like:

Notice how a small part of each curve (for different values of c in the distribution’s function) has high values and the majority has smaller values. When you pick your step length at random from, say, the red curve, you have higher odds of of picking a smaller step length than a longer one. This means in a Lévy flight, most of the step lengths will be short but a small number of steps will be long. Thus the ‘flight’ looks like this:

Sharks and many other animals have been known to follow a Lévy flight when foraging. To quote from an older post:

Research has shown that the foraging path of animals looking for food that is scarce can be modelled as a Lévy flight: the large steps correspond to the long distances towards food sources that are located far apart and the short steps to finding food spread in a small area at each source.

Brownian motion is a more famous kind of random walk. It’s the name for the movement of an object that’s following the Wiener process. This means the object’s path needs to obey the following five rules (from the same post):

(i) Each increment of the process is independent of other (non-overlapping) increments;

(ii) How much the process changes over a period of time depends only on the duration of the period;

(iii) Increments in the process are randomly sampled from a Gaussian distribution;

(iv) The process has a statistical mean equal to zero;

(v) The process’s covariance between any two time points is equal to the lower variance at those two points (variance denotes how quickly the value of a variable is spreading out over time).

Thus Brownian motion models the movement of pollen grains in water, dust particles in the air, electrons in a conductor, and colloidal particles in a fluid, and the fluctuation of stock prices, the diffusion of molecules in liquids, and population dynamics in biology. That is, all these processes in disparate domains evolve at least in part according to the rules of the Wiener process.

Still doesn’t mean a shark understands what a Lévy flight is. By saying “sharks use a Lévy flight”, we also discard in the process how the shark makes its decisions — something worth learning about in order to make more complete sense of the world around us rather than force the world to make sense only in those ways we’ve already dreamt up. (This is all the more relevant now with #sharkweek just a week away.)

I care so much because metaphors are bridges between language and reality. Even if the statement “sharks employ advanced mathematical concepts” doesn’t feature a metaphor, the risk it represents hews close to one that stalks the use of metaphors in science journalism: the creation of false knowledge.

Depending on the topic, it’s not uncommon for science journalists to use metaphors liberally, yet scientists have not infrequently upbraided them for using the wrong metaphors in some narratives or for not alerting readers to the metaphors’ limits. This is not unfair: while I disagree with some critiques along these lines for being too pedantic, in most cases it’s warranted. As science philosopher Daniel Sarewitz put it in that 2012 article:

Most people, including most scientists, can acquire knowledge of the Higgs only through the metaphors and analogies that physicists and science writers use to try to explain phenomena that can only truly be characterized mathematically.

Here’s The New York Times: “The Higgs boson is the only manifestation of an invisible force field, a cosmic molasses that permeates space and imbues elementary particles with mass … Without the Higgs field, as it is known, or something like it, all elementary forms of matter would zoom around at the speed of light, flowing through our hands like moonlight.” Fair enough. But why “a cosmic molasses” and not, say, a “sea of milk”? The latter is the common translation of an episode in Hindu cosmology, represented on a spectacular bas-relief panel at Angkor Wat showing armies of gods and demons churning the “sea of milk” to producean elixir of immortality.

For those who cannot follow the mathematics, belief in the Higgs is an act of faith, not of rationality.

A metaphor is not the thing itself and shouldn’t be allow to masquerade as such.

Just as well, there are important differences between becoming aware of something and learning it, and a journalist may require metaphors only to facilitate the former. Toeing this line also helps journalists tame the publics’ expectations of them.

Featured image credit: David Clode/Unsplash.