# Fly: I didn’t expect it, I also know advanced mathematics

A fly landed on the table next to Xiao Ming. He slowly approached the fly, intending to give the fly, which was disturbing his meal, a lesson it would never forget. At this point he was less than an arm’s length away from the fly, and just as the thought of action occurred to him, the fly flew away, and Ming stopped in frustration.
The fly then landed a short distance away. Ming tried the same trick again, intending to approach slowly as before. But this time he just moved and the fly flew up again, flew in front of him for a while, and then landed in the distance. Irritated, Ming abandoned any tactics and chose to strike head-on, but the fly was much more agile than he was – once again he fell short and the fly flew away quickly, unharmed.
Ming was unsuccessful because the fly was using one of nature’s most ghostly flights – Levi’s Flight.
The unpredictable Levi’s flight

Named after the French mathematician Paul Levy, Levy’s flight is a random flight movement that often turns suddenly and without any regularity, and its two-dimensional image is shown in Figure 2. The graph of this flight is a kind of fractal geometry, and its characteristic is that it randomly picks the place where the graph has a turn, and then enlarges this piece, no matter how many times it is enlarged, it still looks similar to the original graph. So, according to this graph, we can find that even if the fly flies in front of us, or rests on the hand, its action is still unpredictable.
Speaking of Lévy flight, we have to mention another kind of random motion – Brownian motion, whose two-dimensional image is shown in Figure 1. The law followed by the random motion of the particle is the Brownian motion. According to the image, we can find that although the Brownian motion is random, more of its trajectories are concentrated in a certain area. In other words, the Brownian motion still has traces.
Comparing the two, the total distance of Lévy’s flight is much less than that of Brownian motion, but it covers a wider area than Brownian motion. This means that the exploration efficiency of Lévy’s flight is much higher than that of Brownian motion.
The ubiquitous Lévy motion

To avoid confusion, the following will refer to Levi’s flight collectively as Levi’s motion. In addition to flies on land, top predators in the ocean also use Levi’s motion. Scientists fitted trackers to 14 species of marine animals, including silky sharks, yellowfin tuna and sailfish, and then recorded more than 12 million foraging movements over 5,700 days to study their foraging trajectories.
Scientists found that these top marine predators used Brownian motion foraging when food was more plentiful. However, when food is more scarce, their foraging trajectory will gradually shift to Levi’s motion. The reason for this shift is that, as mentioned earlier, the search efficiency of Lévy motion is much higher than that of Brownian motion. In the case of equal or even less total distance traveled, the search area of Lévy’s movement is much wider than that of Brown’s movement, and thus has the opportunity to obtain more food.

Not only that, soil amoebae, plankton, termites, bumblebees, large terrestrial herbivores, birds, primates, and aborigines have similar patterns in their routes when foraging for food, and Levi’s flight seems to be a common law for organisms to survive in an environment with scarce resources.
There’s no such thing as a free lunch

In 2015, Japanese scientists further studied the Levi’s motion of flies. Although Levi’s motion is mathematically graphically very efficient, when it comes to reality, the situation becomes a bit more complicated. They found the following three scenarios.
In the first case, flies that used Levi’s motion to forage for food when there was no predator attack did end up with an extremely rich diet. The search was very efficient.
The second is when there is a predator, but the predator is a “lazy” hunter like a spider. They take their time and rarely strike, waiting for their prey to approach, fall into a trap, or deliver themselves. In this case, the fly’s search is the least efficient, even less efficient than the Brownian motion. And flies in this situation have the highest mortality rate. The flies die quickly, and therefore the search efficiency is low.
The third type remains the case where there are predators or natural enemies. This time it is a fast-moving animal, such as a human. The search efficiency of flies in this environment is reduced, but still higher than that of Brownian motion. It is not difficult to imagine that a spider with a large web is more likely to catch a fly when it flies by than a human with a fly swatter. Finally, the scientists found that Levi’s motion also increased the probability that the fly would encounter a predator.
Now, let’s go back a little to the ocean’s top predators. If the Lévy movement is always taken to feed, then this increases the chances of the predator meeting other competitors or more powerful predators, when it can be said to be directly into a life and death crisis. In the case of sufficient food, they do not need to take the extra risk.
In summary, although the Levi’s movement is very efficient, in real life, using it is not without any cost.