Fly: Surprised?! I also understand advanced mathematics

A fly landed on the dining table next to Xiao Ming. He approached the fly slowly, intending to teach the fly that disturbed his meal a lesson he would never forget. At this time, he was less than an arm’s distance away from the fly. Just when he had the idea of moving, the fly flew away, and Xiao Ming stopped in frustration.

The fly then landed not far away. Xiao Ming repeated his old tricks, intending to approach slowly as before. But this time as soon as he moved, the fly flew again, flew in front of him for a while, and then landed in the distance. Xiao Ming became angry, abandoned any tactics, and chose to attack head-on, but the fly’s reaction was much faster than his – he missed again, and the fly flew away quickly without being hurt at all.

Xiao Ming failed because the fly was using one of nature’s most spooky flights – Levi’s flight.

Unpredictable Levi flight

Levy flight, named after the French mathematician Paul Levy, is a random flight movement, often turning abruptly without regularity. Its two-dimensional image is shown in Figure 2. This flying figure is a kind of fractal geometry, and its characteristic is that the place where the figure has a turning point is randomly selected, and then this place is enlarged, no matter how many times it is enlarged, it still looks similar to the original figure. So, according to this graph, we can find that even if the fly is flying in front of it or parked on the hand, its action is still unpredictable.

When it comes to Levi’s flight, we have to mention another random motion – Brownian motion. Its two-dimensional image is shown in Figure 1. The law of random motion of particles is Brownian motion. According to the image, we can find that although Brownian motion is random, more of its trajectories are concentrated in a certain area. In other words, Brownian motion is still traceable.

In comparison, the total distance of Levi’s flight is much less than that of Brownian motion, but its coverage is wider than that of Brownian motion. This means that the exploration efficiency of Levi’s flight is much higher than that of Brownian motion.

The ubiquitous Levi movement

To avoid confusion, Levi’s flight will be collectively referred to below as Levi’s motion. In addition to flies on land, top predators in the ocean also use Levi Motion. Scientists fitted trackers to 14 marine animals including silky sharks, yellowfin tuna and sailfish, and then recorded more than 12 million feeding movements over 5,700 days to study their feeding trajectories.

Scientists have found that these top marine predators use Brownian motion to forage when food is plentiful. However, when food was scarcer, their foraging trajectories gradually shifted to Levy movements. The reason for this shift is that, as mentioned earlier, the search efficiency of Levi’s motion is much higher than that of Brownian motion. Under the condition that the total distance of motion is equal or even less, the search area of ​​Levi’s motion is much wider than that of Brownian motion, so there is a chance to obtain more food.

Not only that, but the routes of amoeba, plankton, termites, bumblebees, large terrestrial herbivores, birds, primates, and aboriginal people in the soil also have similar patterns when they forage for food. Levi flight seems to be a biological resource. The common law of survival in a scarce environment.

There is no free lunch

In 2015, Japanese scientists further studied the Levi’s movement in flies. Although Levi’s motion is mathematically very efficient, in reality, things get a little more complicated. They found the following three cases.

The first is that flies that use Levie motion to forage do end up with extremely plentiful food in the absence of predator attack. Search is very efficient.

The second is where there are predators, but the predators are “slack” hunters like spiders. They walk slowly and rarely strike, waiting for their prey to approach, fall into a trap, or deliver themselves to the door. In this case, the search efficiency of flies is the lowest, even not as efficient as the search efficiency of Brownian motion. And in this case the flies have the highest mortality rate. The flies die fast, so the search efficiency is low.

The third is still the case of predators or natural enemies. This time it’s a fast-moving animal, like a human. Although the search efficiency of flies in this environment is reduced, it is still higher than the search efficiency of Brownian motion. It’s not hard to imagine that when a fly flies by, a spider with a large web is more likely to catch the fly than a human with a fly swatter. Finally, the scientists found that Levi’s motion also increased the probability that flies would encounter predators.

Now, let’s go back a little bit to the top predators of the ocean. If the foraging method of Levi’s movement is always adopted, then this will increase the chances of predators meeting other competitors or more powerful predators, and it can be said that it directly enters a life-and-death crisis. When food is plentiful, they don’t need to take extra risks.

To sum up, although Levi Motion is very efficient, in real life it is not without a cost.

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