A foraging spider monkey appears to follow a specific kind of “random walk” that optimizes its chances for finding food. Now theoretical work in the 5 November Physical Review Letters explores how this type of motion may help these monkeys, as well as other animals, avoid unfriendly encounters with competitors or predators. The theory predicts how long it would take for different foraging animals to meet up, but the formalism could also help in understanding the dynamics of physical systems such as chemical reactions in turbulent environments.
The standard example of a random walk is a drunk man stumbling away from the bar in random directions for each step. Textbooks show calculations of various quantities, such as the time it takes the drunk to travel some distance or the likelihood that he returns to his starting point. In the 1980s, researchers added a sinister twist: they imagined several drunks on the streets, each carrying a gun. If two of these so-called “vicious” walkers met, they would shoot each other. This scenario may mimic real physical situations, as in the random movement of domain boundaries in a magnet. Wherever two boundaries meet, they annihilate each other because there is no longer a domain between them.
Much of the vicious walker research has dealt with Brownian motion, in which the length of each random step is roughly the same. However, there is a class of random walks called Lévy flights, which include occasional long-distance jumps. The distribution of step sizes is described by a power law, which means that there are steps of all sizes and no well-defined “average” step size, at least for one class of Lévy flights. They have been observed in various natural settings, most famously in the search strategy of certain animals when food is scarce. For example, hungry sharks will typically scour back and forth over small areas, but if the search is fruitless, they will intermittently “jump” to new, far-off areas . “People have also [studied] Lévy flights in stock prices, epidemics, and small world networks,” says Ajay Gopinathan, from the University of California, Merced. He and graduate student Igor Goncharenko wondered what would happen if Lévy flights turned vicious.
Although previous work has simulated predators and prey both performing Lévy-type motion  no one has studied the general mathematical problem of vicious Lévy flights. Gopinathan and Goncharenko imagined several groups of independent random walkers all obeying the same Lévy distribution of step sizes. They then investigated how long it would take until one walker encountered another from a different group. The calculations came from field theory, which is well-suited to dealing with the statistics of many particles. Field theory had been used before for vicious walkers obeying Brownian motion  so Goncharenko and Gopinathan extended that formalism to the case of Lévy flights.
As expected, they found that encounters were extremely rare when walkers had three dimensions to explore. In one or two dimensions, the probability that walkers would meet after a certain time was dependent on the distribution of step sizes. Essentially, the more long jumps the walkers made, the more area they roamed, and the longer it took them to meet up. The authors confirmed their field theory calculations with simulations on a one-dimensional lattice. They believe further work in this direction could address optimization, such as what would be the best Lévy flight distribution for a predator stalking prey that moves with a distinct Lévy flight pattern. It might also be used in analyzing the distribution of long-range links for small world networks like Facebook.
“The idea of vicious walkers with Lévy flight statistics is new and appealing,” says Ralf Metzler of the Technical University of Munich. He thinks the work may benefit search algorithms for databases, but he also wonders how accurate discrete lattice simulations are in describing continuous Lévy flight distributions, especially for higher dimensions. David Sims from the Marine Biological Association of the UK, who has studied marine predator foraging, says that applying this theoretical framework to ecology may be possible in the future. “Considering the movements of both predators and prey in the context of Lévy flights is likely to be important for helping to understand how animals in the wild search for resources.”
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