The mathematics of fish schools and flocks of humans

What drives groups of individual animals to act in a coherent manner? Everyone has seen the oddly coordinated behavior exhibited by flocks of birds or schools of fish as they turn, sweep, and rotate seemingly as one. But how does a group of individuals make decisions about how to move and where to go at once? Do they follow some prescribed and describable mathematical behavior? A symposium at this year’s AAAS conference attempted to answer this question.

Professor Ian Couzin from Princeton University opened the symposium by describing his work on modeling the underlying behavior of large groups of individuals. In his work, he describes the equation of motion for any individual entity as governed by three factors: a short-range repulsive behavior, an intermediate range desire to align with neighbors, and a long-range attraction to the group as a whole.

Simulated swarms of creatures that follow these simple rules are able to reproduce the complex motions seen in fish in his laboratory’s aquarium. If any one of his three rules is neglected, then the medium-range coherent motion disappears. The addition of an “avoid the predator” rule turned out to be very successful in mimicking the behavior of his fish when they were attacked by a robotic fish predator that was designed for his research.

Taking the work further, to understand what it takes to lead and to follow, he looked at a slight modification of these rules in which each individual’s motion was defined by two terms: the conventional rules above, plus a linearly weighted “leader” factor that would cause the individual to move towards a goal, or in some specified direction. It turns out that only a few individuals in a group need to know where they are going in order to lead the group, even if they don’t do anything to communicate their leadership role other than move.

Also, the larger a group gets, the smaller the percentage of “knowledgeable” or “leader” individuals that are needed. The limit seems to be about five percent; the remaining 95 percent simply followed the herd. This has interesting implications for evolutionary roles and needs as it sheds light on what a group needs to survive.

Swarming high school students

Bridging the gap between swarming creatures and humans, he reported on an experiment of asking undergrads to evacuate a gym with many exits without talking or communicating with one another. Turns out, much like in his simple models, the group would follow a handful of individuals who were told a specific exit to use ahead of time. Interesting implications/explanations for high school abound here.

The following talk also looked at the spontaneous organization of humans in crowds; Pierre Degond of Paul Sabatier University was motivated by an understanding of crowd safety, and how to design comfortable and efficient areas for crowds to gather in or move through. At high densities—greater than seven people per square meter (gah!)—crowds of humans behave much like incompressible fluids, their motion described by the Navier-Stokes equation. However, at lower, more common densities, there is no single way to model a crowd’s behavior.

For the Full article click the link below

The mathematics of fish schools and flocks of humans