Swarms of insects or schools of fish can suddenly switch direction, seemingly at random. A research team now believes they understand why. In the July Physical Review E they describe experiments with locusts and their mathematical model of the swarm. The model suggests that tiny alignment errors between neighboring individual locusts within the swarm, which normally cancel each other, can accumulate over long times and eventually cause the entire swarm to suddenly move in a new direction.
From ants and bees to fish and starlings, many different species move together in swarms, schools, or flocks. Understanding how the collective motion of these groups emerges from the individual actions of its constituents has been of interest to mathematical biologists for a long time. Over the last several years, researchers have been borrowing techniques from statistical mechanics–the study of systems of many objects, such as molecules in a gas or atoms in a solid–and applying them to swarms.
In 2006, Jerome Buhl from the University of Sydney and his colleagues carried out a series of experiments  with desert locust nymphs, wingless juveniles that walk in “marching bands” that can extend many kilometers. The team set various numbers of locusts into ring-shaped arenas and watched as the bands would move in one direction around the ring and then occasionally, after some time, switch to marching in the opposite direction. These rapid directional switches are a familiar facet of swarm and flock motion.
In 2009, Buhl, along with some new collaborators–including Carlos Escudero of the Autonomous University of Madrid–published a follow-up paper that deepened the mathematical modeling of these experimental results . They used the formalism of the Fokker-Planck equation, a differential equation describing the evolution of the positions or velocities of a large collection of particles. Such an equation was first used to describe the ordinary jiggling and drifting of molecules known as Brownian motion.
In the simplest case, the Fokker-Planck equation reduces to standard diffusion: if a collection of randomly moving particles starts out densely packed in one location, it will tend to spread out over time. The equation contains a term for this “spreading” (diffusion) of the collection and another term for the “drifting” of the average position of the group. But Buhl, Escudero, and their colleagues didn’t model all of the individual locust velocities directly; instead they modeled the spreading and drifting of the probabilities for the average velocity of the entire swarm. For example, if the probability function were a Gaussian (“bell”) curve, it would be centered over the most probable value for the average velocity, and this function would drift and spread over time.
In their 2009 paper, the team studied how the swarm maintains its uniform motion. Now they have taken the Fokker-Planck analysis further: instead of taking the drift and diffusion terms directly from the data, the team started with a data-inspired analytical formula for each term. Next, by solving the Fokker-Planck equation exactly to obtain the average velocity of the swarm as a function of time, they were able to calculate how long it takes the marching horde to reverse its direction.
Their calculations show that the switching times are governed by the Poisson statistical distribution. This distribution applies to a system of many independent individuals that randomly but rarely do something measurable, such as a large set of radioactive nuclei that can each decay at any time. The team says that in this case the rare events are tiny alignment errors between neighboring locusts that can occasionally add up to a direction switch of the whole swarm. Furthermore, each switch is independent of all the others, according to the analysis, which makes them practically impossible to predict. The properties of the calculated Poisson distribution fit well with the actual times measured in the experiment.
Tamas Vicsek, a professor of biological physics at Eötvös Loránd University in Hungary, thinks that the team’s approach to the Fokker-Planck equation is “elegant and new” and that “one is tempted to assume that the results coming from it should be applicable to other systems.” However, he cautions that so far it only applies to motion in one dimension and needs to be checked for realistic three-dimensional swarms.
- J. Buhl, D. J. T. Sumpter, I. D. Couzin, J. J. Hale, E. Despland, E. R. Miller, and S. J. Simpson, “From Disorder to Order in Marching Locusts,” Science 312, 1402 (2006) .
- C. A. Yates, R. Erban, C. Escudero, I. D. Couzin, J. Buhl, I. G. Kevrekidis, P. K. Maini, and D. J. T. Sumpter, “Inherent Noise Can Facilitate Coherence in Collective Swarm Motion,” Proc. Natl. Acad. Sci. USA 106, 5464 (2009).