A flock of birds comprising thousands of individuals moves through the air like a single organism. Although a familiar sight to many people, it fills the alert observer with wonder. How can so many individuals, who are really only able to see their immediate neighbors within the flock, coordinate their motion so perfectly?

The phenomenon of collective motion in nature is not restricted to flocks of birds; different groups of creatures, such as animals, fish, insects, bacteria, and even human crowds, may show similar swarming patterns. The swarm inspired not only artists and writers, but became, in recent decades, the subject of a highly interdisciplinary research field. Biologists, engineers, mathematicians, computer scientists, and physicist search together for answers to fundamental questions. How does collective motion emerge in nature? What advantages does a swarm provide for the individual? Are there universal swarm properties?

Disastrous examples of the collective motion of insects are locust swarms. They might affect the life of millions of people over wide parts of the globe. The formation of these swarms is still not fully understood. Only recently have biologists' experiments shown how the collective motion of locusts and related insects can be driven by cannibalism. Hunger drives the insects to attack each other in order to survive - the individual seeks to survive at the expense of others. In such an ‘aggressive' swarm the motion with the group can be seen as a protective mechanism against being attacked, and, in the final event, against being cannibalized. This finding provides a new perspective on the onset of collective motion in nature. So far, swarm formation was considered as a cooperative behavior; a school of fish offers the individual fish protection against predators; swarming bacteria are able to explore unfavorable surfaces collectively.

These recent research results on insect swarms were the motivation for me and Lutz Schimansky-Geier, theoretical physicists from Humboldt-Universität zu Berlin to develop, together with the biologist Iain Couzin from Princeton, a mathematical model for collective motion with individual aggressive behavior. Their work can be seen as a link between the minimal models of collective motion studied by physicist and the more complex biological models which take into account different behavioral rules.

‘Escape' and ‘Pursuit'

In the model, an individual is described as a so-called ‘active Brownian particle'. Brownian motion refers originally to the random motion of small particles resulting from a large number of collisions with even smaller particles, for example the molecules of a liquid. The concept of active Brownian motion was introduced by physicists from Humboldt-Universität zu Berlin a decade ago in order to describe the random motion of animals. Each individual is considered as a Brownian particle with an internal energy depot.

The internal energy enables the particle to respond actively to external stimuli. In our mathematical model, a solitary individual moves completely randomly. But if it is approached by another individual it increases its speed in the opposite direction in order to escape. However, if there is another individual moving away, the focal individual increases its speed in the direction of the other individual in order to follow it. So each individual has only two simple behavioral rules, ‘escape' and ‘pursuit'. In computer simulations it was shown that at high particle density both interactions ‘escape' and ‘pursuit' lead to collective motion.

But the patterns of collective motion depend strongly on which of these motions dominates. At low density another important difference appears. For behavior dominated by ‘pursuit' the formation of swarms can be observed at any density, whereas for strong ‘escape' a breakdown of collective motion occurs below a critical density of particles.

Well known from Computer Games

The scientist involved think that the ‘escape-pursuit' model can be applied to different swarming phenomena. Whereas for aggressive insects ‘escape' seems to dominate, ‘pursuit' provides a simple mechanism for the onset of collective motion in non-aggressive animal groups, such as sheep or fish. The applications of swarm research go far beyond biology.

Swarms in nature provide a template for the design of simple communicating robots, which are able to perform complex tasks as a group without any external control, such as the exploration of dangerous or impassable environments. In computer science, so-called, swarm-algorithms are investigated for the effective solution of optimization problems. But most people probably come into contact with them through another application. Mathematical models - similar to the one here presented - are used to create realistic animal swarms or even human crowds in computer games or on the movie screen.