How do you model the collective behaviour of a large group of individuals? If those individuals are people, what sort of rules can we come up with to mimic the way individuals with free will can behave? This is far from an academic exercise. Many organisations rely for their operation on being able to control crowds safely and effectively. These include the police, the designers of sports stadia, London Underground, and even the Olympic Committee.

Agent based models (ABMs) provide one way of approaching this problem. In an ABM we have a number of agents (these could represent people or animals, for example) which interact with each other, and which often move under the effects of this interaction. Individual agents are typically assumed to be rational and to be acting in what they perceive as their own interest — such as reproduction, economic benefit, or social status. ABM agents may also learn from their experiences.

Most agent-based models are composed of:

• A number of agents
• A set of decision-making rules for each agent
• A set of learning rules for each agent
• A space in which the agents can move/operate and an environment in which they can interact.

Typically it's assumed that the agents' behaviour over time obeys a set of nonlinear ordinary differential equations: solving these equations will tell you where each agent will be at any given moment in time.

ABMs can produce surprisingly realistic results. The simulation below of a flock of birds is an example. It comes from a model created by Charlotte K. Hemelrijk and Hanno Hildenbrandt in their paper Some causes of the variable shape of flocks of birds. (The movie is reproduced here under a creative commons licence.)

To find out more about ABMs read The dynamics of crowds.