Some of the most mesmerizing examples of collective behavior are seen overhead every day. V-shaped formations of migrating geese, starlings dancing in the evening sky, and hungry seagulls swarming over a fish market, are just some of the wide variety of shapes formed by bird flocks. Fish schools also come in many different shapes and sizes: stationary swarms; predator avoiding vacuoles and flash expansions; hourglasses and vortices; highly aligned cruising parabolas, herds, and balls. These dynamic spatial patterns often provide the examples that first come into our heads when we think of animal groups.
While the preceding three chapters described the dynamics of animal groups, they did not explicitly describe the spatial patterns generated by these groups. For example, the decision-making of insects and fish was studied in situations where individuals have only two or a small number of alternative sites to choose between. In models of these phenomena, space is represented as the number of individuals who have taken each of these alternatives. This approach often simplifies our understanding of the underlying dynamics of these groups, but in doing so it can fail to capture the spatial structure that characterizes them. As a simple consequence of the fact that these groups move, we need to give careful consideration to how they change position in space as well as time.
The main tool I will use in describing the dynamics of flocking are selfpropelled particle (SPP) models (Czirok & Vicsek 2000; Okubo 1986; Vicsek et al. 1995). In SPP models “particles” move in a one-, two-, or three-dimensional space. Each particle has a local interaction zone within which it responds to other particles. The exact form of this interaction varies between models but typically, individuals are repulsed by, attracted to, and/or align with other individuals within one or more different zones. These models allow us to investigate the conditions under which collective patterns are produced by spatially local interactions.