Chalmers Conferences, 9th European Conference on Mathematical and Theoretical Biology

Phase diagram of a fish school model: marches versus circles determined by the relative strength of alignment and cohesion
Yuki Kubo, Yoh Iwasa

Last modified: 2014-03-28


Many species of fish, bird, and insect form groups of individuals that move together, called schools, flocks, or swarms, of characteristic shape and speed. Here, we study a model of animal group formation on a plane, in which each individual changes its swimming angle in response to its neighbors within a radius of interaction. Outside of a short range of separation (or repulsion), each individual changes swimming direction to achieve a similar swimming direction as its neighbors (alignment) and to swim toward them (cohesion). Depending on the relative strength of alignment and cohesion, the model produces groups of two distinct patterns: marches and circles. We introduce several statistics for the swimming patterns of individuals. As the strength of alignment relative to cohesion increases, the shapes of groups change in the following order: (1) circles, (2) mixture of circles and marches, (3) short marches, (4) long marches, (5) wide marches. We derived a formula for the spatial size of circles, which explains that the radius of circles does not change with the number of individuals, but it increases with swimming speed and decreases with the sensitivity of swimming direction to neighbors. We also discuss how the length and width of marches depend on the relative strength of alignment and cohesion.