Phase diagram of a multiple forces model for animal group formation: marches versus circles determined by the relative strength of alignment and cohesion

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 that traces movements of many individuals, in which each individual moves at a constant speed, and changes its movement angle in response to its neighbors within a radius of interaction. Outside of a short range of separation (or repulsion), each individual changes moving direction to achieve a similar moving direction as its neighbors (alignment) and to move toward them (cohesion). Between each pair of individuals within an interaction range, both alignment and cohesion are at work simultaneously (multiple forces model). This is different from many other models for animal group formation in which only one of the two forces is at work (single force model), different forces operating in different zones of between-individual distance. Depending on the relative strength of alignment and cohesion, our model produces groups of two distinct patterns: marches and circles. We showed the phase diagram of group patterns depending on the relative strength of alignment and cohesion. As the strength of alignment relative to cohesion increases, the shapes of groups change gradually 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 moving speed and decreases with the sensitivity of moving direction to neighbors.