Mean Field Trajectories in a Spin Model for Decision Making on the Move

Collective decision making is a key feature during natural motion of animal groups. A model of collective decision making regarding direction of travel was introduced as an extension of the Ising model where the spin-spin interaction is interpreted as a social force. We study trajectories in the mean field approximation of this model for two and three targets. For two targets there are two types of possible directions of movement: A compromise movement between the two targets when the angle between them is small or a movement towards one of the targets when the angle is large. At a critical angle there is a phase transition that manifests itself in a bifurcation point.
For three targets we find that the decision is broken down to a series of binary decisions and moreover, for a wide class of geometries we find an infinite number of bifurcations. In particular, as a result of the geometry of this series of bifurcations, we find that the mechanism of reaching the central target (out of three) is very different than the one which is used to reach the external ones. In addition, the susceptibility along the trajectories diverges at bifurcation points as expected at phase transitions. We support the mean field results by comparison to simulations.