A discrete boltzmann-type model of swarming

A new model for interacting “agents” (organisms, cells, particles etc.) is proposed. We consider the one-dimensional case in which agents are characterized by their position and orientation (+/-) with “majority-based” local (swarming) interaction controlled by a sensitivity parameter (⋎). The model possesses equilibrium solutions corresponding to the diffusive (isotropic) and the aligned (swarming) state. In the space-independent case, for ⋎ > 1 alignment asymptotically occurs while for 0 < ⋎ < 1 alignment is asymptotically destroyed. This behaviour can be interpreted as a phase transition. In the space-dependent case, we provide an existence theory and prove the existence of a Lyapunov functional.