The flocking of animals is often modelled as a dynamical system, in which individuals are represented as particles whose interactions are determined by the current state of the system. Many animals, however, including humans, have predictive capabilities, and presumably base their behavioural decisions—at least partially—upon an anticipated state of their environment. We explore a minimal version of this idea in the context of particles that interact according to a pairwise potential. Anticipation enters the picture by calculating the interparticle forces from linear extrapolation of the positions some time τ into the future. Our analysis shows that for intermediate values of τ the particles rapidly form milling structures, induced by velocity alignment that emerges from the prediction. We also show that for τ>0, any dynamical system governed by an even potential becomes dissipative. These results suggest that anticipation could play an important role in collective behaviour, since it induces pattern formation and stabilises the dynamics of the system.
The impact of anticipation in dynamical systems
P. Gerlee, K. Tunstrøm, T. Lundh, B. Wennberg