How to Get a Model in Pedestrian Dynamics to Produce Stop and Go Waves

Stop and go waves in granular flow can often be described mathematically by a dynamical system with a Hopf bifurcation. We show that a certain class of microscopic, ordinary differential equation-based models in crowd dynamics fulfil certain conditions of Hopf bifurcations. The class is based on the Gradient Navigation Model. An interesting phenomenon arises: the number of pedestrians in the system must be greater than nine for a bifurcation—and hence for stop and go waves to be possible at all, independent of the density. Below this number, no parameter setting will cause the system to exhibit stable stop and go behaviour. The result is also interesting for car traffic, where similar model…