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

Abstract

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 fulfill 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 a stable stop and go behavior. The result is also interesting for car traffic, where similar models exist. Numerical experiments of several parameter settings are used to illustrate the mathematical results.