Reduced model of a reaction-diffusion system for the collective motion of camphor boats

The unidirectional motion of a camphor boat along an annular water channel is observable. When camphor boats are placed in a water channel, both homogeneous and inhomogeneous states occur as collective motions, depending on the number of boats. The inhomogeneous state is a type of congestion, that is, the velocities of the boats change with temporal oscillation, and the shock wave appears along the line of travel of the boats. The unidirectional motion of a single camphor boat and the homogeneous state can be represented by traveling wave solutions in a mathematical model. Because the experimental results described here are thought of as a type of bifurcation phenomenon, the destabilization of traveling wave solutions may indicate the emergence of congestion. We previously attempted to study a linearized eigenvalue problem associated with a traveling wave solution. However, the problem is too difficult to analyze rigorously, even for just two camphor boats. Therefore we developed a center manifold theory and derived a reduced model in our previous work. In the present paper, we study the reduced model and show that the original model and our reduced model qualitatively exhibit the same properties by applying numerical techniques. Moreover, we demonstrate that the numerical results obtained in our models for camphor boats are quite similar to those in a car-following model, the OV model, but there are some different features between our reduced model and a typical OV model.