Abstract:Pattern formation occurs on curved surfaces abundantly especially in biological systems. Previous studies have revealed that the surface curvature affects the pattern dynamics and plays biological functions, however, comprehensive understanding is still elusive. Here by employing reaction-diffusion systems showing Turing pattern, we show for the first time that static pattern on a flat surface can be propagating wave on a curved surface. By numerical and theoretical analyses, it is shown that the pattern propagation is conditioned by the symmetry of surface and pattern. We also show the results of weakly nonlinear analysis applied to the problem, which suggests rich dynamics can arise on curved surface.