Pattern Formation Induced by Fixed Boundary Condition (Vol. 48 No. 2)

Time development of the reaction diffusion dynamics. Fixed boundary condition at x=0 transforms uniform temporal oscillation to stationary periodic pattern

Pattern formation in nonequilibrium systems has been extensively investigated in physical and chemical systems as well as for biological morphogenesis, since the seminal study by Alan Turing: How perturbations to uniform, stationary states are amplified to form a spatially periodic pattern is thoroughly understood with extensive experimental demonstrations. In contrast to this spontaneous pattern formation, however, little is understood how given boundary condition leads to global pattern formation. Here, we demonstrate that the fixed boundary can transform a temporally-periodic, spatially-uniform state to a spatially-periodic, stationary pattern– a novel class of pattern formation mechanism. This pattern formation is not understood by the Fourier-mode linear stability analysis – the standard tool for Turing instability. Rather, by introducing a one-dimensional ‘spatial’ map, the emergent pattern is reproduced well as its periodic attractor, by replacing the time with space. Accordingly, linear dispersion relationship between the period and wavelength is obtained. This provides a general tool to analyze the pattern formation in reaction-diffusion systems, while the boundary-induced pattern formation mechanism will explain several biological morphogenesis, including recent experimental observations.

T. Kohsokabe and K. Kaneko, Boundary-induced pattern formation from temporal oscillation: Spatial map analysis, EPL 116, 48005 (2016).