Abstract
Pattern dynamics triggered by fixing a boundary is investigated. By considering a reaction-diffusion equation that has a unique spatially uniform and limit cycle attractor under a periodic or Neumann boundary condition, and then by choosing a fixed boundary condition, we found three novel phases depending on the ratio of diffusion constants of activator to inhibitor: transformation of temporally periodic oscillation into a spatially periodic fixed pattern, travelling wave emitted from the boundary, and aperiodic spatiotemporal dynamics. The transformation into a fixed, periodic pattern is analyzed by crossing of local nullclines at each spatial point, shifted by diffusion terms, as is analyzed by using recursive equations, to obtain the spatial pattern as an attractor. The generality of the boundary-induced pattern formation as well as its relevance to biological morphogenesis is discussed.
Highlights
Pattern formation in nonlinear-nonequilibrium systems has gathered much attention over decades both theoretically and experimentally
In spite of extensive studies on the initial-condition dependence, boundary-condition dependence is not fully explored, whereas most pattern dynamics in nature generally progress in a finite system, pattern formation following the boundary condition is of crucial importance
The transformation of oscillation into a fixed spatial pattern is analyzed in depth, which should be essential to biological morphogenesis, such as somitogenesis
Summary
Pattern formation in nonlinear-nonequilibrium systems has gathered much attention over decades both theoretically and experimentally. We recently found an alternative mechanism for the transformation from temporal oscillation to fixed spatial pattern, induced by diffusive interaction and boundary condition, even without any input of external gradient.. We recently found an alternative mechanism for the transformation from temporal oscillation to fixed spatial pattern, induced by diffusive interaction and boundary condition, even without any input of external gradient.11,12 This finding can provide possible explanation for recent experimental results on somitogenesis, where cell-cell interaction is suggested to be relevant.. We confirm that the boundary-induced transformation of temporal period spatial pattern works under the existence of diffusive activator.
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