Abstract
We present a model of pattern formation in reaction-diffusion systems that is based on coupling between a propagating wave front and temporal oscillations. To study effects of internal fluctuations on the spatial structure development we use a chemical master equation for our reaction-diffusion model. First, a model with local, uncoupled oscillators is studied. Based on it we show that synchronization of oscillations in neighboring cells is necessary for the formation of regular patterns. We introduce synchronization through diffusion, but then, to get a stable pattern, it is necessary to add an additional species that represents the local state of the system. Numerical simulations of the master equation show that this extended model is resistant to fluctuations.
Highlights
Nonlinear reaction-diffusion systems in conditions out of equilibrium are known to produce stable spatial structures
Spatial patterns in the Belousov-Zhabotinsky reaction were produced in an experiment based on the clock and wavefront mechanism [8]
We examine robustness of the spatial structure to internal fluctuations by studying them at a mesoscopic level
Summary
Nonlinear reaction-diffusion systems in conditions out of equilibrium are known to produce stable spatial structures. The principal example of this behavior are Turing patterns, which are formed in activator-inhibitor systems if mobilities of species are different [1,2,3]. This mechanism can explain formation of many shapes observed in nature, most notably animal coat patterns [4,5,6]. In this paper we present a different type of generic reaction-diffusion scheme that can produce stable, periodic structures. It is inspired by the ideas of Cooke and Zeeman, who proposed a clock and wavefront model as a mechanism of axial segmentation during embryonic development [7].
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