Eur. Phys. J. B (2014) 87: 77
https://doi.org/10.1140/epjb/e2014-50017-5

Regular Article

Reaction-diffusion scheme for the clock and wavefront mechanism of pattern formation

¹ Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland
² Sorbonne Universités, UPMC Univ. Paris 06, Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), 4 place Jussieu, case courrier 121, 75252 Paris Cedex 05, France
³ CNRS UMR 7600, LPTMC, 75252 Paris, France
⁴ Physics Laboratory, Warsaw University of Life Sciences - SGGW, Nowoursynowska 159, 02-776 Warsaw, Poland

^a e-mail: pdziekan@ichf.edu.pl

Received: 8 January 2014
Received in final form: 18 February 2014
Published online: 1 April 2014

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.