Progress of Theoretical Physics Supplement No. 150, 2003 367 The Effect of Noise on Turing Patterns T. Lepp¨ anen, 1 M. Karttunen, 1 R. A. Barrio 1,2 and K. Kaski 1 1 Laboratory of Computational Engineering, Helsinki University of Technology, P.O. Box 9203, FIN–02015 HUT, Finland 2 Instituto de Fisica, UNAM, Apartado Postal 20-364, 01000 M´ exico, D.F., M´ exico The effect of noise on pattern formation in Turing systems is studied. It is shown how robustness of 2D patterns and 3D structures against noise depends on the characteristics of the morphology. The effect of noise is of particular interest since Turing systems are often used for explaining biological patterns or structures, which have to be stable against noise. §1. Introduction In 1952 Alan Turing showed that a system of coupled reaction-diffusion equations can exhibit a finite wavelength instability resulting in spatial patterns. 1) He proposed this mechanism to be biologically relevant, for example in relation to an embryo developing from the nearly spherically symmetrical blastula stage. Turing proved that differences in the diffusion coefficients of the reactants can lead to a symmetry break through a mechanism called diffusion-driven instability. Turing systems have been used in mathematical biology as models for pattern formation in biological systems, 2) e.g. fish, 3), 4) butterflies 5) and lady beetles 6) have been studied. The first experimental evidence of Turing patterns dates only twelve years back. 7) Numerically, however, the effect of inhomogeneous diffusion coeffi- cients, 8) growing domains 9) and curvature of domains 10) have been studied. Recently, we have focused on the formation of Turing structures in three dimensions. 11) To the authors’ knowledge, the effect of random noise on Turing patterns has not been studied before. In contrast, Turing structures may still appear under spatially correlated external forcing provided that the correlation length of the forcing is not too small. 12) This implies that Turing structures possess an error-correcting property. Furthermore, under spatially periodic forcing, defects can be removed and the system can be driven into a more symmetric state. 13) In this paper we study the effect of Gaussian noise on a Turing system. We present results for both 2D and 3D systems and study the dependence of stability on the morphological characteristics and dimensionality of the system. §2. Model The general form for a Turing system describing the evolution of the concentra- tions of two chemicals or morphogens 1) is given by the reaction-diffusion equations U t = D U ∇ 2 U + f (U, V ), V t = D V ∇ 2 V + g(U, V ), (2 . 1)