Stability analysis of reaction-diffusion systems with constant coefficients on growing domains Anotida Madzvamuse* University of Sussex, Department of Mathematics, Mantell Building, Brighton, BN1 9RF, UK FAX: 44 (0)1273 678097 E-mail: a.madzvamuse@sussex.ac.uk *Corresponding author Abstract: This paper seeks to present a detailed linear stability theory for reaction-diffusion sys- tems (RDS) with constant coefficients on continuously deforming domains. By using the arbi- trary Lagrangian-Eulerian formulation (ALE), the model equations on a continuously deforming domain are transformed to a fixed domain at each time, resulting in a conservative system. First we prove that if the domain velocity is divergence free, then the linearised system of RDS reduces to one obtained for the RDS on a fixed domain. Secondly, we derive and show that the diffusion- driven instability conditions for an exponentially growing domain depend on the domain growth rate. More important, the parameter space is a direct shift of the Turing space obtained on a fixed domain in the absence of domain growth. Alternatively, by looking at the eigenvalues, we show that the shifting of the Turing space is equivalent to the standard Turing space of the RDS on a fixed domain but with eigenvalues shifted to the left of the complex plane by a constant factor given by the divergence of the domain velocity. Keywords: Diffusion-driven instability, convection-reaction-diffusion systems, Turing instabity, pattern formation, growing domains, divergence free, mesh movement, ALE formulation. Reference to this paper should be made as follows: Anotida Madzvamuse. (2008). ”Stability analysis of reaction-diffusion systems with constant coefficients on growing domains”, Int J. of Dynamical and Differential Equations. Biographical notes: Anotida Madzvamuse is a lecturer at The University of Sussex, Department of Mathematics, Brighton, England. He completed his D Phil at Oxford University in 2000, became a research fellow at Oxford University until 2003 then he moved to Auburn University, Alabama, USA to take up at tenure-track position until 2006. He joined Sussex University in Au- gust of 2006. His research focuses on the interface between numerical analysis and mathematical biology with particular applications to pattern formation, bifurcation theory and numerical solu- tions of reaction-diffusion systems on growing domains. 1