ANALYSIS OF PATTERN FORMATION IN REACTION DIFFUSION MODELS WITH SPATIALLY INHOMOGENOUS DIFFUSION COEFFICIENTS

Abstract-We consider a reaction diffusion system in one spatial dimension in which the diffusion coefficients are spatially varying. We present a non-standard linear analysis for a certain class of spatially varying diffusion coefficients and show that it accurately predicts the behaviour of the full nonlinear system near bifurcation. We show that the steady state solutions exhibit qualitatively different behaviour to that observed in the usual case with constant diffusion coefficients. Specifi-cally, the modified system can generate patterns with spatially varying amplitude and wavelength. Application to chondrogenesis in the limb is discussed. 1. MODEL EQUATIONS We consider a general reaction diffusion mechanism for pattern formation with spatially hetero-geneous diffusion coefficients. Such a system could arise from a two step patterning process in which the spatial pattern in a control chemical regulates morphogen diffusivity in an overlying reaction diffusion system. Although several authors have considered reaction diffusion systems with spatially varying parameters [l-5], we are not aware of an analytic treatment for the case of spatially varying diffusion coefficients. The model, in one space dimension, is described by three coupled partial differential equations; w = rf(u, v) + (a&(C)~z)z, (l.la) Vt = Y!9( % u)