PATTERN-FORMATION IN FINITE-SIZE NONEQUILIBRIUM SYSTEMS AND MODELS OFMORPHOGENESIS

Two canonical pattern forming systems, the Rayleigh-Benard convection and the Turing mechanism for biological pattern formation, are compare d. The similarity and fundamental differences in the mathematical stru cture of the two systems are addressed, with special emphasis on how t he linear onset of patterns is affected by the finite size and the bou ndary conditions. Our analysis is facilitated by continuously varying the boundary condition, from one that admits simple algebraic solution of the problem but is unrealistic to another which is physically real izable. Our investigation shows that the size dependence of the convec tion problem can be considered generic, in the sense that for the majo rity of boundary conditions the same trend is to be observed, while fo r the corresponding Turing mechanism one will rely crucially on the as sumed boundary conditions to ensure that a particular sequence of patt erns be picked up as the system grows in size. This suggests that, alt hough different systems might exhibit similar pattern forming features , it is still possible to distinguish them by characteristics which ar e specific to the individual models.