Turing pattern with proportion preservation

Although Turing pattern is one of the most universal mechanisms for pattern formation, in its standard model the number of stripes changes with the system size, since the wavelength of the pattern is invariant. It fails to preserve the proportionality of the pattern, i.e. the ratio of the wavelength to the size, that is often required in biological morphogenesis. To get over this problem, we show that the Turing pattern can preserve proportionality by introducing a catalytic chemical whose concentration depends on the system size. Several plausible mechanisms for such size dependence of the concentration are discussed. Following this general discussion, two models are studied in which arising Turing patterns indeed preserve the proportionality. Relevance of the present mechanism to biological morphogenesis is discussed from the viewpoint of its generality, robustness, and evolutionary accessibility.