  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 morphogeneis. 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.
