ON ISOCONCENTRATION SURFACES OF THREE-DIMENSIONAL TURING PATTERNS
Abstract
We consider three-dimensional Turing patterns and their isoconcentration surfaces corresponding to the equilibrium concentration of the reaction kinetics. We call these surfaces equilibrium concentration surfaces (EC surfaces). They are the interfaces between the regions of "high" and "low" concentrations in Turing patterns. We give alternate characterizations of EC surfaces by means of two variational principles, one of them being that they are optimal for diffusive transport. Several examples of EC surfaces are considered. Remarkably, they are often very well approximated by certain minimal surfaces. We give a dynamical explanation for the emergence of Scherk's surface in certain cases, a structure that has been observed numerically previously in [De Wit et al., 1997].
This work was supported by NSF Grant No. IBN-0083653. We acknowledge support from the Center for Applied Mathematics and the Interdisciplinary Center for the Study of Biocomplexity at the University of Notre Dame and the Biocomplexity Institute at Indiana University.
