Spatial resonances and superposition patterns in a reaction-diffusion model with interacting Turing modes

       On the chemical basis for morphogenesis, Turing structures, in some sense, shed light on How does the leopard get its spots? We focus on interacting Turing modes.
The chlorite-iodide-malonic acid (CIMA) reaction exhibits an interesting Turing pattern, black-eye, where the black center (high concentration of iodide) is surrounded by a white ring(low concentration of iodide), and these eye-like spots arranged into a hexagonal lattice. [G. H. Gunaratne, Q. Ouyang, and H. L. Swinney, Phys. Rev. E 50, 2802 (1994)].

A subsequent study[M. G. Gomes, Phys. Rev. E 60, 3741 (1999)] suggested that black-eye patterns are not two-dimensional,but rather projections of a three-dimensional  body-centered cubic lattice. Recent experimental results suggest that the three-dimensional interpretation is not appropriate[C. Zhou, H. Guo, and Q. Ouyang, Phys. Rev. E 65, 036118 (2002)], but to date no model has been developed that reproduces the black-eye patterns in two dimensions.

We propose a model that generates not only black-eye patterns but also a variety of other spatial resonant patterns: “black eye”, “white-eye”, and “Twinkling-eye” resulting from interactions between different Turing modes, or between a single mode and its harmonics. Also, we obtained superposition patterns with different combination of stripe- and spot- Turing patterns.

This work appears in Phys. Rev. Lett. 88, 208303 ( 2002). Our results provide a possible mechanism not only for the recent experimental observation of ¬°¬°lack eye?patterns, but may also provide insights into the chemical basis of morphogenesis.

Stationary internal spatial resonance patterns: black eyes, and white eyes

Twinkling eyes (sub-harmonic resonance)

Supperposition patterns

The original Brusselator model
The coupled Brusselator model in PDF

"Black-eye" in art