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
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)
The original Brusselator model
The coupled Brusselator model in PDF
"Black-eye" in art