Turing Model for Ladybug Beetle Patterns

Zebras, leopards, and giraffes are just a few creatures exhibiting intricate patterns that can be duplicated with models pioneered by the late mathematical genius Alan Turing (Update 80). The models are based on diffusion equations, which are often used to describe the spontaneous mixing of materials over time. As a rule, mathematicians and physicists have studied Turing models of biological patterns as though they were formed on flat surfaces. Of course, few animals tend to be flat, unless they've lingered too long on a highway.

Although Turing models can mimic tiger stripes and cheetah spots fairly well despite this simplifying assumption, a group of researchers from the National Chung-Hsing University in Taiwan decided to consider a slightly more complex shape. When S.S. Liaw (liaw@phys.nchu.edu.tw, 011-8864-2284-0427) and colleagues studied Turing models on a portion of a spherical surface, patterns reminiscent of those on lady bug beetles emerged. There are over 4500 species of ladybugs, most bearing unique, recognizable designs in contrasting colors such as black and red.

By adjusting coefficients in the model's equations and varying the initial distributions of hypothetical compounds that mix to create the colors, the researchers could reproduce the stripes, swirls, and spots that decorate many of these predatory insects. The new model shows that an animal's specific geometry is important in determining it's adornment, and adds weight to Turing's proposal that diffusion is potentially a mechanism that helps generate an endless variety of patterns in nature. (S. S. Liaw; C. W. Yang; R. T. Liu; J. T. Hong, Physical Review E, October 2001.)