Ivars Peterson's MathTrek

October 15, 2001

Ladybugs are among the most familiar of beetles. More than 4,000 species are found throughout the world, ranging in size from 4 to 18 millimeters. Also known as lady beetles or ladybirds, these insects (coccinellids) have rounded bodies and bright red, orange, or yellow wing covers, which usually bear an array of contrasting black spots or other markings.

Different species generally exhibit distinctive, recognizable patterns. Spot patterns are most common, but some species have stripes or a combination of spots and stripes. In every case, the patterns are symmetric. The position, size, and color of spots and stripes are mirrored on the pair of wing covers.

Intrigued by the variety of patterns displayed by lady beetles found in Taiwan, physicist S.S. Liaw of National Chung-Hsing University and his collaborators developed a simple mathematical model that generates patterns similar to those found on ladybugs. They reported their findings in the October Physical Review E.

The physicists turned to a model first proposed in 1952 by mathematician Alan M. Turing (1912-1954). He suggested that biological forms mirror patterns in the concentrations of hypothetical chemicals called morphogens.

Turing postulated that, under appropriate conditions, the reaction of these chemicals and the subsequent diffusion of their reaction products combine to create distinctive patterns from an initially uniform distribution of morphogens. He encapsulated this mechanism in a set of reaction-diffusion differential equations.

By adjusting coefficients in the Turing model's equations, it's possible to vary the initial distribution of the hypothetical morphogens that mix together to create the colors and produce swirls, spots, or stripes.

Researchers had previously used the Turing model to simulate pattern formation in seashells, fish, zebras, leopards, giraffes, and other animals. However, the reaction-diffusion equations were usually solved for a flat rather than a rounded surface.

Liaw and his coworkers worked with a Turing model that involves two interacting morphogens--an activator and a substrate--and takes into account the curvature of the lady beetle's body by using a portion of a spherical surface to approximate the geometry of the wing covers.

"The basic patterns on the curved surface are not different from those on a flat surface," the researchers reported. "However, the exact positions and shapes of the spots and stripes are strongly dependent on the curvature of the surface and the shape of the boundary."

By adjusting parameters in their model, the physicists could reproduce the patterns displayed by different species of lady beetles native to Taiwan.

"Several common lady beetle patterns, such as spots, stripes, and loops, can be obtained by choosing appropriate diffusion coefficients and simple initial distributions of the morphogens," they concluded. "Our calculation offers one more successful example supporting the reaction-diffusion dynamic process as a quite general mechanism in generating biological patterns."

The researchers are now investigating in greater detail the effects of surface geometry on the patterns produced in reaction-diffusion systems.

References:

Goodwin, B. 1994. How the Leopard Changed Its Spots: The Evolution of Complexity. New York: Simon & Schuster.

Liaw, S.S., C.C. Yang, R.T. Liu, and J.T. Hong. 2001. Turing model for the patterns of lady beetles. Physical Review E 64(October):041909.

Peterson, I. 1992. Generating chemical spots and stripes. Science News 141(May 9):311.

Varea, C., J.L. Aragón, and R.A. Barrio. 1999. Turing patterns on a sphere. Physical Review E 60(October):4588-4592.