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This article is a *little* different from the fare you’re used to getting from me: it’s for *SIAM News*, which is the glossy magazine for members of the Society for Industrial and Applied Mathematics (SIAM). The audience for this magazine, in other words, is professional mathematicians and related researchers working in a wide variety of fields. While the article contains equations, I wrote it to be understandable even if you skip over the math.

## Leopard Spots, Frog Eggs, and the Spectrum of Nonlinear Diffusion Processes

For *SIAM News:*

Stripes, spots, or a mix of both appear on the skin of many animals — from tigers to beetles to whale sharks. These patterns are typically unique to individual creatures, and biologists often use them for identification. While distinct patterns may seem random, they obey certain rules that suggest a common underlying description. Striping and spotting occur in many unrelated species, implying that both evolutionary advantages and simple biochemical mechanisms drive such patterns.

As Björn Sandstede of Brown University noted during his invited address at the 2018 SIAM Annual Meeting, held in Portland, Ore., this July, similar patterns appear in certain chemical reactions and granular material under vibration. Nonlinear reactions and diffusion describe biological and non-biological patterns, producing stable concentrations in this space.

Alan Turing—best known for his work in computer science and cryptography—first made the mathematical connection between nonlinear diffusion processes and animal stripes in the 1950s. Many researchers have applied the resulting model to demonstrate how various species get their spots and describe nonlinear waves in chemical reactions.