Date of Award

Spring 2015

Document Type



Mathematics, Engineering & Computer Science


Butterfly wings consist of many patterns and every species has its own combination of patterns to make their wings unique. However, every possible pattern is derived from a common blueprint called a nymphalid groundplan. The cells of butterfly wings each emit a color either through the cell’s structure or through pigments that the cell produces. These cells must therefore work together in order to arrange themselves into the many patterns we see in butterfly wings. As the wings are developing, signals are distributed throughout the cells of the wings. These signals induce the cells of the wings to emit their designated color. In 1952, the mathematician Alan Turing devised a method to model the formation of patterns in nature using reaction-diffusion mechanisms. Turing named the cell signals morphogens and believed that these morphogens diffused through the cells of the organisms, inducing pigmentation and forming patterns. Using Turing’s models along with more recently developed numerical methods for solving partial differential equations, the formation of the many patterns found on butterfly wings can be simulated. In this project, a small region of a butterfly wing was modeled using a system of two morphogens over a two-dimensional trapezoidal surface. This region included an eyespot from which it was assumed one of the morphogens was emitted. After solving this system using the Finite Element Method, a banded pattern is formed over the region. Therefore, repeating this process for the different regions of the wing can reproduce a pattern commonly found on butterflies.