A model mechanism proposed by J. D. Murray for generating wing patterns and eyespots on butterflies and moths is based on a morphogen (S) activated biological switch for a gene product (g). We analyze one of the resulting partial differential equation systems, namely S//t equals D DELTA S minus kS, g//t equals k//1S plus alpha g(g minus k//2)(g//c minus g), where D, k, k//1, k//2, g//c greater than k//2 and alpha are positive constants. We determine analytically the size of the spatial domain where g approaches g//c as t approaches infinity after an influx of S at the origin. This gives the size of the eyespot in terms of the mechanism parameters. The analytical problem is a nontrivial singular perturbation expansion which we discuss in detail.
ASJC Scopus subject areas
- Applied Mathematics