## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 943-955 |

Number of pages | 13 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 45 |

Issue number | 6 |

DOIs | |

State | Published - 1985 |

## ASJC Scopus subject areas

- Applied Mathematics