'The Japan Society for Industrial and Applied Mathematics'
Doi
Abstract
A model mechanism proposed by Murray [Phil. Traps. Roy. Soc. London B, 295 (1981), pp. 473–496] 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 analyse one of the resulting partial differential equation systems, namely S_t = DΔS - kS, g_t = k_tS + αg (g-k_2) (g_c-g ), where D,k,k_1 ,k_2 ,g_c > k_2 and α are positive constants. We determine analytically the size of the spatial domain where g → g_c as t → ∞ 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
