A model mechanism proposed by the second author [On pattern formation mechanisms for lepidopteran wing patterns and mammalian coat markings. Philos. Trans. R. Soc. Lond., B 295, 473-496 (1981)] 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 \[ S_ t=D\Delta S-kS,\quad g_ t=k_ 1S+\alpha g(g-k_ 2)(g_ c-g), \] where D, k, \(k_ 1\), \(k_ 2\), \(g_ c>k_ 2\) and \(\alpha\) are positive constants. We determine analytically the size of the spatial domain where \(g\to g_ c\) as \(t\to \infty\) 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.