Stability of n-dimensional patterns in a generalized Turing system:: implications for biological pattern formation

The stability of Turing patterns in an n-dimensional cube (0, pi)(n) is studied, where n greater than or equal to 2. It is shown by using a generalization of a classical result of Ermentrout concerning spots and stripes in two dimensions that under appropriate assumptions only sheet-like or nodule-like structures can be stable in an n-dimensional cube. Other patterns can also be stable in regions comprising products of lower-dimensional cubes and intervals of appropriate length. Stability results are applied to a new model of skeletal pattern formation in the vertebrate limb.