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We investigate a simple generic model of a reaction–diffusion system consisting of an activator and an inhibitor molecule in the presence of a linear morphogen gradient. We assume that this morphogen gradient is established independently of the reaction–diffusion system and acts by increasing the production of the activator proportional to the morphogen concentration. The model is motivated by several existing models in developmental biology in which a Turing patterning mechanism is proposed and various chemical gradients are known to be important for development. Mathematically, this leads to reaction–diffusion equations with explicit spatial dependence. We investigate how the Turing pattern is affected, if it exists. We also show that in the parameter range where a Turing pattern is not possible, the system may nevertheless produce 'Turing-like' patterns.

The stability of Turing patterns in an n-dimensional cube (0 ,π ) n is studied, where n 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.