  We analyze an inhomogeneous system of coupled reaction-diffusion equations
representing the dynamics of gene expression during differentiation of nerve
cells. The outcome of this developmental phase is the formation of distinct
functional areas separated by sharp and smooth boundaries. It proceeds through
the competition between the expression of two genes whose expression is driven
by monotonic gradients of chemicals, and the products of gene expression
undergo local diffusion and drive gene expression in neighboring cells. The
problem therefore falls in a more general setting of species in competition
within a non-homogeneous medium. We show that in the limit of arbitrarily small
diffusion, there exists a unique monotonic stationary solution, which splits
the neural tissue into two winner-take-all parts at a precise boundary point:
on both sides of the boundary, different neuronal types are present. In order
to further characterize the location of this boundary, we use a blow-up of the
system and define a traveling wave problem parametrized by the position within
the monotonic gradient: the precise boundary location is given by the unique
point in space at which the speed of the wave vanishes.
