  Spatial pattern formation is a key feature of many natural systems in
physics, chemistry and biology. The essential theoretical issue in
understanding pattern formation is to explain how a spatially homogeneous
initial state can undergo spontaneous symmetry breaking leading to a stable
spatial pattern. This problem is most commonly studied using partial
differential equations to model a reaction-diffusion system of the type
introduced by Turing. We report here on a much simpler and more robust model of
spatial pattern formation, which is formulated as a novel type of coupled map
lattice. In our model, the local site dynamics are coupled through a
competitive, rather than diffusive, interaction. Depending only on the strength
of the interaction, this competitive coupling results in spontaneous symmetry
breaking of a homogeneous initial configuration and the formation of stable
spatial patterns. This mechanism is very robust and produces stable pattern
formation for a wide variety of spatial geometries, even when the local site
dynamics is trivial.
