  This paper investigates the competition between both simple (e.g. stripes,
hexagons) and ``superlattice'' (super squares, super hexagons) Turing patterns
in two-component reaction-diffusion systems. ``Superlattice'' patterns are
formed from eight or twelve Fourier modes, and feature structure at two
different length scales. Using perturbation theory, we derive simple analytical
expressions for the bifurcation equation coefficients on both rhombic and
hexagonal lattices. These expressions show that, no matter how complicated the
reaction kinectics, the nonlinear reaction terms reduce to just four effective
terms within the bifurcation equation coefficients. Moreover, at the hexagonal
degeneracy -- when the quadratic term in the hexagonal bifurcation equation
disappears -- the number of effective system parameters drops to two, allowing
a complete characterization of the possible bifurcation results at this
degeneracy. The general results are then applied to specific model equations,
to investigate the stability of different patterns within those models.
