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We investigate the response of two-dimensional pattern forming systems with abroken up-down symmetry, such as chemical reactions, to spatially resonantforcing and propose related experiments. The nonlinear behavior immediatelyabove threshold is analyzed in terms of amplitude equations suggested for a$1:2$ and $1:1$ ratio between the wavelength of the spatial periodic forcingand the wavelength of the pattern of the respective system. Both sets ofcoupled amplitude equations are derived by a perturbative method from theLengyel-Epstein model describing a chemical reaction showing Turing patterns,which gives us the opportunity to relate the generic response scenarios to aspecific pattern forming system. The nonlinear competition between stripepatterns and distorted hexagons is explored and their range of existence,stability and coexistence is determined. Whereas without modulations hexagonalpatterns are always preferred near onset of pattern formation, single modesolutions (stripes) are favored close to threshold for modulation amplitudesbeyond some critical value. Hence distorted hexagons only occur in a finiterange of the control parameter and their interval of existence shrinks to zerowith increasing values of the modulation amplitude. Furthermore depending onthe modulation amplitude the transition between stripes and distorted hexagonsis either sub- or supercritical.Comment: 10 pages, 12 figures, submitted to Physical Review

Stripe-hexagon competition in forced pattern-forming systems with broken up-down symmetry