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A lattice Boltzmann model for reaction-diffusion systems is developed. The method provides an efficient computational scheme for simulating a variety of problems described by the reaction-diffusion equations. Diffusion phenomena, the decay to a limit cycle, and the formation of Turing patterns are studied. The results of lattice Boltzmann calculations are compared with the lattice gas method and with theoretical predictions, showing quantitative agreement. The model is extended to include velocity convection in chemically reacting fluid flows.