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A three-component reaction-diffusion model is proposed as the first example to exhibit chemical turbulence with multiaffine fractal structures, the underlying mechanism being the same as for similar turbulence discovered recently in some nonlocally coupled oscillator systems. The role played by the strongly diffusive component can be substituted by a long-wave random force, and this idea leads to our proposal of the second, far simpler reaction-diffusion model given by the randomly driven FitzHugh-Nagumo nerve conduction equation.