  Sufficient conditions for the wave instability in general three-component
reaction-diffusion systems are derived. These conditions are expressed in terms
of the Jacobian matrix of the uniform steady state of the system, and enable us
to determine whether the wave instability can be observed as the mobility of
one of the species is gradually increased. It is found that the instability can
also occur if one of the three species does not diffuse. Our results provide a
useful criterion for searching wave instabilities in reaction-diffusion systems
of various origins.
