sentences:  An important idea that was first proposed by Alan Turing is that a state that is stable in the local system can become unstable in the presence of diffusion.A linear stability analysis however shows that when linearizing the general two-component system
  
  a plane wave perturbation
  
  of the stationary homogeneous solution will satisfy
  
  Turing's idea can only be realized in four equivalence classes of systems characterized by the signs of the Jacobian of the reaction function.In particular, if a finite wave vector is supposed to be the most unstable one, the Jacobian must have the signs
  
  This class of systems is named "activator-inhibitor system" after its first representative