On the origin of Turing instability

A reaction–diffusion system consisting of one, two or three chemical species and taking place in an arbitrary number of spatial dimensions cannot exhibit Turing instability if none of the reaction steps express cross-inhibition. A corollary of this result – obtained by elementary calculations – underlines the importance of nonlinearity in the formation of stationary structures, a kind of self-organization on a chemical basis. Relations to global stability of reaction–diffusion systems, and results on multispecies systems are also mentioned. The statements are not restricted to mass action type models. As a by-product, the solution of a basic inverse problem of formal kinetics is also presented which extends a previous result by Hárs and Tóth (1981) to models with arbitrary – including rational – functions as reaction rates so often occurring, e.g., in enzyme kinetics.