Transient dynamics and pattern formation: reactivity is necessary for Turing instabilities

The theory of spatial pattern formation via Turing bifurcations – wherein an equilibrium of a nonlinear system is asymptotically stable in the absence of dispersal but unstable in the presence of dispersal – plays an important role in biology, chemistry and physics. It is an asymptotic theory, concerned with the long-term behavior of perturbations. In contrast, the concept of reactivity describes the short-term transient behavior of perturbations to an asymptotically stable equilibrium. In this article we show that there is a connection between these two seemingly disparate concepts. In particular, we show that reactivity is necessary for Turing instability in multispecies systems of reaction–diffusion equations, integrodifference equations, coupled map lattices, and systems of ordinary differential equations.