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.
Neubert
Michael G
MG
Biology Department, MS #34, Woods Hole Oceanographic Institution, Woods Hole, MA 02543-1049, USA. mneubert@whoi.edu
Caswell
Hal
H
Murray
J D
JD
D000818:Animals; D004463:Ecology; D008954:Models, Biological; D017711:Nonlinear Dynamics