Transient growth induces unexpected deterministic spatial patterns in the Turing process

¹ DITIC, Politecnico di Torino - Corso Duca Abruzzi 24, 10129, Turin, Italy, EU
² Department of Environmental Sciences, University of Virginia - Charlottesville VA, USA

^a luca.ridolfi@polito.it

Received: 22 March 2011
Accepted: 18 May 2011

Abstract

Turing models are often invoked to explain spatial pattern formation in a number of physical, chemical and biological processes. Pattern occurrence is generally investigated through a classical eigenvalue analysis, which evaluates the asymptotic stability of the homogeneous state of the system. Here we show that deterministic patterns may emerge in a Turing model even when the homogeneous state is stable. In fact, the non-normality of the eigenvectors is able to generate transient (long-lasting) patterns even in the region of the parameter space where the dynamical system is asymptotically stable (i.e., the eigenvalues are negative). Moreover, non-normality–induced patterns usually display an interesting multiscale structure that can be investigated analytically.