Regular patterns in dichotomically driven activator-inhibitor dynamics

Abstract

We investigate Turing pattern formation in the presence of additive dichotomous fluctuations in the context of an extended system with diffusive coupling and FitzHugh-Nagumo kinetics. The fluctuations vary in space and/or time. Depending on the realization of the dichotomous switching the system is, at a given time (for spatial disorder at a given position) in one of two possible excitable dynamical regimes. Each of the two excitable dynamics for itself does not support pattern formation. With proper dichotomous fluctuations, however, the homogeneous steady state is destabilized via a Turing instability. We investigate the influence of different switching rates (different correlation length of the spatial disorder) on pattern formation. We find three distinct mechanisms: For slow switching existing boundaries become unstable, for high rates the system exhibits "effective bistability" which allows for a Turing instability. For medium rates the fluctuations create spatial structures via a new mechanism where the influence of the fluctuations is twofold. First they produce local inhomogeneities, which then grow (again caused by fluctuations) until the whole space is covered. Utilizing a nonlinear map approach we show bistability of a period-one and a period-two orbit being associated with the steady homogeneous and the Turing pattern state, respectively. Finally, for purely static dichotomous disorder we find destabilization of homogeneous steady states for finite nonzero correlation length of the disorder resulting again in Turing patterns.