Pattern formation in systems with one spatially distributed species (Q1363035)

The authors study mechanisms for pattern-formation in reaction-diffusion systems, where only one species diffuses (or, more generally, allows for spatial interactions). The scenario is bifurcation from a spatially homogeneous equilibrium. The spatial interaction in the first species is assumed to stabilize a spatially homogeneous distribution of concentration. The goal is, as in the classical scenario of Turing instabilities, to look for situations, where the equilibrium becomes unstable with respect to perturbations with a nonzero, finite wave number. The results show that this instability can never be a stationary instability, like the Turing instability. Moreover, the Hopf-Turing instability cannot occur in a two-species system, whereas it actually does occur in a three-species system. The Hopf-Turing instability in this context is then illustrated by means of two examples, from ecology and neural networks. Both standing and traveling waves may emerge.