Turing patterns formed by activator-inhibitor systems on networks are considered. The linear stability analysis shows that the Turing instability generally occurs when the inhibitor diffuses sufficiently faster than the activator. Numerical simulations, using a prey-predator model on a scalefree random network, demonstrate that the final, asymptotically reached Turing patterns can be largely different from the critical modes at the onset of instability, and multistability and hysteresis are typically observed. An approximate mean-field theory of nonlinear Turing patterns on the networks is constructed.PACS numbers: 82.40. Ck, 89.75.Fb, 87.23.Cc Turing instability in activator-inhibitor systems is one of the most important classical concepts of nonequilibrium pattern formation [1], with diffusion destabilizing a uniform stationary state of the system and leading to formation of stationary spatial patterns. Turing patterns in ordinary continuous media have been extensively investigated in theoretical [2] and experimental [3,4] studies. Their properties have also been studied for spatially heterogeneous systems [5] and in the presence of fractional diffusion [6].In this Letter, we analyze Turing patterns exhibited by activator-inhibitor systems on networks. In such systems, the nodes are populated by activator and inhibitor species which are diffusively transported over the edges that form a network. As examples, multicellular systems [7], networks of coupled chemical reactors [8], and transportation networks that facilitate spreading of animals or insects [9, 10] can be mentioned. If several species are present, their mobilities on a network may significantly differ, and a situation is possible where the inhibitor species is much more easily transported over a network than the activator. Previously, it has been shown by linear stability analysis that the uniform stationary state in such networks can be unstable [7,8]. However, detailed numerical and analytical investigations have so far been performed only for lattices [7] and for small network systems [8].Here, we provide a complete linear stability analysis valid for any activator-inhibitor system and perform a detailed numerical study of Turing patterns in the Mimura-Murray model of interacting prey-predator populations [11]. Our numerical results reveal that the actual nonlinear behavior of network-based activator-inhibitor systems may show significant differences from the predictions of the linear stability analysis, with coexistence of many different structures and hysteresis being typically observed. As we further show, the properties of network Turing patterns can often be well understood in the framework of a mean-field approximation.Activator-inhibitor systems on networks are described by equationsUwhere U i and V i are concentrations of the activator and the inhibitor species on node i = 1, · · · , N , and functions f (U, V ) and g(U, V ) determine their local dynamics on a single node. In these equations, ∇ 2 represents the diffusion (or Laplacian) ...

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Library of Congress Cataloging-in-Publication Data. Mikhailov, A. S. (Alexander S.), 1950-Foundations of synergetics / A. S. Mikhailov. p. cm.-(Springer series in synergetics; v. 51) Includes bibliographical references and index. Contents: 1. Distributed active systems.

Winfree turbulence of scroll waves is a special kind of spatiotemporal chaos that exists exclusively in three-dimensional excitable media and is currently considered one of the principal mechanisms of cardiac fibrillation. A chaotic wave pattern develops through the negative-tension instability of vortex filaments, which tend to spontaneously stretch, bend, loop, and produce an expanding tangle that fills up the volume. We demonstrate that such turbulence can readily be controlled by weak nonresonant modulation of the medium excitability. Depending on the forcing frequency, either suppression or induction of turbulence can be achieved.

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