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Turing instability in reaction-diffusion models on complex networks. (English) Zbl 1400.92060
Summary: In this paper, the Turing instability in reaction-diffusion models defined on complex networks is studied. Here, we focus on three types of models which generate complex networks, i.e. the Erdos-Rényi, the Watts-Strogatz, and the threshold network models. From analysis of the Laplacian matrices of graphs generated by these models, we numerically reveal that stable and unstable regions of a homogeneous steady state on the parameter space of two diffusion coefficients completely differ, depending on the network architecture. In addition, we theoretically discuss the stable and unstable regions in the cases of regular enhanced ring lattices which include regular circles, and networks generated by the threshold network model when the number of vertices is large enough.

MSC:
92C15 Developmental biology, pattern formation
05C80 Random graphs (graph-theoretic aspects)
05C82 Small world graphs, complex networks (graph-theoretic aspects)
35R02 PDEs on graphs and networks (ramified or polygonal spaces)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35K57 Reaction-diffusion equations
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