Impact of non-diffusive interactions on Turing instability

Abstract

Observational studies of the chemical reactions have shown that the uniform state, driven by the diffusion differences of activator and inhibitor species, can seed the spontaneous emergence of spatially organized patterns—also known as Turing instability. The presence of diffusive interaction between the species in discrete media can naturally form the structure of the network Laplacian, thereby resulting in that the Turing instability can be better understood via the eigenvectors and eigenvalues of the graph Laplacian. However, it is still unclear whether and under which conditions the Turing instability can be induced in the absence of diffusive transport. In this work, to overcome the limitation of network Laplacian matrix that only has non-positive eigenvalues, we investigate the effects of non-diffusive (or no-local) coupling between the species on the Turing instability for the directed and undirected networks, respectively, where the interaction matrix may produce the positive and negative eigenvalues. In particular, we first derive the critical conditions that the Turing instability occurs by employing the linear stability analysis. Then we apply them to the networked systems governed by the FitzHugh–Nagumo dynamics and obtain the critical range of eigenvalues of the interaction matrix. Finally, we use the Girko’s circular law to predict the eigenvalues’ distribution, and show, via extensive numerical simulations, the transition process of the Turing instability depending on network size, network connectivity and connection weights (intraspecific interaction strength). Our results reveal that in the absence of diffusive transport, the Turing instability still can arise, offering a new perspective on the generation mechanism of the Turing instability.