Characterization of spatial patterns produced by a Turing instability in coupled dynamical systems

We quantify the degree of spatial order of patterns at fixed time generated by lattices of coupled dynamical systems, using correlation-based and recurrence-based numerical diagnostics. These patterns are obtained through numerical integration of differential equations describing the interplay between activator and inhibitor species generating Turing patterns. We consider different types of coupling: linear (diffusive) interaction with nearest-neighbors, global (all-to-all) coupling and intermediate (nonlocal) coupling. Numerical simulations are performed in one and two spatial dimensions. The effects of noise are briefly discussed. We introduce a recurrence-based quantity (recurrence-rate matrix) to characterize two-dimensional spatial patterns.