Pattern formation and Turing instability in an activator–inhibitor system with power-law coupling

• Activator–inhibitor systems with non-local coupling are studied in two dimensions.
• Linear stability analysis of spatial modes gives conditions for Turing instability.
• Numerical simulations are performed to study non-linear saturation of linearly growing modes.

We investigate activator–inhibitor systems in two spatial dimensions with a non-local coupling, for which the interaction strength decreases with the lattice distance as a power-law. By varying a single parameter we can pass from a local (Laplacian) to a global (all-to-all) coupling type. We derived, from a linear stability analysis of the Fourier spatial modes, a set of conditions for the occurrence of a Turing instability, by which a spatially homogeneous pattern can become unstable. In nonlinear systems the growth of these modes is limited and pattern formation is possible. We have studied some qualitative features of the patterns formed in non-local coupled activator–inhibitor systems described by the Meinhardt–Gierer equations.