  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 scale-free
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.
