A new necessary condition for Turing instabilities

Reactivity (a.k.a initial growth) is necessary for diffusion driven instability (Turing instability). Using a notion of common Lyapunov function we show that this necessary condition is a special case of a more powerful (i.e. tighter) necessary condition. Specifically, we show that if the linearised reaction matrix and the diffusion matrix share a common Lyapunov function, then Turing instability is not possible. The existence of common Lyapunov functions is readily checked using semi-definite programming. We apply this result to the Gierer–Meinhardt system modelling regenerative properties of Hydra, the Oregonator, to a host-parasite-hyperparasite system with diffusion and to a reaction–diffusion-chemotaxis model for a multi-species host-parasitoid community.

► Common Lyapunov functions (CLFs) are used to study Turing instability.
► If the diffusion and reaction matrices have a CLF, then no Turing instability.
► Existence of CLFs is verified using semi-definite programming.
► The CLF technique is applied to models of diffusive and chemotactic growth.