A new necessary condition for Turing instabilities

作者：Elragig Aiman; Townley Stuart^*

来源：Mathematical Biosciences, 2012, 239(1): 131-138.

DOI：10.1016/j.mbs.2012.04.006

• Lyapunov function
• Diffusion driven (Turing) instability
• Reactivity
• Semi-definite programming

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.

• 出版日期2012-9