title: A new necessary condition for Turing instabilities.
abstract: 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.
authors.affiliation: Mathematics Research Institute, College of Engineering, Mathematics & Physical Sciences, University of Exeter, Exeter, UK.
authors.forename: Aiman
authors.initials: A
authors.lastname: Elragig
authors.forename: Stuart
authors.initials: S
authors.lastname: Townley
mesh_terms: D000818:Animals; D002633:Chemotaxis; D006790:Host-Parasite Interactions; D006801:Humans; D006829:Hydra; D008954:Models, Biological; D008956:Models, Chemical