A new necessary condition for Turing instabilities Aiman Elragig a , Stuart Townley b,⇑ a Mathematics Research Institute, College of Engineering, Mathematics & Physical Sciences, University of Exeter, UK b Environment and Sustainability Institute, College of Engineering, Mathematics and Physical Sciences, University of Exeter Cornwall Campus, UK article info Article history: Received 24 June 2011 Received in revised form 28 April 2012 Accepted 30 April 2012 Available online 19 May 2012 Keywords: Lyapunov function Diffusion driven (Turing) instability Reactivity Semi-definite programming 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 exis- tence 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. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction Turing’s analysis of reaction–diffusion systems [1] has had sig- nificant impact on several branches of science. Turing suggested that a system of reacting and diffusing chemical species, termed morphogens, could lead to spatial varieties (patterns) of chemical abundances from an initially uniform state. This phenomenon is called diffusion-driven instability (DDI) or Turing instability. In biol- ogy, morphogenesis (i.e. the development in form or structure of an organism during its life cycle) is often assumed to occur in two, possibly concurrent, stages – the chemical and the mechani- cal. In the chemical (pre-pattern) reaction–diffusion (RD) stage it is hypothesised that the heterogeneous concentration of the mor- phogens caused by DDI enhances cell differentiation according to the intensity of chemicals they are exposed to. In the mechanical stage, it is assumed, owing to the physical interaction of cells with their environment, that they form spatial patterns and conse- quently cells in high density, aggregate and then differentiate [2,3]. Turing’s work focused on the chemical part, although he recognised that mechanics is crucial. RD models have subsequently been widely applied to various biological patterning phenomena [2,3]. An early application of Turing theory was to patterning of the body segment in fruit fly Drosophila [4,5]. RD systems have been used to model complex pattern formation of certain animal skins [6,7]. Reaction diffusion theory has been utilised to examine the spatio-temporal pattern formation on the surface of tumour spheroids [8]. A Turing model is considered in [9] to understand the origination of Escherichia coli biofilm development. Pattern formation via diffusion driven instability plays an important role in chemistry [10,11] and physics [12]. Ecologists use RD models to understand spatial patterns in populations and communities [13–19], where for instance, a very fast prey (predator) would intu- itively drive the density of the whole population to be spatially dependent. In studying pattern formation in RD systems, the key first step is to determine the Turing space for a given model, i.e. the parameter set for the model on which pattern formation can be triggered [20,21]. This can then be followed by bifurcation analyses of spe- cific pattern formations [22]. A typical approach to determining this parameter set is to compute principle minors [23–25] of line- arised reaction–diffusion matrices. However, for high dimensional systems with several parameters, this task is at best unwieldy so that most studies are restricted to low-dimensional models. Neu- bert et al. [26] observed a subtle relationship between diffusion driven instability (DDI) and system reactivity. Specifically, they showed that reactivity of the linearised reaction matrix is neces- sary for DDI. One clear advantage of this result is that it can be ap- plied very easily, even to high dimensional systems involving numerous parameters. We show that this result is in fact a special case of a tighter, more powerful, necessary condition for DDI which can be expressed in terms of common Lyapunov functions (CLFs). If the linearised reaction matrix and the diffusion matrix share a CLF, then DDI is not possible. Reactivity of the linearised reaction ma- trix is equivalent to the identity matrix not being the CLF. Deter- mining the existence of CLFs is achieved using semi-definite programming [27]. This CLF approach extends to other movement processes such as chemotaxis which result in non-diagonal, diffu- sion-chemotaxis coefficient matrices. 0025-5564/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.mbs.2012.04.006 ⇑ Corresponding author. E-mail address: S.B.Townley@exeter.ac.uk (S. Townley). Mathematical Biosciences 239 (2012) 131–138 Contents lists available at SciVerse ScienceDirect Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs