Turing Pattern in the Oregonator Revisited Elragig Aiman, Dreiwi Hanan, Townley Stuart, Elmabrook Idriss Abstract—In this paper we reconsider the analysis of the Oregonator model. We highlight an error in this analysis which leads to an incorrect depiction of the parameter region in which diffusion driven instability is possible. We believe that the cause of the oversight is the complexity of stability analyses based on eigenvalues and the dependence on parameters of matrix minors appearing in stability calculations. We regenerate the parameter space where Turing patterns can be seen, and we use the common Lyapunov function (CLF) approach, which is numerically reliable, to further confirm the dependence of the results on diffusion coefficients intensities Keywords—Diffusion driven instability, common Lyapunov function (CLF), turing pattern, positive-definite matrix. I. I NTRODUCTION T URING theory of pattern formation [9] has had a tremendous impact on various branches of science. According to Turing analysis a systems of reacting and diffusion chemical species, termed as morphogens, could lead to a spatial heterogenieties (patterns) of chemical densities from an intial uniform state. This phnomenon is known as diffusion-driven-instability (DDI) or Turing instability [10]. In other words Turing explanation of pattern formation is based on using a reaction diffuion (RD) system. RD models have subsequently been widely applied to various biological patterning phenomena [10], [11]. An early application of Turing’s theory was to patterning of the body segment in fruity Drosophila [12], [13]. RD systems have been used to model complex pattern formation of certain animal skins [14], [15]. Reaction diffusion theory has been also utilised to examine the spatio-temporal pattern formation on the surface of tumour spheroids [16]. Pattern formation via diffusion driven instability plays an important role in chemistry [17]–[19] and physics [19]. Ecologists use RD models to understand spatial patterns in populations and communities [20]–[26], where for instance, a very fast prey (predator) would intuitively drive the density of the whole population to be spatially dependent. Despite all the promising successes of Turing mechanism to replicate many patterns in nature, as mentioned above, existence of morphogens has not yet been proved for definite. However, there do exist very close candidates for morphogens. Calcium as morphogen leading to hair spacing in Acetabularia [27], and Fibronectin as a morphogen for cartilage formation [28]. Nevertheless, there is no definitive assertion that they are interacting as suggested by Turing. For details see [29]. A. Elragig is a Lecturer with the Department of Mathematics, faculty of science, University of Benghazi, Libya (e-mail: aimen732003@yahoo.com). H. Dreiwi is a lecturer with the Department of Mathematics, faculty of science, University of Benghazi, Libya (e-mail: ahs2412006@yahoo.com). I. Elmabrook is a Professor in Applied Mathematics with the Department of Mathematics, faculty of science, University of Benghazi, Libya S. Townley is a Professor in Applied Mathematics with the University of Exeter, the UK (e-mail: S.B.Townley@exeter.ac.uk). In chemical systems, Turing structure has been shown by a group in Bordeaux led by De Kepper [30], [31]. The chemical reaction they used was the CIMA reaction. This paper is organised as follows. In Section II we present a classical approach for diffusion driven instability. Section III will focus on the error made during the analysis of the Oregonator model as developed by Qian et. al [1]. II. ACLASSICAL APPROACH TO DETERMINING DIFFUSION DRIVEN I NSTABILITY A reaction diffusion (RD) system is a system of the form ∂u ∂t = f (u)+ D∇ 2 u. (1) The function f ( we assume it is regular) describes the reaction dynamics and D is a diagonal matrix of diffusion coefficients. Here u(t, x) : [0, ∞) × R n → [0, ∞) is an n-tuple vector of densities at spatial position x and time t on a domain Ω, which typically bounded, with zero flux boundary conditions (i.e. ∇.u| Ω =0). Imposing such boundary conditions is due to their neutral nature as they do not pump the space with any additional material and this makes ”self-organization” plausible. Taking other boundary conditions can influence the predictions where this can drive forming different patterns, see [36]. 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 [37], [38]. This can then be followed by bifurcation analysis of specific pattern formations [39]. Pattern formation is trigged by Turing instability. Turing instability, or diffusion driven instability(DDI), is a concept first proposed by Turing [9]. This concept is defined as follows. Definition: We say that a system of the form (1) exhibits Turing instability, or DDI, if the system without diffusion, i.e., ∂u ∂t = f (u). (2) has locally stable equilibrium state which becomes unstable in the presence of diffusion. To analyse DDI mathematically, we use linearised stability analysis. If ˆ u is a spatially uniform equilibrium of (2), then small disturbances w away from ˆ u are governed, qualitatively, by the linear system dw dt = Aw. Here A, the Jacobian matrix of f evaluated at ˆ u, is the linearised reaction matrix. If A is stable (all its eigenvalues have negative real parts), which we assume for the remainder of this chapter, then ˆ u is an asymptotically stable equilibrium for (2). The equilibrium ˆ u is also a spatially homogeneous World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:11, No:7, 2017 305 International Scholarly and Scientific Research & Innovation 11(7) 2017 scholar.waset.org/1307-6892/10008068 International Science Index, Mathematical and Computational Sciences Vol:11, No:7, 2017 waset.org/Publication/10008068