UK Nonlinear News, May 2002

Flow-and-diffusion structures (FDS) or how Turing patterns come to life

By Razvan Satnoianu

A new instability mechanism for pattern formation, published recently in Physica D (160 (2001), 79-102), extends the celebrated Turing mechanism [1] in a novel way. It involves the combined action of diffusion, flow and local nonlinear interaction and leads to the formation of stationary, space periodic waves. It is referred to as the flow- and- diffusion-structures or FDS mechanism [2,5].

FDS has remarkable properties [2] compared with the Turing scenario. First, FDS are supported in a parameter space that is very large when compared to the Turing space to which it adjoins (see figures 1,2 illustrating this for the typical cases of Gierer-Meinhardt and Schnakenberg type nonlinear cubic kinetics). Second the classical Turing case appears as a limit of the FDS scenario, when the flow rate vanishes (pure diffusion limit), see figures 3,4. The third feature is that FDS patterns are robust to parameter changes when compared to the Turing scenario (figure 5). Also FDS patterns exhibit a richer pattern structure, at least in the one-dimensional setting, for example showing the property of phase differences between the amplitudes of the interacting species (figure 6), as well as patterns with variable amplitude in the spatial domain (figure 7). All these properties are lacking in a typical Turing system (figure 8). We have explored these issues by analyzing the FDS patterns in systems of reaction-diffusion- advection systems with a number of different nonlinear kinetics: Gray-Scott, Gierer- Meinhardt, Schnakenberg, Thomas, CIMA, BZ, etc. [2].

FDS patterns exist also in the purely advective regime [5]. This limit was studied before [3] and was shown to exhibit stationary patterns by perturbing an oscillatory steady state. This type of pattern, called flow-distributed oscillations FDO [4,7] has recently attracted considerable interest [3-9]. FDO represent the pure advective limit behaviour of the more general FDS mechanism.

Our main conclusion is that the FDS mechanism holds much promise in terms of ease and robustness - hence it should be readily verifiable experimentally [10]. In fact it may well be that FDS are the best opportunity to date for the Turing patterns to come to life. Already it is envisaged that the FDS mechanism plays a key role in pattern formation of axially growing organisms, for example during the segmentation of embryos [6,7].

The figures are available as a pdf file.


  1. A.M. Turing, The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237 (1952), 37-72.
  2. Razvan Satnoianu, Philip Maini and Michael Menzinger: Parameter space analysis, pattern sensitivity and model comparison for Turing and flow-distributed waves (FDS), Physica D 160 (2001), 79-102.
  3. P. Andresen, M Bache, E. Mosekilde, G. Dewel, P. Borckmans, Stationary space-periodic structures with equal diffusion coefficients. Phys. Rev. E 60 (1999), 297-301.
  4. M. Kaern, M. Menzinger, Experimental observation of stationary chemical waves in a flow system, Phys. Rev. E 60 (1999), 3471-3474.
  5. R.A. Satnoianu, M. Menzinger, Non-Turing stationary patterns: flow-distributed stationary structures with general diffusion and flow rates, Phys. Rev. E 62 (2000), 113-119.
  6. M. Kaern, M. Menzinger, R.A. Satnoianu and A. Hunding, Chemical waves in open flows of active media: their relevance to axial segmentation in biology. Faraday Discussion 120 (2001) NONLINEAR CHEMICAL KINETICS: Complex Dynamics and Spatiotemporal Patterns, The Royal Society of Chemistry, Faraday Division, 295-312.
  7. M.Kaern, M.Menzinger, A.Hunding, Segmentation and somitogenesis derived from phase dynamics in growing oscillatory media, J.Theor. Biol. 207 (2000) 473- 493.
  8. J.R.Bamforth, J,H,Merkin S.K.Scott, Flow-distributed oscillation patterns in the Oregonator model, PCCP (2001), 1435-1438.
  9. M. Sheintuch, O.A. Nekhamkina, Catalysis Today 70 (2001), 383-391.
  10. M. Kaern, Razvan Satnoianu, A.P. Munuzuri and M. Menzinger, Controlled pattern formation in the CDIMA reaction with a moving boundary of illumination, PCCP (2002), DOI 10.139/b109387h.

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