Morphology of Experimental and Simulated Turing Patterns

of the experimental quasi-2D patterns is analyzed in Ref. [12, 46, 55], where alsospatio-temporal effects due to interactions of different Turing patterns in the threedimensional system are analyzed and a simple resonance of patterns has also beendiscussed in Ref. [17] to explain the formation of rhombic and black-eyed patterns.The possibility of spatio-temporal patterns in the LE model is discussed in Ref. [53],where spatio-temporal patterns are found as oscillating hexagonal patterns in avery small region of the parameter space. However no theoretical model whichreproduces the turbulent patterns reported in Ref. [47] has been found yet. Althoughspatio-temporal patterns could be obtained in coupled reaction-diffusion systems noevidence of similar turbulence is found in any model based on deterministic partialdifferential equations [67].The effect of noise on Turing pattern formation in a generic reaction-diffusion systemhas been studied by Leppänen in Ref. [33] and patterns have been found to be veryrobust against noise. The robustness of pattern formation against noise is alsoconfirmed for the LE model in chapter 6 in this thesis.The usual approaches to obtain information about spatial structures such as Turingpatterns are Fourier transformations and autocorrelation functions [47]. Whereasthese methods provide useful information about ordered patterns like the characteristicwavelength they can be insufficient for characterizing more irregular structures.Figure 1.3 shows two hexagonal concentration profiles and a one-dimensional crosssection,where the grey-value is shown. Although both patterns have the samewavelength there are obvious morphological differences.u2502001501005000 10 20 30 40 50xu2502001501005000 10 20 30 40 50x(a)(b)Figure 1.3: Two artificial hexagonal patterns with the same wavelength but morphologicallydifferent concentration profiles. The line in the images indicates the cross-sectionshown below.13