"Morphological diversity and robustness of Turing structures"
MORPHOLOGICAL DIVERSITY AND ROBUSTNESS OF TURING STRUCTURES The reaction-diffusion systems discovered by Alan Turing in 1952 have been shown to possess the ability to imitate natural patterns, e.g. animal coatings (mammals, fish, butterflies etc.). Conclusive evidence connecting the Turing mechanism to biology is still missing, however. The recent growth in computational resources has enabled extensive numerical studies, which have both brought a great deal of new insight complementing the knowledge obtained from chemical experiments and facilitated the development of complex biological growth models. We have numerically simulated the evolution of structures in two and three dimensions arising from the Turing instability . The numerical approach has made it possible to study morphological transitions as the bifurcation parameter  or the spatial dimensions of the system  are varied. We have also studied the effect of additive Gaussian random noise on the developing structures (melting/stabilization) . In addition to the numerical simulations, we have analytically investigated the pattern selection in a generic Turing model by approximating the chemical dynamics by a standard amplitude equation presentation obtained using center manifold reduction . The parameters of our generic Turing model may be adjusted in such a way that three stationary states exist instead of only one. The relative stability of these states, and the type of the unstable modes combined with their nonlinear coupling, determines the complex spatio-temporal behavior of the system. The fact that a Turing model may exhibit oscillations, and these oscillations may couple with Turing patterns of fixed characteristic length scale, may be of great interest in biological context. For example, skin hair follicle formation, which is closely related to skin pigmentation, occurs in cycles. REFERENCES (available at http://www.lce.hut.fi/research/polymer/turing.shtml)  T. Leppänen, M. Karttunen, K. Kaski, R. A. Barrio, and L. Zhang, "A new dimension to Turing patterns", Physica D 168-169C, 35-44 (2002).  T. Leppänen, M. Karttunen, R.A. Barrio, and K. Kaski, "Morphological transitions and bistability in Turing systems", submitted to PRE, 2003.  T. Leppänen, M. Karttunen, K. Kaski, and R. A. Barrio, "Dimensionality effects in Turing pattern formation", Int. J. Mod. Phys. B 17, 5541-5553 (2004).  T. Leppänen, M. Karttunen, K. Kaski, and R. A. Barrio, "The effect of noise on Turing patterns", Prog. Theor. Phys. (Suppl.) 150, 367-370 (2003).  T. Leppänen, "The theory of Turing pattern formation", submitted 2004.