Regensburg 2013 – wissenschaftliches Programm

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DY: Fachverband Dynamik und Statistische Physik

DY 7: Poster I

DY 7.25: Poster

Montag, 11. März 2013, 17:30–19:30, Poster C

Patterns in anisotropic reaction diffusion systems — •Fabian Bergmann, Alexei Krekhov, and Walter Zimmermann — Universität Bayreuth, Theoretische Physik, 95440 Bayreuth, Germany

In 1952, Turing suggested a hypothetical reaction-diffusion model showing spatially periodic pattern. This class of models was recently identified as an important framework for understanding Biological Pattern Formation, such as the formation of pigment stripes in the skin of fishes [1]. Recently, anisotropic diffusion [2] and cross diffusion have been identified as two important extensions of Turing’s model [3].

We investigate bifurcation scenarios in anisotropic reaction-diffusion systems including cross diffusion. We find for this class of systems a wide range of parameters, whereby two orthogonally oriented stripe patterns have the same threshold (codimension-two bifurcation). In this case, a nonlinear competition between both stripe orientations takes place beyond threshold. Near threshold this competition is analyzed in terms of amplitude equations, which are derived from the basic reaction-diffusion model. We find three different scenarios: (i) In a wide range of parameters either one or the other stripe pattern is stable, depending on the initial condition. (ii) In a second range the coexistence of both stripe patterns (rectangular patterns) is preferred. (iii) Both cases are separated by a parameter range, where surprisingly, only one of both stripe orientations is stable.

[1] S. Kondo and T. Miura, Reaction-Diffusion Model as Framework for Understanding Biological Pattern Formation, Science 329, 1616 (2010)

[2] H. Shoji, Y. Iwasa, A. Mochizuki and S. Kondo, Directionality of Stripes Formed by Anisotropic Reaction-Diffusion Models, J. Theor. Biol. 214, 349 (2002)

[3] N. Kumar and W. Horsthemke, Effects of cross diffusion on Turing bifurcations in two species reaction-transport systems, Phys. Rev. E 83, 036105 (2011)