Bifurcation and Periodic Triangular Pattern Formation in Reaction-Diffusion with Anisotropic Diffusion
Abstract
Turing demonstrated a coupled reaction-diffusion equation with two components produced steady-state heterogeneous spatial patterns, under certain conditions. The instability found by Turing is now called a diffusion-driven instability or Turing instability. Systems in two dimensions produce spot and stripe patterns, and these systems have been applied as models to explain patterns observed in biological and chemical fields and to develop image information processing tools. Previously, we developed a method that utilizes a reaction-diffusion system with anisotropic diffusion that exhibits triangular patterns, thereby introducing a certain anisotropic strength. In this chapter, we discuss the effects of anisotropic diffusion on the generation of triangular patterns. By defining the statistical index characterizing the spatial patterns, we investigated the parameter range over which the triangular patterns were obtained. We determined the explanatory variable based on the relative distance of the pitchfork bifurcation point between the maximum and minimum anisotropic diffusion functions. Its relevance to diffusion instability is also discussed.
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