In 1952 mathematician Alan Turing proposed that a system of reaction-diffusion equations describing chemical reactions and diffusion could act as plausible simplified model for morphogenesis, i.e., the development of patterns, shapes and structures found in nature. These complex systems have been shown to possess the ability to imitate natural patterns, e.g. animal coatings (mammals, fish, butterflies etc.). However, the conclusive evidence connecting Turing mechanism to biology is still missing.
The recent growth in computing resources has enabled extensive numerical studies, which have brought a great deal of new insight into Turing systems completing the knowledge obtained from chemical experiments. We have numerically simulated the evolution of structures generated by the Turing mechanism in two- and three-dimensional domains under different kind of constraints. In addition, we have analytically investigated the stability and pattern selection of the model and employed linear analysis to obtain the dispersion relation in terms of the Fourier modes. To study the nonlinear effects in the model we have used weakly nonlinear bifurcation analysis, where the dynamics of the chemical system is mapped to an equivariant amplitude space using the center manifold reduction.