Computational studies of pattern formation in Turing systems

This thesis is an analytical and computational treatment of Turing models, which are coupled partial differential equations describing the reaction and diffusion behavior of chemicals. Under particular conditions, such systems are capable of generating stationary chemical patterns of finite characteristic wave lengths even if the system starts from an arbitrary initial configuration. The characteristics of the resulting dissipative patterns are determined intrinsically by the reaction and diffusion rates of the chemicals, not by external constraints. Turing patterns have been shown to have counterparts in natural systems and thus Turing systems could provide a plausible way to model the mechanisms of biological growth. Turing patterns grow due to diffusion-driven instability as a result of infinitesimal perturbations around the stationary state of the model and exist only under non-equilibrium conditions. Turing systems have been studied using chemical experiments, mathematical tools and numerical simulations. In this thesis a Turing model called the Barrio-Varea-Aragon-Maini (BVAM) model is studied by employing both analytical and numerical methods. In addition to the pattern formation in two-dimensional domains, also the formation of three-dimensional structures is studied extensively. The scaled form of the BVAM model is derived from first principles. The model is then studied using the standard linear stability analysis, which reveals the parameter sets corresponding to a Turing instability and the resulting unstable wave modes. Then nonlinear bifurcation analysis is carried out to find out the stability of morphologies induced by two-dimensional hexagonal symmetry and various three-dimensional symmetries (SC, BCC, FCC). This is realized by employing the center manifold reduction technique to obtain the amplitude equations describing the reduced chemical dynamics on the center manifold. The main numerical results presented in this thesis include the study of the Turing pattern selection in the presence of bistability, and the study of the structure selection in three-dimensional Turing systems depending on the initial configuration. Also, the work on the effect of numerous constraints, such as random noise, changes in the system parameters, thickening domain and multistability on Turing pattern formation brings new insight concerning the state selection problem of non-equilibrium physics.