For a uniquely defined subset of phase space, solutions of non-linear, coupled reaction-diffusion equations may converge to heterogeneous steady states, organic in appearance. Hence, many theoretical models for pattern formation, as in the theory of morphogenesis, include the mechanics of reaction-diffusion equations. The standard method of simulation for such pattern formation models does not facilitate reproducibility of results, or the verification of convergence to a solution of the problem via the method of mesh refinement. In this thesis we explore a new methodology circumventing the aforementioned issues, which is independent of the choice of programming language. While the new method allows more control over solutions, the user is required to make more choices, which may or may not have a determining effect on the nature of resulting patterns. In an attempt to quantify the extent of the possible effects, we study heterogeneous steady states for two well known reaction-diffusion models, the Gierer-Meinhardt model and the Schnakenberg model. / Alexander Graham Bell Canada Graduate Scholarship provides financial support to high calibre scholars who are engaged in master's or doctoral programs in the natural sciences or engineering. / Natural Sciences and Engineering Research Council of Canada

In an attempt to describe how patterns emerge in biological systems, Alan Turing proposed a mathematical model encapsulating the properties of such processes. It details a partial differential equation governing the dynamics of two or more substances, called morphogens, reacting and diffusing in a specific manner, in turn generating what has now come to be denoted as Turing patterns. In recent years, evidence has accumulated to support Turing's claim and it has been proposed that it is responsible for the dynamical characteristics of phenomena such as skin pigmentation and branching of lungs in vertebrates. The aim of this paper is to study how the choice of model parameters and reaction kinetics influence the nature of patterns generated, as well as explore how boundary control can be employed to generate pre-defined patterns and the efficiency of this procedure. To simulate the patterns, the differential equation is solved in Python by means of a spectral method using discretized space and time domains. The model parameters were then studied to try to gain insight in their effects on the patterns yielded. The boundary control was implemented in MATLAB using a difference method. The metric used for efficiency was taken to be the energy expenditure of the boundary cells. The complex dynamics of the studied systems make it difficult to draw valuable conclusions on the influence of the parameters, but the results support the expected characteristics of the models used. The efficiency of the pattern generation is deemed to be closely related to the amount of boundary control utilized.