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