This thesis presents through a number of applications a self-contained and robust methodology for exploring mathematical models of pattern formation from the perspective of a dynamical system. The contents of this work applies the methodology to investigate the influence of the domain-size and geometry on the evolution of the dynamics modelled by reaction-diffusion systems (RDSs). We start with deriving general RDSs on evolving domains and in turn explore Arbitrary Lagrangian Eulerian (ALE) formulation of these systems. We focus on a particular RDS of activator-depleted class and apply the detailed framework consisting of the application of linear stability theory, domain-dependent harmonic analysis and the numerical solution by the finite element method to predict and verify the theoretically proposed behaviour of pattern formation governed by the evolving dynamics. This is achieved by employing the results of domain-dependent harmonic analysis on three different types of two-dimensional convex and non-convex geometries consisting of a rectangle, a disc and a flat-ring.