This work explores the influence of domain-size on the evolution of pattern formation modelled by an activator-depleted reactiondiffusion system on a flat-ring (annulus). A closed form expression is derived for the spectrum of the Laplace operator on the domain. Spectral method is used to depict the close form solution on the domain. The bifurcation analysis of activator-depleted reactiondiffusion system is conducted on the admissible parameter space under the influence of domain-size. The admissible parameter space is partitioned under a set of proposed conditions relating the reactiondiffusion constants with the domain-size. Finally, the full system is numerically simulated on a two dimensional annular region using the standard Galerkin finite element method to verify the influence of the analytically derived domain-dependent conditions.

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.

In this work an activator-depleted reaction-diffusion system is investigated on polar coordinates with the aim of exploring the relationship and the corresponding influence of domain size on the types of possible diffusion-driven instabilities. Quantitative relationships are found in the form of necessary conditions on the area of a disk-shape domain with respect to the diffusion and reaction rates for certain types of diffusion-driven instabilities to occur. Robust analytical methods are applied to find explicit expressions for the eigenvalues and eigenfunctions of the diffusion operator on a disk-shape domain with homogenous Neumann boundary conditions in polar coordinates. Spectral methods are applied using chebyshev non-periodic grid for the radial variable and Fourier periodic grid on the angular variable to verify the nodal lines and eigensurfaces subject to the proposed analytical findings. The full classification of the parameter space in light of the bifurcation analysis is obtained and numerically verified by finding the solutions of the partitioning curves inducing such a classification. Spatio-temporal periodic behaviour is demonstrated in the numerical solutions of the system for a proposed choice of parameters and a rigorous proof of the existence of infinitely many such points in the parameter plane is presented under a restriction on the area of the domain, with a lower bound in terms of reaction-diffusion rates.

This paper explores the classification of parameter spaces for reaction-diffusion systems of two chemical species on stationary rectangular domains. The dynamics of the system are explored both in the absence and presence of diffusion. The parameter space is fully classified in terms of the types and stability of the uniform steady state. In the absence of diffusion the results on the classification of parameter space are supported by simulations of the corresponding vector-field and some trajectories of the phase-plane around the uniform steady state. In the presence of diffusion, the main findings are the quantitative analysis relating the domain-size with the reaction and diffusion rates and their corresponding influence on the dynamics of the reaction-diffusion system when perturbed in the neighbourhood of the uniform steady state. Theoretical predictions are supported by numerical simulations both in the presence as well as in the absence of diffusion. Conditions on the domain size with respect to the diffusion and reaction rates are related to the types of diffusion-driven instabilities namely Turing, Hopf and Transcritical types of bifurcations. The first condition is a lower bound on the area of a rectangular domain in terms of the diffusion and reaction rates, which is necessary for Hopf and Transcritical bifurcation to occur. The second condition is an upper bound on the area of domain in terms of reaction-diffusion rates that restricts the diffusion-driven instability to Turing type behaviour, whilst forbidding the existence of Hopf and Transcritical bifurcation. Theoretical findings are verified by the finite element solution of the coupled system on a two dimensional rectangular domain.