  This work explores the influence of domain size of a non-compact two
dimensional annular domain on the evolution of pattern formation that is
modelled by an \textit{activator-depleted} reaction-diffusion system. A closed
form expression is derived for the spectrum of Laplace operator on the domain
satisfying a set of homogeneous conditions of Neumann type both at inner and
outer boundaries. The closed form solution is numerically verified using the
spectral method on polar coordinates. The bifurcation analysis of
\textit{activator-depleted} reaction-diffusion system is conducted on the
admissible parameter space under the influence of two bounds on the parameter
denoting the thickness of the annular region. The admissibility of Hopf and
transcritical bifurcations is proven conditional on the domain size satisfying
a lower bound in terms of reaction-diffusion parameters. The admissible
parameter space is partitioned under the proposed conditions corresponding to
each case, and in turn such conditions are numerically verified by applying a
method of polynomials on a quadrilateral mesh. 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 conditions on the domain size for all types of admissible bifurcations.
