  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 eigen-surfaces 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. Furthermore, analytical
results are found relating the area of (disk-shape) domain with
reaction-diffusion rates in the form of necessary conditions for the different
types of bifurcations. These results are on one hand presented in the form of
mathematical theorems with rigorous proofs, and, on the other hand using finite
element method, each claim of the corresponding theorems are verified by
obtaining the theoretically predicted behaviour of the dynamics in the
numerical simulations.
