  In this article we present a computational framework for isolating spatial
patterns arising in the steady states of reaction-diffusion systems. Such
systems have been used to model many different phenomena in areas such as
developmental and cancer biology, cell motility and material science. Often one
is interested in identifying parameters which will lead to a particular
pattern. To attempt to answer this, we compute eigenpairs of the Laplacian on a
variety of domains and use linear stability analysis to determine parameter
values for the system that will lead to spatially inhomogeneous steady states
whose patterns correspond to particular eigenfunctions. This method has
previously been used on domains and surfaces where the eigenvalues and
eigenfunctions are found analytically in closed form. Our contribution to this
methodology is that we numerically compute eigenpairs on arbitrary domains and
surfaces. Here we present various examples and demonstrate that mode isolation
is straightforward especially for low eigenvalues. Additionally we see that if
two or more eigenvalues are in a permissible range then the inhomogeneous
steady state can be a linear combination of the respective eigenfunctions.
Finally we show an example which suggests that pattern formation is robust on
similar surfaces in cases that the surface either has or does not have a
boundary.
