  In this paper we present computational techniques to investigate the
solutions of two-component, nonlinear reaction-diffusion (RD) systems on
arbitrary surfaces. We build on standard techniques for linear and nonlinear
analysis of RD systems, and extend them to operate on large-scale meshes for
arbitrary surfaces. In particular, we use spectral techniques for a linear
stability analysis to characterize and directly compose patterns emerging from
homogeneities. We develop an implementation using surface finite element
methods and a numerical eigenanalysis of the Laplace-Beltrami operator on
surface meshes. In addition, we describe a technique to explore solutions of
the nonlinear RD equations using numerical continuation. Here, we present a
multiresolution approach that allows us to trace solution branches of the
nonlinear equations efficiently even for large-scale meshes. Finally, we
demonstrate the working of our framework for two RD systems with applications
in biological pattern formation: a Brusselator model that has been used to
model pattern development on growing plant tips, and a chemotactic model for
the formation of skin pigmentation patterns. While these models have been used
previously on simple geometries, our framework allows us to study the impact of
arbitrary geometries on emerging patterns.
