  The Gray-Scott (GS) model represents the dynamics and steady state pattern
formation in reaction-diffusion systems and has been extensively studied in the
past. In this paper, we consider the effects of anomalous diffusion on pattern
formation by introducing the fractional Laplacian into the GS model. First, we
prove that the continuous solutions of the fractional GS model are unique. We
then introduce the Crank-Nicolson (C-N) scheme for time discretization and
weighted shifted Gr\"unwald difference operator for spatial discretization. We
perform stability analysis for the time semi-discrete numerical scheme, and
furthermore, we analyze numerically the errors with benchmark solutions that
show second-order convergence both in time and space. We also employ the
spectral collocation method in space and C-N scheme in time to solve the GS
model in order to verify the accuracy of our numerical solutions. We observe
the formation of different patterns at different values of the fractional
order, which are quite different than the patterns of the corresponding
integer-order GS model, and quantify them by using the radial distribution
function (RDF). Finally, we discover the scaling law for steady patterns of the
RDFs in terms of the fractional order $1<\alpha \leq 2 $.
