  The reaction-diffusion model can generate a wide variety of spatial patterns,
which has been widely applied in chemistry, biology, and physics, even used to
explain self-regulated pattern formation in the developing animal embryo. In
this work, a second-order stabilized semi-implicit time-stepping Fourier
spectral method is presented for the reaction-diffusion systems of equations
with space described by the fractional Laplacian. We adopt the temporal-spatial
error splitting argument to illustrate that the proposed method is stable
without imposing the CFL condition, and we prove an optimal L2-error estimate.
We also analyze the linear stability of the stabilized semi-implicit method and
obtain a practical criterion to choose the time step size to guarantee the
stability of the semi-implicit method. Our approach is illustrated by solving
several problems of practical interest, including the fractional Allen-Cahn,
Gray-Scott and FitzHugh-Nagumo models, together with an analysis of the
properties of these systems in terms of the fractional power of the underlying
Laplacian operator, which are quite different from the patterns of the
corresponding integer-order model.
