  We study a fractional reaction-diffusion system with two types of variables:
activator and inhibitor. The interactions between components are modeled by
cubical nonlinearity. Linearization of the system around the homogeneous state
provides information about the stability of the solutions which is quite
different from linear stability analysis of the regular system with integer
derivatives. It is shown that by combining the fractional derivatives index
with the ratio of characteristic times, it is possible to find the marginal
value of the index where the oscillatory instability arises. The increase of
the value of fractional derivative index leads to the time periodic solutions.
The domains of existing of periodic solutions for different parameters of the
problem are obtained. A computer simulation of the corresponding nonlinear
fractional ordinary differential equations is presented. It is established by
computer simulation that there exists a set of stable spatio-temporal tructures
of the one-dimensional system under the Neumann and periodic boundary
condition. The features of these solutions consist in the transformation of the
steady state dissipative structures to homogeneous oscillations or space
temporary structures at a certain value of fractional index and the ratio of
characteristic times of system.
