The generalized activator-inhibitor model with cubic nonlinearity, in which the classical Laplacian is replaced by fractional operator has been studied. The fractional operator reflects the nonlocal behavior of superdiffusion. A spatially homogeneous, time independent solution has been found and its linear stability was studied. We have also performed a weakly nonlinear analysis and obtained a system of amplitude equations that are the basis for analysing pattern formation as well as parameter regimes for which various steady-state patterns would exist.

- Henry B. I., Wearne S. L. Existence of Turing instabilities in a two-species fractional reaction-diffusion system. SIAM J. Appl. Math.
**62**, n. 3, 870–887 (2002). - Datsko B., Luchko Y., Gaﬁychuk V. Pattern formation in fractional reaction-diffusion systems with multiple homogeneous states. Int. J. Bifurcation Chaos.
**22**, 1250087 (2012). - Datsko B., Gaﬁychuk V., Podlubny I. Solitary travelling auto-waves in fractional reaction–diffusion systems. Communications in Nonlinear Science and Numerical Simulation.
**23**(1), 378–387 (2015). - Nec Y., Ward M. J. The stability and slow dynamics of two-spike patterns for a class of reaction-diffusion system. Math. Model. Nat. Phenom.
**8**(5), 206–232 (2013). - Fomin S., Chugunov V., Hashida T. Mathematical modeling of anomalous diffusion in porous media. Fract. Different. Calc.
**1**, n. 1, 1–28 (2011). - Farago J., Meyer H., Semenov A. N. Anomalous Diffusion of a Polymer Chain in an Unentangled Melt. Phys. Rev. Lett.
**107**(17), 178301 (2011). - Carcione J. M., Sanchez-Sesma F. J., Luzon F., Gavilan J. J. P. Theory and simulation of time-fractional ﬂuid diffusion in porous media. J. Phys. A: Math. Theor.
**46**, 345501 (2013). - Aarão Reis F. D. A., di Caprio D. Crossover from anomalous to normal diffusion in porous media. Phys. Rev. E.
**89**, 062126 (2014). - Garra R. Fractional-calculus model for temperature and pressure waves in ﬂuid-saturated porous rocks. Phys. Rev. E.
**84**, 036605 (2011). - Roubinet D., de Dreuzy J. R., Tartakovsky D. M. Particle-tracking simulations of anomalous transport in hierarchically fractured rocks. Computers & Geosciences.
**50**, 52–58 (2013). - Carreras B. A., Lynch V. E., Zaslavsky G. M. Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model. Phys. Plasmas.
**8**(12), 5096–5103 (2001). - Priego M., Garcia O. E., Naulin V., Rasmussen J. J. Anomalous diffusion, clustering, and pinch of impurities in plasma edge turbulence. Phys. Plasmas.
**12**(6), 062312 (2005). - Krivolapov Y., Levi L., Fishman Sh., Segev M., Wilkinson M. Super-diffusion in optical realizations of Anderson localization. New J. Phys.
**14**, 043047 (2012). - Barkai E., Jung Y., Silbey R. Theory of single-molucule spectroskopy: beyond the ensemble average. Annu. Rev. Phys. Chem.
**55**, 457–507 (2004). - Golovin A. A., Matkowsky B. J., Volpert V. A. Turing pattern formation in the Brusselator model with superdiffusion. J. Appl. Math.
**69**, n. 1, 251–272 (2008). - Zhang L., Tian C. Turing pattern dynamics in an activator-inhibitor system with superdiffusion. Phys. Rev. E.
**90**, 062915 (2014). - Duﬁet V., Boissonade J. Dynamics of Turing pattern monolayers close to onset. Phys. Rev. E.
**53**, 4883 (1996). - Prytula Z. Mathematical modelling of nonlinear dynamics in activator-inhibitor systems with superdiffusion. The Bulletin of Lviv Polytechnic National University titled “Computer Sciences and Information Technologies”.
**826**, 230–237 (2015). - Samko S. G., Kilbas A. A., Marichev O. I. Fractional integrals and derivatives, theory and applications. Gordon and Breach, Amsterdam (1993).
- Uchaikin V. Method of fractional derivatives. Artishok-Press (2008), (in Russian).
- Petráš I. Fractional-Order Nonlinear Systems. Modeling, Analysis and Simulation. Springer (2011).
- Podlubny I. Fractional Differential Equations. San Diego: Acad. Press (1999).
- Walgraef D. Spatio-Temporal Pattern Formation. Springer, New York (1997).

Math. Model. Comput. Vol. 3, No. 2, pp. 191-198 (2016)