The Gierer–Meinhardt model with Lévy flights is shown to give rise to patterns of spikes with algebraically decaying tails. The spike shape is given by a solution to a fractional differential equation. Near an equilibrium formation the spikes drift according to the differential equations of the form known for Fickian diffusion, but with a new homoclinic. A nonlocal eigenvalue problem of a new type is formulated and studied. The system is less stable due to the Lévy flights, though the behavior of eigenvalues is changed mainly quantitatively.