Spike‐Type Solutions to One Dimensional Gierer–Meinhardt Model with Lévy Flights

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.