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Spike‐Type Solutions to One Dimensional Gierer–Meinhardt Model with Lévy Flights
Author(s) -
Nec Y.
Publication year - 2012
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2012.00550.x
Subject(s) - eigenvalues and eigenvectors , spike (software development) , homoclinic orbit , mathematics , type (biology) , differential equation , stochastic differential equation , diffusion , mathematical analysis , lévy flight , physics , mathematical physics , bifurcation , random walk , thermodynamics , nonlinear system , statistics , quantum mechanics , ecology , management , economics , biology
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.