Year: 2022
Author: Jun-Cheng Wei, Wen Yang, Daniel Gomez, Jun-Cheng Wei, Wen Yang
Numerical Mathematics: Theory, Methods and Applications, Vol. 15 (2022), Iss. 4 : pp. 938–989
Abstract
In this paper we consider the existence and stability of multi-spike solutions to the fractional Gierer-Meinhardt model with periodic boundary conditions. In particular we rigorously prove the existence of symmetric and asymmetric two-spike solutions using a Lyapunov-Schmidt reduction. The linear stability of these two-spike solutions is then rigorously analyzed and found to be determined by the eigenvalues of a certain $2 × 2$ matrix. Our rigorous results are complemented by formal calculations of $N$-spike solutions using the method of matched asymptotic expansions. In addition, we explicitly consider examples of one- and two-spike solutions for which we numerically calculate their relevant existence and stability thresholds. By considering a one-spike solution we determine that the introduction of fractional diffusion for the activator or inhibitor will respectively destabilize or stabilize a single spike solution with respect to oscillatory instabilities. Furthermore, when considering two-spike solutions we find that the range of parameter values for which asymmetric two-spike solutions exist and for which symmetric two-spike solutions are stable with respect to competition instabilities is expanded with the introduction of fractional inhibitor diffusivity. However our calculations indicate that asymmetric two-spike solutions are always linearly unstable.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2022-0003s
Numerical Mathematics: Theory, Methods and Applications, Vol. 15 (2022), Iss. 4 : pp. 938–989
Published online: 2022-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 52
Keywords: Gierer-Meinhardt system eigenvalue stability fractional Laplacian localized solution.