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Volume 15, Issue 4
Stability of Spike Solutions to the Fractional Gierer-Meinhardt System in a One-Dimensional Domain

Daniel Gomez, Jun-Cheng Wei & Wen Yang

Numer. Math. Theor. Meth. Appl., 15 (2022), pp. 938-989.

Published online: 2022-10

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  • 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.

  • AMS Subject Headings

35R11, 35B32, 60K50, 35B25

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COPYRIGHT: © Global Science Press

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@Article{NMTMA-15-938, author = {Gomez , DanielWei , Jun-Cheng and Yang , Wen}, title = {Stability of Spike Solutions to the Fractional Gierer-Meinhardt System in a One-Dimensional Domain}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2022}, volume = {15}, number = {4}, pages = {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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2022-0003s}, url = {http://global-sci.org/intro/article_detail/nmtma/21086.html} }
TY - JOUR T1 - Stability of Spike Solutions to the Fractional Gierer-Meinhardt System in a One-Dimensional Domain AU - Gomez , Daniel AU - Wei , Jun-Cheng AU - Yang , Wen JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 938 EP - 989 PY - 2022 DA - 2022/10 SN - 15 DO - http://doi.org/10.4208/nmtma.OA-2022-0003s UR - https://global-sci.org/intro/article_detail/nmtma/21086.html KW - Gierer-Meinhardt system, eigenvalue, stability, fractional Laplacian, localized solution. AB -

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.

Daniel Gomez, Jun-Cheng Wei & Wen Yang. (2022). Stability of Spike Solutions to the Fractional Gierer-Meinhardt System in a One-Dimensional Domain. Numerical Mathematics: Theory, Methods and Applications. 15 (4). 938-989. doi:10.4208/nmtma.OA-2022-0003s
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