Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T23:24:57.602Z Has data issue: false hasContentIssue false

Reduced wave Green's functions and their effect on the dynamics of a spike for the Gierer–Meinhardt model

Published online by Cambridge University Press:  24 October 2003

THEODORE KOLOKOLNIKOV
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, Canada V6T 1Z2
MICHAEL J. WARD
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, Canada V6T 1Z2

Abstract

In the limit of small activator-diffusivity $\varepsilon$, a formal asymptotic analysis is used to derive a differential equation for the motion of a one-spike solution to a simplified form of the Gierer–Meinhardt activator-inhibitor model in a two-dimensional domain. The analysis, which is valid for any finite value of the inhibitor diffusivity $D$ with $D\,{\gg}\,\varepsilon^2$, is delicate in that two disparate scales $\varepsilon$ and ${-1/\ln\varepsilon}$ must be treated. This spike motion is found to depend on the regular part of a reduced-wave Green's function and its gradient. Limiting cases of the dynamics are analyzed. For $D$ small with $\varepsilon^2 \,{\ll}\, D \,{\ll}\, 1$, the spike motion is metastable. For $D\,{\gg}\, 1$, the motion now depends on the gradient of a modified Green's function for the Laplacian. The effect of the shape of the domain and of the value of $D$ on the possible equilibrium positions of a one-spike solution is also analyzed. For $D\,{\ll}\,1$, stable spike-layer locations correspond asymptotically to the centres of the largest radii disks that can be inserted into the domain. Thus, for a dumbbell-shaped domain when $D\,{\ll}\,1$, there are two stable equilibrium positions near the centres of the lobes of the dumbbell. In contrast, for the range $D\,{\gg}\,1$, a complex function method is used to derive an explicit formula for the gradient of the modified Green's function. For a specific dumbbell-shaped domain, this formula is used to show that there is only one equilibrium spike-layer location when $D\,{\gg}\,1$, and it is located in the neck of the dumbbell. Numerical results for other non-convex domains computed from a boundary integral method lead to a similar conclusion regarding the uniqueness of the equilibrium spike location when $D\,{\gg}\,1$. This leads to the conjecture that, when $D\,{\gg}\, 1$, there is only one equilibrium spike-layer location for any convex or non-convex simply connected domain. Finally, the asymptotic results for the spike dynamics are compared with corresponding full numerical results computed using a moving finite element method.

Type
Papers
Copyright
© 2003 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)