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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.

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Papers
Copyright
© 2003 Cambridge University Press

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