Spike Pinning for the Gierer-Meinhardt Model
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Abstract:
The pinning eect induced by two dierent types of spatial inhomogeneities on the dynamics and equilibria of a one-spike solution to the one-dimensional Gierer-Meinhardt (GM) activatorinhibitor model of morphogenesis is studied. The rst problem that is treated is the shadow problem that results from taking the innite inhibitor diusivity limit in the GM model. For this problem, we show that an exponentially weak spatially varying activator diusivity can stabilize an equilibrium spike-layer solution that would necessarily be unstable when the activator diusivity was spatially uniform. The second problem that is treated is the full GM model in the presence of a spatially varying inhibitor decay rate. For this problem, we show that the equilibrium location of a one-spike solution depends on certain global properties of the inhibitor decay rate over the domain. 1
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