We study the dynamics of an interior spike of the Gierer--Meinhardt system. Under certain assumptions on the domain size, the diffusion coefficients, and the decay rates, we prove that the velocity of the center of the spike is proportional to the negative gradient of R(ξ,ξ), where R(x,ξ) is the regular part of the Green's function of the Laplacian with the Neumann boundary condition. Hence, an interior spike moves towards local minima of R(ξ,ξ) and therefore stays as an interior spike forever. This dynamics is fundamentally different from that of the shadow Gierer--Meinhardt system where an interior spike moves towards the closest point on the boundary.
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Chen et al. (2001) studied this question.