  We consider aspects of the population dynamics, inside a bound domain, of
diffusing agents carrying an attribute which is stochastically destroyed upon
contact with the boundary. The normal mode analysis of the relevant Helmholtz
equation under the partially absorbing, but uniform, boundary condition
provides a starting framework in understanding detailed evolution dynamics of
the attribute in the time domain. In particular, the boundary-localized
depletion has been widely employed in practical applications that depend on
geometry of various porous media such as rocks, cement, bones, and cheese.
While direct relationship between the pore geometry and the
diffusion-relaxation spectrum forms the basis for such applications and has
been extensively studied, relatively less attention has been paid to the
spatial variation of the boundary condition. In this work, we focus on the way
the pore geometry and the inhomogeneous depletion strength of the boundary
become intertwined and thus obscure the direct relationship between the
spectrum and the geometry. It is often impossible to gauge experimentally the
degree to which such interference occur. We fill this gap by perturbatively
incorporating classes of spatially-varying boundary conditions and derive their
consequences that are observable through numerical simulations or controlled
experiments on glass bead packs and artificially fabricated porous media. We
identify features of the spectrum that are most sensitive to the inhomogeneity
and apply the method to the spherical pore with a simple hemi-spherical binary
distribution of the depletion strength and obtain bounds for the induced change
in the slowest relaxation mode.
