Summary: The asymptotic behaviour of effective diffusion for a parabolic equation in \(\mathbb{R}^ n\) with periodic potential and small initial diffusion is discussed. The potential is assumed to be localized in periodic islands - - sets around points of the integer lattice in \(\mathbb{R}^ n\), where the density of diffusing particles increases. Off these islands the particles are annihilated. Logarithmic asymptotics of the effective diffusion are found when the initial diffusion tends to zero in terms of the geometrical characteristics of the given potential. These results rely on the large deviation technique for the diffusion of respective particles. For symmetric islands, the logarithmic asymptotics of the effective diffusion depend only on the distance between the islands.