  This paper is devoted to the study of spatial propagation dynamics of species
in locally spatially inhomogeneous patchy environments or media. For a lattice
differential equation with monostable nonlinearity in a discrete homogeneous
media, it is well-known that there exists a minimal wave speed such that a
traveling front exists if and only if the wave speed is not slower than this
minimal wave speed. We shall show that strongly localized spatial inhomogeneous
patchy environments may prevent the existence of transition fronts (generalized
traveling fronts). Transition fronts may exist in weakly localized spatial
inhomogeneous patchy environments but only in a finite range of speeds, which
implies that it is plausible to obtain a maximal wave speed of existence of
transition fronts.
