  We propose a spatially continuous and temporally discrete model for pattern
formation in vertically vibrated granular layers. The grain transfer and the
grain mobility transitions are introduced qualitatively, but explicitly. This
model reproduces experimentally observed f/2 patterns, including squares in
high-mobility conditions and stripes in low-mobility conditions. Hexagons are
obtained when a high-mobility condition and a low-mobility condition are
alternately applied. We discuss this hexagon formation in connection with the
formation of squares and stripes. In addition to the experimentally observed
localized oscillon structures, this model also exhibits localized hexagonal
structures. These localized hexagonal structures grow by nucleating oscillons
of equal phase.
