  Vegetation patterns are a characteristic feature of semi-deserts occurring on
all continents except Antarctica. In some semi-arid regions, the climate is
characterised by seasonality, which yields a synchronisation of seed dispersal
with the dry season or the beginning of the wet season. We reformulate the
Klausmeier model, a reaction-advection-diffusion system that describes the
plant-water dynamics in semi-arid environments, as an integrodifference model
to account for the temporal separation of plant growth processes during the wet
season and seed dispersal processes during the dry season. The model further
accounts for nonlocal processes involved in the dispersal of seeds. Our
analysis focusses on the onset of spatial patterns. The Klausmeier partial
differential equations (PDE) model is linked to the integrodifference model in
an appropriate limit, which yields a control parameter for the temporal
separation of seed dispersal events. We find that the conditions for pattern
onset in the integrodifference model are equivalent to those for the continuous
PDE model and hence independent of the time between seed dispersal events. We
thus conclude that in the context of seed dispersal, a PDE model provides a
sufficiently accurate description, even if the environment is seasonal. This
emphasises the validity of results that have previously been obtained for the
PDE model. Further, we numerically investigate the effects of changes to seed
dispersal behaviour on the onset of patterns. We find that long-range seed
dispersal inhibits the formation of spatial patterns and that the seed
dispersal kernel's decay at infinity is a significant regulator of patterning.
