  We consider the spatial Lambda-Fleming-Viot process model for frequencies of
genetic types in a population living in R^d, in the special case in which there
are just two types of individual, labelled 0 and 1. At time zero, everyone in
the half-space consisting of points whose first coordinate is non-positive is
type 1, whereas everyone in the complementary half-space is of type 0. We are
concerned with patterns of frequencies of the two types at large space and time
scales. We consider two cases, one in which the dynamics of the process are
driven by purely `local' events and one incorporating large-scale extinction
recolonisation events. We choose the frequency of these events in such a way
that, under a suitable rescaling of space and time, the ancestry of a single
individual in the population converges to a symmetric stable process of index
alpha in (1,2] (with alpha=2 corresponding to Brownian motion). We consider the
behaviour of the process of allele frequencies under the same space and time
rescaling. For alpha=2 and d>1 it converges to a deterministic limit. In all
other cases the limit is random and we identify it as the indicator function of
a random set. In particular, there is no local coexistence of types in the
limit. We characterise the set in terms of a dual process of coalescing
symmetric stable processes, which is of interest in its own right. The complex
geometry of the random set is illustrated through simulations.
