  We consider localized states in a discrete bistable Allen-Cahn equation. This
model equation combines bistability and local cell-to-cell coupling in the
simplest possible way. The existence of stable localized states is made
possible by pinning to the underlying lattice; they do not exist in the
equivalent continuum equation.
  In particular we address the existence of 'isolas': closed curves of
solutions in the bifurcation diagram. Isolas appear for some non-periodic
boundary conditions in one spatial dimension but seem to appear generically in
two dimensions. We point out how features of the bifurcation diagram in 1D help
to explain some (unintuitive) features of the bifurcation diagram in 2D.
