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

Snaking and isolas of localized states in bistable discrete lattices