  Drylands are pattern-forming systems showing self-organized vegetation
patchiness, multiplicity of stable states and fronts separating domains of
alternative stable states. Pattern dynamics, induced by droughts or
disturbances, can result in desertification shifts from patterned vegetation to
bare soil. Pattern-formation theory suggests various scenarios for such
dynamics; an abrupt global shift involving a fast collapse to bare soil, a
gradual global shift involving the expansion and coalescence of bare-soil
domains, and an incipient shift to a hybrid state consisting of stationary
bare-soil domains in an otherwise periodic pattern. Using models of dryland
vegetation we address the question which of these scenarios can be realized. We
found that the models can be split into two groups: models that exhibit
multiplicity of periodic-pattern and bare-soil states, and models that exhibit,
in addition, multiplicity of hybrid states. Furthermore, in all models we could
not identify parameter regimes in which bare-soil domains expand into vegetated
domains. The significance of these findings is that while models belonging to
the first group can only exhibit abrupt shifts, model belonging to the second
group can also exhibit gradual and incipient shifts. A discussion of open
problems concludes the paper.
