  A feature common to many models of vegetation pattern formation in semi-arid
ecosystems is a sequence of qualitatively different patterned states, "gaps ->
labyrinth -> spots", that occurs as a parameter representing precipitation
decreases. We explore the robustness of this "standard" sequence in the generic
setting of a bifurcation problem on a hexagonal lattice, as well as in a
particular reaction-diffusion model for vegetation pattern formation.
Specifically, we consider a degeneracy of the bifurcation equations that
creates a small bubble in parameter space in which stable small-amplitude
patterned states may exist near two Turing bifurcations. Pattern transitions
between these bifurcation points can then be analyzed in a weakly nonlinear
framework. We find that a number of transition scenarios besides the standard
sequence are generically possible, which calls into question the reliability of
any particular pattern or sequence as a precursor to vegetation collapse.
Additionally, we find that clues to the robustness of the standard sequence lie
in the nonlinear details of a particular model.
