  The dryland vegetation model proposed by Rietkerk and collaborators has been
explored from a bifurcation perspective in several previous studies. Our aim
here is to explore in some detail the bifurcation phenomena present when the
coefficients of the model are allowed to vary in a wide range of parameters. In
addition to the primary bifurcation parameter, the precipitation, we allow the
two infiltration rate parameters to vary as well. We find that these two
parameters control the size and stability of nonhomogeneous biomass states in a
way that can be predicted. Further, they control when certain homogeneous and
inhomogeneous (in space) periodic (in time) orbits exist. Finally, we show that
the model possesses infinitely many unphysical steady state branches. We then
present a modification of the model which eliminates these unphysical
solutions, and briefly explore this new model for a fixed set of parameters.
