  The process of desertification is usually modeled as a first order
transition, where a change of an external parameter (e.g. precipitation) leads
to a catastrophic bifurcation followed by an ecological regime shift. However,
vegetation elements like shrubs and trees undergo a stochastic birth-death
process with an absorbing state; such a process supports a second order
continuous transition with no hysteresis. We present a numerical study of a
minimal model that supports bistability and catastrophic shift on spatial
domain with demographic noise and an absorbing state. When the external
parameter varies adiabatically the transition is continuous and the front
velocity renormalizes to zero at the extinction transition. Below the
transition one may identify three modes of desertification: accumulation of
local catastrophes, desert invasion and global collapse. A catastrophic regime
shift occurs as a dynamical hysteresis, when the pace of environmental
variations is too fast. We present some empirical evidence, suggesting that the
mid-holocene desertification of the Sahara was, indeed, continuous.
