The process of desertification is usually modeled as a first order transition, where a change of an external parameter (e.g. participation) leads to a discontinuous saddle-node bifurcation followed by an ecological regime shift. However, a stochastic birth-death process of shrubs or trees with an absorbing state supports only a second order continuous transition with no hysteresis. We study a minimal model of first order transition in which one of the locally stable states is absorbing. When the external parameter varies adiabatically the transition is indeed continuous and belongs to the directed percolation universality class 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 may occur as dynamical hysteresis, if the rate of environmental variations is too fast.