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.PACS numbers: 87.10. Mn,87.23.Cc,64.60.Ht,05.40.Ca The catastrophic bifurcation and its statistical mechanics analog, the first order transition, play a central role in the physical sciences. In these processes a tiny change in the value of an external parameter leads to a sudden jump of the system from one phase to another. This change is irreversible and is accompanied by hysteresis: once the system relaxed to its new phase, it will not recover even when the external parameters are restored.The relevance of these processes to the ecology of population and communities has been established while ago [1]. Recently, there is a growing concern about the possible occurrence of regime shifts in ecological systems [2][3][4][5]. The anthropogenic changes of local and global environmental parameters from habitat fragmentation to the increasing levels of CO2 in the atmosphere -raise anxiety about the possibility of an abrupt and irreversible catastrophe that may be destructive to the functions and the stability of an ecosystem [6]. This concern triggered an intensive search for empirical evidence that may allow one to identify an impending tipping point, where the most popular suggestion is to use the phenomenon of critical slowing down [5,[7][8][9][10][11]. Other suggested early warning indicators, especially for sessile species, deal with spatial patterns and the level of aggregation [4,12,13] Of particular importance is the process of desertification, which is considered as an irreversible shift from the "active" vegetation state to the "inactive" bare soil state, resulting from an increased pressure (e.g., overgrazing, declines in precipitation). As drylands cover about 41% of Earth land surface, desertification affects about 250 million people around the world [14]. Various models show that, when the vegetation state has a positive feedback, like an increased runoff interception or reduced evaporation close to vegetation pa...