Stochastic desertification

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. Here we study a minimal model of a first-order transition with an absorbing state. When the external parameter varies adiabatically the transition is indeed continuous, and we present some empirical evidence that supports this scenario. The front velocity renormalizes to zero at the extinction transition, leaving a finite "quantum" region where domain walls are stable and the desertification takes place via accumulation of local extinctions. A catastrophic regime shift may occur as a dynamical hysteresis, if the pace of environmental variations is too fast.