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...
Catastrophic shifts are known to pose a serious threat to ecology, and a reliable set of early warning indicators is desperately needed. However, the tools suggested so far have two problems. First, they cannot discriminate between a smooth transition and an imminent irreversible shift. Second, they aimed at predicting the tipping point where a state loses its stability, but in noisy spatial system the actual transition occurs when an alternative state invades. Here we suggest a cluster tracking technique that solves both problems, distinguishing between smooth and catastrophic transitions and to identify an imminent shift in both cases. Our method may allow for the prediction, and thus hopefully the prevention of such transitions, avoiding their destructive outcomes.
The demographic (shot) noise in population dynamics scales with the square root of the population size. This process is very important, as it yields an absorbing state at zero field, but simulating it, especially on spatial domains, is a nontrivial task. Here, we analyze two similar methods that were suggested for simulating the corresponding Langevin equation, one by Pechenik and Levine and the other by Dornic, Chaté, and Muñoz (DCM). These methods are based on operator-splitting techniques and the essential difference between them lies in which terms are bundled together in the splitting process. Both these methods are first order in the time step so one may expect that their performance will be similar. We find, surprisingly, that when simulating the stochastic Ginzburg-Landau equation with two deterministic metastable states, the DCM method exhibits two anomalous behaviors. First, the stochastic stall point moves away from its deterministic counterpart, the Maxwell point, when decreasing the noise. Second, the errors induced by the finite time step are larger by a significant factor (i.e., >10×) in the DCM method. We show that both these behaviors are the result of a finite-time-step induced shift in the deterministic Maxwell point in the DCM method, due to the particular operator splitting employed. In light of these results, care must be exercised when computing quantities like phase-transition boundaries (as opposed to universal quantities such as critical exponents) in such stochastic spatial systems.
The process of desertification in the semi-arid climatic zone is considered by many as a catastrophic regime shift, since the positive feedback of vegetation density on growth rates yields a system that admits alternative steady states. Some support to this idea comes from the analysis of static patterns, where peaks of the vegetation density histogram were associated with these alternative states. Here we present a large-scale empirical study of vegetation dynamics, aimed at identifying and quantifying directly the effects of positive feedback. To do that, we have analyzed vegetation density across 2.5 × 106 km2 of the African Sahel region, with spatial resolution of 30 × 30 meters, using three consecutive snapshots. The results are mixed. The local vegetation density (measured at a single pixel) moves towards the average of the corresponding rainfall line, indicating a purely negative feedback. On the other hand, the chance of spatial clusters (of many “green” pixels) to expand in the next census is growing with their size, suggesting some positive feedback. We show that these apparently contradicting results emerge naturally in a model with positive feedback and strong demographic stochasticity, a model that allows for a catastrophic shift only in a certain range of parameters. Static patterns, like the double peak in the histogram of vegetation density, are shown to vary between censuses, with no apparent correlation with the actual dynamical features. Our work emphasizes the importance of dynamic response patterns as indicators of the state of the system, while the usefulness of static modality features appears to be quite limited.
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