Bistability and Catastrophic Collapse: Thermodynamics Analysis of Desertification



Extended abstract
1- Introduction
Gradual approach to desertification has been caused the means of desertification involves the land degradation in arid and semi-arid areas. In this view the border of equilibrium, non-equilibrium, stable and unstable of ecosystem to detect threshold and ecosystem collapse are undetermined. The opposite of this view arises from thermodynamic approach to desertification which backs from flow-energy changes and equilibrium. In this paper has been explained the system theory to desertification based on thermodynamics analysis. The main of this paper is expression of ecosystem collapse and thresholds based on tipping points and critical transitions. The results of this research have been led to new epistemology for desertification analysis.

Catastrophic Changes and Tipping Points
The equilibrium state of a system can respond in different ways to changes in conditions such as exploitation pressure or temperature rise (Fig 2 a, b, c). If the equilibrium curve is folded backwards
(Fig 2 c, d), three equilibria can exist for a given condition. The grey dotted arrows in the plots indicate the direction in which the system moves if it is not in equilibrium (that is, not on the curve). It can be seen from these arrows that all curves represent stable equilibria, except for the dashed middle section in Fig 2 c, d. If the system is driven slightly away from this part of the curve, it will move further away instead of returning.
Hence, equilibria on this part of the curve are unstable and represent the border between the basins of attraction of the two alternative stable states on the upper and lower branches. If the system is very close to a fold bifurcation point (for example point F1 or point F2), a tiny change in the condition may cause a large shift in the lower branch (Fig 2 c). Also, close to such a bifurcation a small perturbation can drive the system across the boundary between the attraction basins (Fig 2 d).
Thus, those bifurcation points are tipping points at which a tiny perturbation can produce a large transition.

Slowing Down Theory
A simple way to understand why we should expect early warnings before critical transitions is to think of the behavior of a system as the motion of a ball in a landscape of valleys and hilltops (Fig 3).
Balls represent the state of the system. Valleys correspond to the basins of attraction of the two alternative stable states of the system. The width and the steepness of the basin of attraction determine the capacity of the system to absorb a perturbation without shifting to an alternative state, and reflect the resilience of the state of the system. As conditions bring the system close to a critical transition (critical threshold1), the basin of attraction of the current states of the system shrinks and so does its resilience: even a tiny perturbation is enough to shift the sphere to the alternative valley. At the same time, the steepness of the basin of attraction becomes lower: this means that the same perturbation that may not tip the system, it will definitely take longer to dissipate due to the phenomenon of critical slowing down (Fig 3 b, c). Mathematically, critical slowing down is connected to the fact that close to the critical transition the dominant Eigen value of the system at equilibrium vanishes. Practically, this approach enables us to probe the dynamics of the system in order to assess its resilience and the risk of an upcoming transition.

The hydrographic, vegetation and ripple mark patterns are same regular and irregular patterns found in the arid ecosystems. There are particular spatial patterns that can arise before a critical transition.
In dry regions self-organization can lead to particular spatial patterns under some conditions. Here the complete loss of vegetation is an important transition, as recovery from the barren state may require more rain than is needed to preserve the last patches. There is good evidence to support the idea that a regular pattern characterized by spots of vegetation signals the proximity of a threshold to such catastrophic desertification.
Vegetation pattern dynamics can be a sign to forecast desertification and ecosystem stability. In arid ecosystems, the number of vegetation patches appears as a straight line when plotted as a function of their size on logarithmic scales. These ecosystems exhibit many small patches and progressively rarer large ones, and they show no characteristic patch size. However, arid ecosystems that experienced high grazing pressures show a deficiency in large patches compared with power laws. Therefore, it has been proposed that deviations from pure power laws towards fewer large patches could serve as indicators of the proximity to a desertification threshold. Despite the potential practical relevance of patch size distributions for the management and preservation of spatially organized ecosystems, the ecological mechanisms underlying the wide emergence of power laws and their deviations are not yet fully understood. Desertification and emerging desertified landscape is due to responses of ecosystem to environmental perturbations. This catastrophic process is with formation of regular and irregular patterns which are early warning signals to forecast desertification.