(4th section of Main Article - Vegetation patterns and desertification)
The mathematical model presents a novel view of desertification and relates it to the phenomenon of hysteresis.
Hysteresis in physical systems
When ice is warmed or water is cooled down the material changes its state from solid to liquid or vice versa at a temperature of 0°C. This transition temperature does not depend on the direction of change - warming ice or cooling water. There are other circumstances where the transition point does depend on the direction of change. A known example is the magnetization of a ferromagnetic material by an external magnetic field. Here the transition point between "magnetization up" and "magnetization down" depends on whether the external field is increased or decreased. The property of a material by which the transition point between two phases or states depends upon the direction of change is known as hysteresis. This property is characteristic in various areas of the natural sciences, and is even present in the
art of M.C. Escher.
Hysteresis in vegetation systems
The phenomenon of hysteresis as it relates to vegetation is illustrated in Fig. 5. When precipitation is sufficiently small, bare soil is the only stable state - any attempts to grow vegetation will not succeed (i.e., planted seeds or saplings will dry up). If the availability of water is gradually raise, say by initiating irrigation, it will eventually reach a critical level, p+, at which vegetation will grow spontaneously and form a spot pattern. This vegetation pattern will be sustained not only if we further increase moisture but even if we reduce it. By gradually decreasing water availability a second critical level is reached, p- (smaller than p+), where the vegetation will dry up and there will be a return to bare soil. The phenomenon of hysteresis - namely, the different precipitation levels representing the transition from bare soil to spotted pattern and of spotted pattern to bare soil - occurs due to the coexistence of the two stable system states, bare soil and spotted pattern, over the precipitation range from p- to p+.
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Figure 5. A schematic illustration of hysteresis in vegetative systems. Solid and dashed lines represent stable and unstable states, respectively. The transition point between the two states "bare soil" and "spotted pattern" depends on the direction of change: upon increasing precipitation bare soil transforms to a spotted pattern at p+ (green arrow) whereas upon decreasing precipitation a spotted pattern transforms back to bare soil at a lower value, p- (brown arrow). |
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Desertification induced by climatic changes
Fig. 5 can help us understand the appearance of desertification following climatic changes, such as drought. Let us take a system characterized by a spot pattern and a precipitation level within the range of coexistence of spots and bare soil. Let us also imagine a situation in which precipitation drops below p-. Under this condition, the vegetation dries out, leaving bare soil. If now, the drought passes and precipitation returns to its previous level, the hysteresis effect will not allow vegetation to recover. Renewal of plant growth will only occur when the precipitation significantly improves and reaches the level of p+.
The picture of desertification made possible by the mathematical model clearly shows that this phenomenon derives from the coexistence of different stable vegetation patterns at identical precipitation levels. The susceptibility of the system to desertification and the degree of its reversibility are dependent on this understanding. The closer the initial precipitation level is to p- the more susceptible the area is to desertification. The further the initial precipitation level is from p+, the more irreversible the desertification.
Desertification induced by human activities
Desertification also arises as a result of human activity, such as livestock grazing. Animation 2 illustrates the existence of a limit of human induced disturbances over which a disturbed spotted pattern will not self-restore. Here too, the condition for this phenomenon is the coexistence of different stable states.
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