Soil degradation

Dual stability of eqn [1] may not only be caused by the particular functional response of the grazer, but also by the shape of the vegetation growth function, which may depend on soil conditions or degradation. W. H. Schlesinger formulated at the beginning of the 1990s a conceptual model of biological feedbacks in desertification. Grazing and trampling may initiate feedback processes in water redistribution and infiltration which transform formerly homogeneous vegetation to heterogeneous vegetation distributed in patches of nutrients (islands of fertility) surrounded by a matrix of bare soil. A similar idea was followed in the end of 1990 by J. van den Koppel, M. Rietkerk, and colleagues. They developed a soil water dynamics submodel to derive a biomass growth function G(V) depending on precipitation, several soil properties, and standing growth biomass V. Under the assumption that the dynamics of soil water acts on a much faster timescale than plant growth they found

where PPT stands for rainfall, W0 is the minimum water infiltration, which is reached in the absence of plants, and k, u, rw, h, and l are site-specific constants regulating the water dynamics. The two most important parameters within this model are PPT and W0. If the value of W0 is low, biomass growth is a hump-shaped function of standing biomass. For example, in sparsely vegetated areas (low biomass), water is lost due to runoff and, consequently,

Figure 3 Multiple stable states due to soil degradation causing reduced biomass growth at small biomass values (eqn [3]). Herbivore consumption is linearly related to standing biomass (green lines). With no or moderate soil degradation, only a single stable equilibrium exists. However, under severe soil degradation growth is strongly limited at low biomass values which results in a positive feedback causing multiple stability (gray solid line). Open circles indicate unstable equilibria and filled circles stable equilibria.

Figure 3 Multiple stable states due to soil degradation causing reduced biomass growth at small biomass values (eqn [3]). Herbivore consumption is linearly related to standing biomass (green lines). With no or moderate soil degradation, only a single stable equilibrium exists. However, under severe soil degradation growth is strongly limited at low biomass values which results in a positive feedback causing multiple stability (gray solid line). Open circles indicate unstable equilibria and filled circles stable equilibria.

biomass growth is close to zero. Biomass growth is maximal at intermediate standing biomass and decreases at high biomass due to increased competition (bold gray line in Figure 3). In the case of greater W0, little water is lost due to run off. Therefore, growth is assumed to be maximal at low standing biomass and decreases thereafter due to density-dependent regulation (Figure 3).

Assuming a linear relation between consumption and biomass, the resulting model shows two equilibria, one at the unvegetated state and one at a vegetated state. Without soil degradation, only the vegetated state is possible (Figure 3) and vegetation changes are always reversible. However, irreversible vegetation shifts may occur under intermediate stocking rates and at a hump-shaped growth function assumed for degraded soils (Figure 3 , gray solid line). The underlying mechanism for dual stability is a negative feedback between grazing and plant growth in degraded soils. If standing biomass is severely reduced due to overgrazing, plant growth cannot equilibrate consumption and the system approaches the stable unvegetated equilibrium.

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