Diversity of Vegetation Patterns and Desertification

Continuum Models and Discrete Systems, 2004

A large-scale view of arid regions often shows that the vegetation grows in patterns. These are related to the amount of precipitation as well as to the topography. A model is presented that reproduces the wide range of patterns observed in water-limited regions, from bare soil at very low precipitation to uniform cover at high precipitation, through intermediate states of spot-, stripe-and hole-patterns. The model predicts the coexistence of more than one stable state in a given range of precipitation. The results of the model lead to an understanding of the hysteretic nature of desertification, and to a new approach to the classification of aridity.

A continuum model for vegetation patterns in water limited systems is presented. The model involves two variables, the vegetation biomass density and the soil water density, and takes into account positive feedback relations between the two. The model predicts transitions from bare-soil at low precipitation to homogeneous vegetation at high precipitation through intermediate states of spot, stripe and gap patterns. It also predicts the appearance of ring-like shapes as transient forms toward asymptotic stripes. All these patterns have been identified in observations made on two types of perennial grasses in the Northern Negev. Another prediction of the model is the existence of wide precipitation ranges where different stable states coexist, e.g. a bare soil state and a spot pattern, a spot pattern and a stripe pattern, and so on. This result suggests the interpretation of desertification followed by recovery as an hysteresis loop and sheds light on the irreversibility of desertification.

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2013

Drylands are pattern-forming systems showing self-organized vegetation patchiness, multiplicity of stable states and fronts separating domains of alternative stable states. Pattern dynamics, induced by droughts or disturbances, can result in desertification shifts from patterned vegetation to bare soil. Pattern-formation theory suggests various scenarios for such dynamics; an abrupt global shift involving a fast collapse to bare soil, a gradual global shift involving the expansion and coalescence of bare-soil domains, and an incipient shift to a hybrid state consisting of stationary bare-soil domains in an otherwise periodic pattern. Using models of dryland vegetation we address the question which of these scenarios can be realized. We found that the models can be split into two groups: models that exhibit multiplicity of periodic-pattern and baresoil states, and models that exhibit, in addition, multiplicity of hybrid states. Furthermore, in all models we could not identify parameter regimes in which bare-soil domains expand into vegetated domains. The significance of these findings is that while models belonging to the first group can only exhibit abrupt shifts, model belonging to the second group can also exhibit gradual and incipient shifts. A discussion of open problems concludes the paper.

Journal of Geophysical Research, 2007

Abrupt transitions between large-scale grassland and desert in arid and semiarid regions have been observed in nature and reproduced by modeling studies. Observations also show the existence of nonuniform fine-scale vegetation patterns along the transition zone. This paper attempts to better understand these observations from two very different spatial scales. By explicitly introducing horizontal interaction terms into our previous dynamical grassland model, vegetation patterns with high diversities are found in the transition zone, and the system possesses an infinite number of equilibrium states in response to a given climatic forcing. The transition can be elucidated in two ways. In terms of the vegetation formations, the ecosystem undergoes the transition from uniform grassland to regular and irregular vegetation patterns, and then to pure desert as the moisture index (i.e., the ratio of precipitation over potential evaporation) decreases. In terms of biomass, the transition from grassland to desert goes through a narrow range of moisture index under which grassland is most fragile, as indicated by erratic vegetation patterns and large variation of average biomass. The existence of this range, however, has not been reported in previous modeling studies, and still needs to be validated using observational data.

Hydrology and Earth System Sciences, 2007

The interaction between vegetation and hydrologic processes is particularly tight in water-limited environments where a positive-feedback links soil moisture and vegetation. The vegetation of these systems is commonly patterned, that is, arranged in a two phase mosaic composed of patches with high biomass cover interspersed within a lowcover or bare soil component. These patterns are strongly linked to the redistribution of runoff and resources from source areas (bare patches) to sink areas (vegetation patches) and play an important role in controlling erosion.

International Journal of Quantum Chemistry, 2004

The large-scale vegetation patterns observed in many arid regions are due to the existence of facilitative and competitive interactions that affect the communal development of plants. For the patterns to form, it is necessary that competitive interactions be of longer range than facilitative interactions. Aridity affects the pattern's symmetry properties. As it increases, one first finds patterns constituted of spots of sparser vegetation, which then transform into an alternation of stripes of sparser and thicker vegetation, and finally into a pattern of vegetation spots separated by bare ground. The model and nonlinear analysis presented below explains the observations.

This work proposes a new mathematical model for reproducing desertification and vegetation patterns. The model consists of two nonlinear partial differential equations. One of them describes the Spatio-temporal dynamic of vegetation in an analogous way to Lefever’s model, while the precipitation dynamic is given by one equation of Hardenberg’s model. The model’s equations are solved using a numerical-functional difference method for the Spatio-temporal terms. The numerical results reproduce various bi-dimensional (2D) patterns observed in water-limited regions, including stripes, spots, hollows, and labyrinths. 2D patterns with these morphologies are characterized by their Fourier spectra and quantified their dimension fractal. The numerical solutions of the model also predict transitions from bare soil at low precipitation to homogeneous vegetation at high rainfall. These results reveal an underlying mechanism for the local desertification process and the vegetation self-organizati...

Mathematical Modelling of Natural Phenomena, 2011

The discovery of nearly periodic vegetation patterns in arid and semi-arid regions motivated numerous model studies in the past decade. Most studies have focused on vegetation pattern formation, and on the response of vegetation patterns to gradients of the limiting water resource. The reciprocal question, what resource modifications are induced by vegetation pattern formation, which is essential to the understanding of dryland landscapes, has hardly been addressed. This paper is a synthetic review of model studies that address this question and the consequent implications for inter-specific plant interactions and species diversity. It focuses both on patch and landscape scales, highlighting bottom-up processes, where plant interactions at the patch scale give rise to spatial patterns at the landscape scale, and top-down processes, where pattern transitions at the landscape scale affect inter-specific interactions at the patch scale.

Ecological Studies, 1992