Self-organized vegetation patterns: Early warning signals for predicting ecosystem transitions

The significance of spatial heterogeneity in understanding ecological processes has been recognized long ago. One of the earliest expressions of this recognition is the habitat heterogeneity hypothesis that links spatial  heterogeneity to niche vegetation patterns formation and species coexistence. Yet, an  important if not crucial  aspect of landscape heterogeneity has escaped deep consideration, that is, the possible occurrence of spatial instabilities leading to self-organized heterogeneity. Self-organized heterogeneity or pattern formation, is ubiquitous in the nature. In theory, spatial patterns may provide more powerful leading indicators, as they contain more information than a single data point in a time-series. For systems that have self-organized patterns formation, there are specific signals. However, these signals tend to be specific to the particular mechanism involved and cannot be generalized to other systems. The interaction between vegetation and hydrologic processes is particularly tight in water-limited environments where a positive feedback links water redistribution 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 low-cover 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 (runoff-run-on mechanism). Disturbances of  such as overgrazing or aridity, can alter the structure of vegetation patterns reducing its density and or size which leads to a “leaky” system. A leaky system is less efficient at trapping  runoff and sediments and loses valuable water and nutrient resources, inducing a positive-feedback loop that reinforces the degradation process. The most common vegetation pattern found in arid and semi-arid ecosystems is usually referred to as spotted or stippled and consists of dense vegetation clusters that are irregular in shape and surrounded by bare soil. Another common pattern is banded vegetation, also known as “tiger bush”, in which the dense biomass patches form bands, stripes or arcs. Banded vegetation is usually aligned along contour lines and is effective in limiting hillslope erosion. Banded patterns commonly act as closed  hydrological systems, with little net outflow and sediment coming out of the system . The effect of spotted vegetation on erosion is more complex and depends on the connectivity of the bare soil areas. Depending on the spatial mechanisms that dominate in arid ecosystems, particular changes in spatial patterns may signal whether vegetation is close to collapsing into bare ground. In during the past few decades have used from mathematical countinume models for evaluation tend of vegetation pattern as, an early warning singnal for predicting desertification transitions in the arid ecosystems. In present paper, describes interaction between vegetations nonlinear dynamics, environmental disturbances and different vegetation patterns according to countinume model of GILAD. Analysis of vegetation patterns can be helpful in understanding desertification.
1-      Materials & Methods
2-1- Vegetation dynamic Models
There is a variety of models for the simulation of vegetation dynamics in water-limited ecosystems. Recent models that capture the interaction between spatial water redistribution and vegetation patterns can be divided in two main groups: First models, discrete or individual-based models and Second models, continuum models or partial-differential-equations (PDEs) models. Discrete models are numerical algorithms that go down to the level of individual plants and often describe them in great detail. Continuum models, consist of spatially continuous variables satisfying sets of coupled PDEs. This models capable of describing continuous processes such as overland water flow, soil-water dynamics, erosion-deposition processes, etc. PDE models are reaction – diffusion equations, that is, systems of PDE that combine linear diffusion with nonlinear interactions. From the implementation point of view discrete models are formulated in terms of algorithms that are executed by numerical computations, whereas continuum PDEs models are amenable to mathematical analysis besides numerical computations. The continuum PDEs  models, are powerful tools for studying of pattern formation theory.
2-2- Vegetation Pattern formation Based on the GILAD model
Vegetation pattern formation Based on the GILAD model is a result of positive feedbacks operating at local scale. We focus here on two important feedbacks. The first, is a positive feedback between above-ground and below-ground biomass (hereafter the root-augmentation feedback) and is related to the root-to-shoot ratio, a  characteristic trait of plant species (Fig.1a). As a plant grows its root zone extends to new soil regions where  water can be taken up from. As a result the amount of water available to the plant increases and the  plant grows even further. The second feedback, is a positive feedback between biomass and water (hereafter the infiltration feedback) (Fig. 2b). Bare soils in arid regions are often covered by biological soil crusts which  reduce the infiltration rate of surface water into the soil relative to the infiltration rate in vegetation patches. As a consequence vegetation patches act as sinks for runoff water generated by their crusted neighborhoods. This accelerates their growth, sharpens the infiltration contrast and increase even further the soil moisture in the patch areas. Soil erosion in bare areas and deposition in vegetation  patches is another mechanism that can induce or enhance infiltration contrast.
 
Fig. 1: Schematic illustrations of  the root-augmentation feedback (a) and the infiltration feedback  (b).
1-       Discussion &  Results
The model discussed this article, shows that the resource concentration mechanism predicts global bistability associated with catastrophic shifts at large spatial scales and self-organized patterns. A variety of mechanisms in ecosystems lead to resource concentration through consumer-resource feedback. The consumers, harvest resources from their surroundings and harvest are facilitated by mass flow of resources toward consumers, triggered by the consumers themselves. Furthermore, consumers spread relatively slowly as compared to flow of resources. A general pattern emerging from these interactions is that consumers are positively associated with resource abundance at short spatial range, but negatively at long spatial range. Thus, a common principle applies to these locally reinforced consumers, in that there is a positive feedback effect that is short-ranged and a negative feedback effect that is long-ranged. This is a necessary condition for self-organized patterns to form. Such scale-dependent feedback can explain a diversity of patterns in ecosystems. The differences in structure and scale of patterns are the result only of varying strength and scale of feedback influence, illustrating the general nature of the underlying scale-dependent mechanisms explaining self-organized patterns in ecosystems. The notion of scale-dependent feedback controlled by the resource concentration mechanism is crucial for a predictive theory of catastrophic shifts in ecosystems. This suggests that catastrophic shifts can be predicted by self-organized patterns. Therefore, the concepts of catastrophic shifts and self-organized patterns are tightly linked, whereby a scale-dependent feedback is triggered by resource concentration. This mechanism predicts global bistability and catastrophic shifts between spotted and uniform states. Vegetation pattern formation theory and  appearance of spatially mixed and intermediate patterns in bistability systems of uniform and spatially  periodic states )The multitude of intermediate states in the bistability range of bare soil and a periodic  spot pattern( according to GILAD model, suggests that desertification may not necessarily be abrupt, but rather a gradual process.
2-      Conclutions
Geomorphic systems are typically nonlinear, owing largely to their threshold-dominated nature. Nonlinear geomorphic systems may exhibit complex behaviors not possible in linear systems, including dynamical instability and deterministic chaos. Linking self-organized patterns with catastrophic shifts by the resource concentration mechanism may help to bridge the present gaps among theory, observation, and management. The self-organized patterns may indicate imminent shifts if resource input decreases in time. For instance, the spotted state may develop only when resource input is decreased, not when it is increased. This means that a snapshot in time of a spotted state would already indicate imminent catastrophic shift. Applications of the pattern formation approach to water-limited landscapes predict the possible emergence of spatial heterogeneity as a self-organization phenomenon. The predicted spatial patterns can be periodic (spots, stripes and gaps), irregular with a characteristic length scale, or scale free. The pattern formation approach provides clear criteria for the realizations of these different pattern types in terms of environmental conditions, such as precipitation rate, infiltration rate, water-ground  friction force, topography and disturbances, and in terms of species traits, such as biomass growth rate, uptake rate and root-to-shoot ratio. Three values of the pattern formation modeling approach have been  emphasized: (1) It  reveals  universal elements such as instabilities, bistability ranges, and resonant  responses, for which a great deal of  knowledge is already available. (2) It captures processes across  different length scales and organization levels and adaptive response to environmental changes. (3) It provides an integrative framework for studying problems in spatial ecology, coupling aspects of landscape, population, community and restoration ecology.