Highly organized vegetation patterns can be found in a number of landscapes around the world. In recent years, several authors have investigated the processes underlying vegetation pattern formation. Patterns that are induced neither by heterogeneity in soil properties nor by the local topography are generally explained as the result of spatial self‐organization resulting from “symmetry‐breaking instability” in nonlinear systems. In this case, the spatial dynamics are able to destabilize the homogeneous state of the system, leading to the emergence of stable heterogeneous configurations. Both deterministic and stochastic mechanisms may explain the self‐organized vegetation patterns observed in nature. After an extensive analysis of deterministic theories, we review noise‐induced mechanisms of pattern formation and provide some examples of applications relevant to the environmental sciences.

[1] Positive feedbacks between vegetation and soil moisture may induce, in arid ecosystems, the emergence of two alternative stable states, corresponding to bare and completely vegetated soil. A new analytical model is developed to investigate this behavior and to study the effects of interannual rainfall variability on the dynamics of bistable ecosystems. When dichotomic Markov noise is used to account for the effect of random rainfall fluctuations, the long-term dynamics of the system can be investigated through an analytical solution of the model. It is found that in a broad class of bistable ecosystems, random rainfall fluctuations may induce an ordered state in the dynamics, i.e., by turning the bistable deterministic system into a stochastic system with only one statistically stable state. This effect is enhanced by increases in noise intensity, whereas the stochastic dynamics become bistable (i.e., the noise-induced ordered state disappears) as the noise intensity decreases below a critical (nonnull) value. This effect of noiseinduced stability is found in association with an enhancement of ecosystem resilience, indicating that the likelihood of catastrophic shifts to the desert state decreases as the noise intensity increases.

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Soil water limitations cause water-stressed conditions in vegetation because of temporary and/or permanent damage to plant tissues. Vegetation water stress is here assumed to increase when soil moisture is below a threshold level, s Ł , and to decrease otherwise. The crossing properties of s Ł allow one to describe the temporal evolution of water stress, , as a dichotomic Markov process. The available analytical solutions for dichotomic processes are used to determine the probability density function of  and the mean first-passage times (MFPT) of a water stress threshold. MFTP is used as a measure of the variability of water stress. The use of MFPT in conjunction with the average stress levels allows us to provide a more complete characterization of health/stress conditions in vegetation. We investigate the influence on the dynamics of vegetation water stress of different climatic, pedological, and plant physiological parameters. From these analyses, we find, for example, that resistant species are favoured in relatively humid environments, whereas resilient shallow-rooted plants have an advantage in drier conditions.