Vegetation clustering and self-organization in inhomogeneous environments

Abstract: Due to climatic changes, excessive grazing, and deforestation, semi-arid and arid ecosystems are vulnerable to desertification and land degradation. Adversely affected biological productivity has a negative impact on social, economic, and environmental factors. The term 'arid ecosystems' refers not only to water-scarce landscapes but also to nutrient-poor environments. Specifically, the vegetation cover loses spatial homogeneity as aridity increases, and the self-organized heterogeneous vegetation patterns developed could eventually collapse into a bare state. It is still unclear whether this transition would be gradual or abrupt, leading to the often-called catastrophic shift of ecosystems. Several studies suggest that environmental inhomogeneities in time or space can promote a gradual transition to bare soil, thus avoiding catastrophic shifts. Environmental inhomogeneities include non-uniformities in the spatial distribution of precipitation, spatial irregularities in topography and other external factors. Employing a generic mathematical model including environmental inhomogeneities in space, we show how two branches of vegetation patterns create a hysteresis loop when the effective mortality level changes. These two branches correspond to qualitatively distinct vegetation self-organized responses. In an increasing mortality scenario, one observes an equilibrium branch of high vegetation biomass forming self-organized patterns with a well-defined wavelength. However, reversing the mortality trend, one observes a low biomass branch lacking a wavelength. We call this phenomenon the clustering of vegetation patches. This behavior can be connected to historically significant trends of climate change in arid ecosystems.

• The paper demonstrates that environmental inhomogeneities induce bifurcation in vegetation patterns, leading to both self-organized periodic structures and aperiodic clustering without a characteristic wavelength.
• It integrates empirical observations, Fourier analyses, and reaction-diffusion modeling to map hysteresis in vegetation response and predict transitions toward ecosystem collapse.
• The findings suggest that incorporating spatial disorder in ecological models provides improved tools for early-warning indicators and desertification mitigation strategies.

Vegetation Clustering and Self-Organization in Inhomogeneous Environments

Introduction

This work presents a combined empirical, numerical, and theoretical investigation into the mechanisms driving the emergence of patchy vegetation patterns in semi-arid and arid landscapes, focusing on the impact of environmental inhomogeneities. The authors analyze field data, remote sensing, and mathematical models to elucidate how spatial disorder—such as topographic variation and precipitation heterogeneity—facilitates two distinct forms of spatial organization: self-organized periodic patterns and clustered, disordered arrangements, the latter lacking a characteristic wavelength. The study offers strict evidence that environmental inhomogeneities fundamentally alter the bifurcation structure of the vegetation cover response to aridity, producing a broad hysteresis between self-organization and clustering, and, crucially, smoothing the transition toward ecosystem collapse.

Field Observations and Global Classification

Empirical analyses were conducted in arid highland systems in Morocco and Argentina, focusing on plant species from the Poaceae family, Stipa tenacissima L. and Festuca orthophylla. These systems, despite similar climatic and structural metrics (notably, the facilitation-to-competition length scale ratio near $1/3$), exhibit markedly different patterning. Spatial data are processed through binarization and Fourier analysis to distinguish ordered from clustered patterns. Stipa tenacissima exhibits vegetation clustering with a dominant $q=0$ Fourier mode, while Festuca orthophylla displays clear periodic structure with a pronounced ring in the 2D spectrum—demonstrating the impact of fine-scale topographic irregularities and microclimate variability on biomass spatial organization.

Figure 1: Examples of both clustered and self-organized vegetation patches in Morocco and Argentina, with Fourier analyses indicating the presence or absence of a characteristic wavelength.

A global survey utilizing high-resolution remote sensing extends this classification. Eight ecosystems worldwide are systematically analyzed (U.S., Argentina, Zambia, Mozambique, India, Australia, Morocco). By examining the mean profile of the 2D Fourier spectrum and pair correlation functions, the paper segments sites into those supporting self-organized periodic patterns and those exhibiting aperiodic clustering, providing a mechanistic global framework for understanding pattern type as a function of both present and historical aridity.

Figure 2: Global classification of vegetation landscapes utilizing remote sensing, Fourier analysis, and pair correlation—distinguishing between self-organized and clustered states.

Mapping Environmental Drivers of Pattern Transitions

To connect observed pattern types to environmental history, the study harnesses climate databases (1979–2018) to derive aridity trends, using mean annual precipitation and reference potential evapotranspiration. It is established that increasing aridity drives ecosystems toward structured self-organization, while decreasing aridity is associated with disorderly clustering. Exceptions (e.g., Mozambique) are discussed, emphasizing the sensitivity of pattern type to additional meteorological and ecological variables.

Figure 3: Mapping the spatial rate of aridity change; blue regions denote decreasing aridity associated with clustering, while red regions highlight increasing aridity supporting self-organization.

Theoretical Modelling: From Homogeneous to Inhomogeneous Regimes

The authors extend the reaction-diffusion "interaction-redistribution" PDE framework to incorporate spatial disorder, promoting key model parameters (especially mortality $\eta$) to stochastic fields. The mean-field, homogeneous limit recapitulates classical sequence and abrupt, catastrophic collapse as adversity increases. Pattern transitions are characterized through bifurcation diagrams of mean biomass versus mortality, revealing the expected sequence: homogeneous cover, labyrinthine gaps, hexagonal patches, and bare state. Spatial organization in this regime is always periodic, evident in both direct imaging and Fourier spectra.

Figure 4: Bifurcation structure for homogeneous environments, progressing from homogeneous to gap, stripe, and patch patterns prior to abrupt collapse.

Introducing static spatial disorder in mortality ($\eta(\mathbf{r})$) induces two solution branches comprising a hysteresis loop—a primary result of the paper. Increasing mean mortality leads to periodic, self-organized patch formation, characterized by a dominant nonzero wavenumber. However, as mortality is reduced (or aridity decreases), the system transitions to a low-density, aperiodically clustered state, devoid of characteristic wavelength, closely matching empirical observations in field data and remote sensing.

Figure 5: Numerical exploration of pattern regimes with spatially disordered mortality. Upper branch: periodic, self-organized patterns; lower branch: aperiodic clustering, with corresponding Fourier and correlation signatures.

A crucial assertion is that inhomogeneities not only broaden the range of pattern persistence under adversity (rendering the transition to bare soil continuous) but enable aperiodic clustering as a robust equilibrium outcome, not only as a transient as in homogeneous systems. The interaction between local replication dynamics and variable environmental favorability leads to patch pinning and disordered spatial organization in the clustering regime.

Implications and Perspectives

The theoretical–empirical correspondence demonstrated, including the emergence of a hysteresis loop and a smoothening of catastrophic shifts, imposes a reinterpretation of resilience and critical transitions in arid ecosystem models. The observation that environmental disorder fundamentally modifies the pattern bifurcation structure implies that standard early-warning indicators (reliant on periodicity or abruptness) are insufficient in naturally disordered terrains. Additionally, the use of 2D Fourier and pair-correlation analysis as a rapid, noninvasive classification tool for landscape state is validated.

Practical implications center on the assessment of ecosystem vulnerability and the formulation of strategies for desertification mitigation. The ability to infer environmental history from present spatial patterning offers a route to diagnose the process and reversibility of land degradation. Theoretically, the explicit treatment of environmental disorder suggests directions for the development of ecosystem models capable of more robust predictions under real-world inhomogeneities.

Future research should address multi-parameter spatial disorder, explore the influence of correlated disorder, and integrate further biologically relevant feedbacks (e.g., species interactions, temporal climate variability).

Conclusion

The paper establishes a rigorous link between environmental inhomogeneity and the spatial organization of dryland vegetation, demonstrating that static disorder induces robust clustering and precludes abrupt transitions to ecosystem collapse. The integrative methodology—combining field study, remote sensing, and generalized reaction-diffusion modeling—provides a framework to classify, understand, and potentially predict landscape transitions under climate change scenarios. The findings necessitate revised strategies in both fundamental ecosystem theory and applied management for dryland resilience.

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