Network-mediated diffusion produces disordered self-organization in vegetation

Vegetation patterns in arid and semiarid regions emerge as a result of a self-organization process triggered by water scarcity. While highly regular patterns are well reproduced through reaction-diffusion models, the origin of commonly observed disordered patterns remains debated. Several studies have attributed them either to spatial heterogeneity, or to a global competition regime induced by domain scale water transport. Here, we propose a new model in which biomass and water diffuse through networks constructed by adding random links (shortcuts) onto a regular lattice. By reproducing the naturally occurring spatially heterogeneous co-existence of different transport scales, our model incorporates both aspects of previous explanations for the origin of disordered patterns. It also allows us to analyze our results through the lens of network theory. In fact, as the density of shortcuts increases, the diffusion networks transition from a regular lattice through a small world network to a random network, resulting in different pattern formation behaviors. On a regular lattice, high-regularity patterns develop reflecting local diffusion processes. On a random network, the system is dominated by domain scale diffusion yielding either a uniform state or a single patch. In the intermediate shortcut density range, on a small world network topology, the interaction between the two scales of diffusion generates two kinds of disordered patterns, both coherent with observations: low-regularity patterns with a well-defined characteristic wavelength, and irregular patterns characterized by a broad patch size distribution. Our model predicts that these two pattern types would vary significantly in their resilience to environmental pressure.