Dryland vegetation patterns: the impact of diffusion network topology

In numerous dryland regions, the vegetation is not distributed uniformly in space. Rather, it is organized into patterns with varying degrees of regularity. These patterns may be explained as the result of a self-organization process driven by water scarcity. In this framework, the ability to form patterns represents an important resource for dryland resilience, as it allows the ecosystem to circumvent a “tipping point” transition from uniform vegetation to desert. Reaction-diffusion vegetation models are often employed to reproduce highly regular patterns. The factors determining the emergence of patterns with lower regularities remain unclear.

Recent studies in the field of network theory have shown that similar reaction-diffusion mathematical models generate patterns on idealized networks. Taking inspiration from these theorethical works, we studied the formation of vegetation patterns in relation to the topologies of the networks through which water and biomass diffuse. To do so, we employed a physical reaction-diffusion vegetation model, and gradually modified the topology of the diffusion networks by adding random shortcuts over a 2-dimensional grid (representing either soil heterogeneities or seed dispersal mechanisms), thus interpolating between a regular lattice and a random network. 

We found that network topology strongly shapes both the resulting vegetation patterns and the precipitation range that supports them. Three behavioral regimes emerge. On a regular lattice, highly regular patterns develop reflecting local diffusion processes. On a random network, the system is dominated by global pressure towards homogenization yielding either a uniform state or a single patch. In the intermediate shortcut density range, as the network topology resembles a small world network, the interaction between the two scales of diffusion generates two kinds of disordered patterns: low-regularity patterns with a well-defined characteristic wavelength, and irregular patterns characterized by a broad patch size distribution. These disordered patterns resemble real-world observations and, in our model, they show different responses to changing precipitation. Although we focused on dryland vegetation, we suggest that network-mediated diffusion could lead to similar mechanisms in a wide variety of pattern-forming systems.