Vegetation patterning dynamics induced by non-local connections

Spatial self-organization is a common response of arid and semi-arid ecosystems to water stress. It may result in periodic patterns such as dots, gaps and labyrinths, or in more irregular arrangements such as scale-free patterns, characterised by a power law distribution of patch sizes. As pattern formation occurs over large spatial domains, in the order of km² , it is often subject to heterogeneous environmental and soil conditions, which may lead to the anisotropic diffusion of resources.

In our project, we study the effects of anisotropic diffusion on pattern formation through the modelling of vegetation dynamics on complex network topologies. 

 

We employ the well-known reaction-diffusion vegetation model by Gilad et al. [1], in its simplified two-equation version by Zelnik et al. [2]. Two partial differential equations describe the dynamics of soil water and biomass densities.

In our implementation, the diffusive terms refer to network Laplacia, which allows us to the modify the topology on which the model operates.

When the diffusion networks of both water and biomass are regular two-dimensional lattices, we reproduce the observed progression of periodic patterns from gaps to labyrinths to dots for decreasing precipitation. 

To increase the complexity and connectivity of the network we implement the Watts-Strogatz small-world network model [3], in which a controlled number of random shortcuts is drawn over the two-dimensional lattice. Thus the number of shortcuts in the water and biomass diffusion networks become model parameters which may be used as proxies of heterogenous conditions affecting the diffusion of water and biomass respectively.

 

Our preliminary results show that an increase in anisotropic diffusion (number of shortcuts) has similar effects to an increase in isotropic diffusion in regards to the global variables of the ecosystem, such as average water and biomass densities. However, a small-world network topology induces the formation of steady-state non-periodic patterns, included scale free patterns, in a certain interval of network connectedness. 

Further, these steady-state scale free patterns appear unstable to the expansion of the largest gaps, leading to rapid desertification following a disturbance that may originate from grazing or human intervention. Hence, we uncover the existance of a bistability between two non-periodic patterns with very different ecological value. 

 

[1] E. Gilad, J. von Hardenberg, A. Provenzale, M. Shachak, and E. Meron. Ecosystem Engineers: From Pattern Formation to Habitat Creation. Physical Review Letters, 93(9):098105, 2004.

[2] Y. R. Zelnik, E. Meron, and G. Bel. Gradual regime shifts in fairy circles. Proceedings of the National Academy of Sciences, 112(40):12327–12331, 2015. 

[3] M. E. J. Newman and D. J. Watts. Scaling and percolation in the small-world network model. Physical Review E, 60(6):7332–7342, 1999.