An Investigation Of Percolation In Ecological Systems

Abstract

Vegetation cover in certain ecological habitats display the naturally occurring presence of non-uniform “patchy” clusters of dense vegetation; with the steady-state distribution of these cluster sizes exhibiting a “power-law” distribution. Thought to be a product of underlying facilitative interactions between individual plants, these features have piqued extensive interest among ecologists as a way to gauge the resilience of the overarching vegetation system. Even so, most ecological studies delving into spatial pattern formation have focused on ‘static’ averaged properties of the clusters, rather than their dynamic behaviour & variations. Concurrently, the mathematical properties of a common computational framework used to study pattern-formation — the Stochastic Cellular Automata (SCA) — remain ill-characterised for relevant models that incorporate facilitative interactions.

In contrast, the rapid expansion of readily accessible remote sensing data has opened up fertile ground in understanding the dynamic behaviour of vegetation clusters in real-world ecosystems. Thus, building on previous work, we test a dynamical metric which indicates that the distribution in cluster size change at equilibrium shows a fattened tail near the geometric critical threshold at ecologically relevant time-scales, and may conceivably serve as a field tool to assess the closeness of ecosystems to such thresholds. We also show that a range of SCA with facilitative feedback— including the model used in the above metric —display finite-sized scaling at their geometric threshold. We expect our presented work will add to the growing understanding and assessment of resilience in ecosystems, as well as foment further theoretical insights into the mathematical nature of SCA.