Origin of Pareto-like Spatial Distributions in Ecosystems

Abstract

Recent studies of cluster distribution in various ecosystems revealed Pareto statistics for the size of spatial colonies. These results were supported by cellular automata simulations that yield robust criticality for endogenous pattern formation based on positive feedback. We show that this patch statistics is a manifestation of the law of proportionate effect. Mapping the stochastic model to a Markov birth-death process, the transition rates are shown to scale linearly with cluster size. This mapping provides a connection between patch statistics and the dynamics of the ecosystem; the "first passage time" for different colonies emerges as a powerful tool that discriminates between endogenous and exogenous clustering mechanisms. Imminent catastrophic shifts (such as desertification) manifest themselves in a drastic change of the stability properties of spatial colonies.

Critical desertification transition in semi-arid ecosystems: The role of local facilitation and colonization rate

“…Our numerical study was based on a stochastic cellular automaton (SCA) model [9], largely used in the literature [10,[12][13][14][15][16][17][26][27][28][29] and validated by the comparison with field data collected in Mediterranean arid ecosystems [10]. In a wide region of the model parameter space, the SCA model predicts the existence of a threshold value, m c , for the mortality rate m (a parameter measuring the strength of external stress, taken as control parameter of the transition) above which there is a complete extinction of the vegetation and a nearly complete degradation of the soil (full desertification).…”

Periodic versus scale-free patterns in dryland vegetation

“…Recent studies, based on simple cellular-automaton models (Kéfi et al 2007a,b;Scanlon et al 2007;Manor & Shnerb 2008a), have suggested that scale-free vegetation patterns are a result of self-organization under conditions of local facilitation and global resource competition. In another recent model study (Manor & Shnerb 2008b), global competition and scale-free patterns have been obtained in the limit of large 'water diffusion'.…”

Eluding catastrophic shifts

“…In particular, ingredients such as spatial heterogeneity and mobility (e.g., diffusion) have been incorporated into those models, leading to interesting consequences and a much richer phenomenology that includes patchiness and pattern formation (31,(36)(37)(38)(39)(40)(41)(42)(43)(44). Moreover, it has been suggested that emerging spatiotemporal patterns could be potentially used as early indicators of tipping points or that transitions in these improved models can become more gradual (15)(16)(17)(45)(46)(47).…”