From companies to colonies: The origin of Pareto-like distributions in ecosystems

TL;DR

This paper explains the emergence of Pareto-like distributions in ecosystems through a stochastic model based on the law of proportionate effect, linking cluster dynamics to criticality and ecosystem stability.

Contribution

It introduces a Markov birth-death process model that connects patch statistics with ecosystem dynamics, highlighting the role of self-organized criticality and stability changes.

Findings

Cluster sizes follow Pareto distributions due to proportional growth.

Transition rates scale linearly with cluster size in the model.

Ecosystem stability shifts drastically near catastrophic thresholds.

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 self-organized criticality is a manifestation of the law of proportionate effect, first discovered in the context of business firm size. 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 (like desertification) manifest themselves in a drastic change of the stability properties of spatial colonies, as the chance of a cluster to disappear depends logarithmically, rather than linearly, on its size.