A Theoretical Framework for Abundance Distributions in Complex Systems Stephan R.P. Halloy
Abstract
	A theoretical framework is proposed to explain how and where complex systems break up into agents or species. Splits lead to diversification Êand to abundance distributions which are similar to power Êfunctions on a rank-abundance representation, Êand to lognormal Êfunctions on a frequency-abundance representation. The combined manifestation of power and lognormal functions is a polo distribution, Êa situation toward which there seems to be a widespread tendency in complex systems (a polo pattern attractor). Minimal complex system Êorganisation requires three integrated hierarchical Êlevels, the system, agents and particles. The tendency to polo emerges, or can be explained by, resource particle interaction, in which particles are attracted to each other according to their size and inversely to their distances. Simulation of this simple rule on a preliminary model leads toward polo abundance distributions. The level of abstraction allows the theoretical framework to be applicable to all fields where complex systems are found to have polo distributions. A clearer understanding of the rules and forces leading to diversification can have a range of applications in planning and management for conservation, agriculture, business, health and other areas dealing with complex systems.

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