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Derivations of the Core Functions of the Maximum Entropy Theory of Ecology

1
Department of Biomathematics, David Geffen School of Medicine, University of California, Los Angeles, CA 90095, USA
2
Department of Ecology and Evolutionary Biology, University of California, Los Angeles, CA 90095, USA
3
Department of Ecology and Evolutionary Biology, University of Arizona, Tucson, AZ 85721, USA
*
Author to whom correspondence should be addressed.
Entropy 2019, 21(7), 712; https://doi.org/10.3390/e21070712
Received: 6 May 2019 / Revised: 15 July 2019 / Accepted: 15 July 2019 / Published: 21 July 2019
(This article belongs to the Special Issue Information Theory Applications in Biology)
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Abstract

The Maximum Entropy Theory of Ecology (METE), is a theoretical framework of macroecology that makes a variety of realistic ecological predictions about how species richness, abundance of species, metabolic rate distributions, and spatial aggregation of species interrelate in a given region. In the METE framework, “ecological state variables” (representing total area, total species richness, total abundance, and total metabolic energy) describe macroecological properties of an ecosystem. METE incorporates these state variables into constraints on underlying probability distributions. The method of Lagrange multipliers and maximization of information entropy (MaxEnt) lead to predicted functional forms of distributions of interest. We demonstrate how information entropy is maximized for the general case of a distribution, which has empirical information that provides constraints on the overall predictions. We then show how METE’s two core functions are derived. These functions, called the “Spatial Structure Function” and the “Ecosystem Structure Function” are the core pieces of the theory, from which all the predictions of METE follow (including the Species Area Relationship, the Species Abundance Distribution, and various metabolic distributions). Primarily, we consider the discrete distributions predicted by METE. We also explore the parameter space defined by the METE’s state variables and Lagrange multipliers. We aim to provide a comprehensive resource for ecologists who want to understand the derivations and assumptions of the basic mathematical structure of METE. View Full-Text
Keywords: information entropy; information theoretics; macroecology; metabolic theory; scaling; species abundance distribution; species-area relationship information entropy; information theoretics; macroecology; metabolic theory; scaling; species abundance distribution; species-area relationship
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Brummer, A.B.; Newman, E.A. Derivations of the Core Functions of the Maximum Entropy Theory of Ecology. Entropy 2019, 21, 712.

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