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Ecological Modelling

journal homepage: www.elsevier.com/locate/ecolmodel

Degrees of freedom: Definitions and their minimum and most meaningfulcombination for the modelling of ecosystem dynamics with the help ofphysical principles

Ricardo A. Rodrígueza,⁎, Rodrigo Rierab,c, Ada M. Herreraa, Janelle M. Duncand,Michael J. Vannid, Juan D. Delgadoe, María J. Gonzálezd

a Independent Researcher, Calle Cartero Casiano Díaz 19, Bajo B, La Gallega, Sta. Cruz de Tenerife, 38107, Tenerife, Canary Islands, Spainb Centro de Investigaciones en Biodiversidad y Ambientes Sustentables (CIBAS), Universidad Católica de la Santísima Concepción, Concepción, Chilec Departmento de Ecología, Facultad de Ciencias, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, ChiledDepartment of Biology, Miami University, Oxford, OH, 45056, USAe Department of Physical, Chemical and Natural Systems, Faculty of Experimental Sciences, University Pablo de Olavide, E-41013, Seville, Spain

A R T I C L E I N F O

Keywords:BiodiversityInterdisciplinary modellingMetabolic theory of ecology (MTE)Organic biophysics of ecosystems (OBEC)Society-nature interactionStatistical mechanics

A B S T R A C T

There is a neglected old schism in ecosystem ecology (EE): the foundations of crucial concepts and principles ofEE lie in thermodynamics, but the current mainstream of ecological thought is significantly biased towardscontingent mathematical models disconnected from physics. Frequently, these models have weak theoreticalsupport in ecology itself, as well as a limited empirical validation. This situation emerged when some ecologistsbecame aware that, seemingly, thermodynamics (devoted to study the dynamics of closed systems in equili-brium) should be useless to understand ecosystems (far-from-equilibrium open systems). The solution was, eitherdeveloping a sort of “new physics” weakly linked to the principles and methods of conventional physics, or adirection change towards an astonishing diversification of analytical ways. In practice, both things have hap-pened simultaneously. One of the many expressions of this controversial decision was a sort of rigmarole in theuse of the concept of “degrees of freedom”. This article, based on a recent proposal (organic biophysics of eco-systems, OBEC): (i) contributes to resolve the dilemma physics vs. non-physics in EE; (ii) proposes a plausible andempirically-backed approach to the meaning, interaction and use of the concept of “degrees of freedom” in EE byreducing them to an inseparable triad of indicators (species diversity, dispersal intensity, and fresh biomass orbody weight per individual) valid for any kind of ecosystem (non-contingency) and backed by six essential traits(simplicity, universality, evolvability, empirical manageability, inter-model inclusivity, and interdisciplinaryscope); and (iii) explores the aftermaths of the aforementioned approach to propose a complementary ex-planation to the metabolic theory of ecology, as well as the cornerstone of an analytical framework commonlyshared by economics and EE, in order to develop a new way of getting reliable results in regard to the interactionbetween society and nature. In summary, the results included in these three analytical axes (from i to iii) arebased on previous publications including empirical field data from 12 different kinds of taxocenes involving atotal of 1649 plots and 8.874× 107 individuals belonging to 1280 species. Besides, this article includes in itselfadditional data from 638 species of mammals, 97 samples of ruderal vegetation, 26 samples of zooplankton, aswell as data in reference to a significant fraction of the U.S.A. population as a whole(x̄ =2.973× 108±8.657×106 −S.D.− individuals per year) in combination with abiotic environmental data(mean temperature and emission of greenhouse gases at the country level) over 12 consecutive years.

1. Introduction

There are five main domains within the realm of ecological re-search: (a) proposal of untested but engaging hypotheses; (b) develop-ment of holistic theoretical approach solidly supported by empirical

field data; (c) development of pure mathematical models with the helpof software and computing devices; (d) techno-methodological issues;(e) empirical applications in order to reduce our impact either on nat-ural systems or on metastable man-made systems, including cities andtheir surroundings. The concept of “degrees of freedom” (df) plays an

https://doi.org/10.1016/j.ecolmodel.2018.11.021Received 11 August 2018; Received in revised form 11 November 2018; Accepted 28 November 2018

⁎ Corresponding author.E-mail address: arodveg@gmail.com (R.A. Rodríguez).

Ecological Modelling 392 (2019) 226–235

0304-3800/ © 2018 Elsevier B.V. All rights reserved.

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essential, although subtle role, in all these domains.In its less strict meaning, the number of df is an indicator of com-

plexity (e.g.: “the atmosphere is a complex dynamical system with manydegrees of freedom”; Buizza, 2015). From a statistical point of view,mainly linked to domains (b) and (c), the number of df reflects theamount of observations which are free to vary after certain restrictionshave been placed on the data. For example, if the data for 50 cases areclassified into two categories, then as soon as we know that, say, 35cases fall into one category, we also know that 15 must fall into theother. For this example, df=1 because, with two categories and anyfixed value of N, as soon as the number of cases in one category isascertained, then the number of cases in the other category is de-termined (Siegel and Castellan, 1988).

In classical mechanics, a rigid physical body without ties deployssix degrees of freedom since it: 1st) moves up and down; 2nd) moves leftand right; 3th) moves forward and backward; 4th) swivels left and right;5th) tilts forward and backward; and 6th) pivots side to side (Tipler andMosca, 2004). In physical chemistry, the df of a system are the numberof intensive state variables (e.g., pressure, temperature, concentration)of the components that can be arbitrarily and independently variedwithout altering the number of phases in the system (Fegley, 2012).

In statistical mechanics or thermostatistics (the statistical under-standing of classical thermodynamics), the number of df is the numberof ways in which a molecule retains aliquots from a total input of en-ergy. For example, a linear diatomic molecule has six df (Halliday et al.,2011): 3 degrees of translational motion (x, y, z axes) + 2 degrees ofrotational motion (y, z axes) + 1 degree of vibrational motion of theatoms to each other (x axis), since molecules are not rigid physicalbodies. Absolute temperature (T) only depends on translational motion,and the kinetic energy (E) per each translational degree of freedom is E= ½·kB·T, where kB is the Boltzmann’s constant = (m·vx2)/T=1.38064852E–23 J·K−1 per molecule; being m: molecular mass(constant for a given kind of gas), and vx: mean molecular velocity onthe x axis. So, since there are 3 degrees of freedom of translationalmotion (x, y, z), and by assuming the equipartition theorem (i.e.: that inthermal equilibrium energy is, on the average, shared equally among allof the degrees of freedom), the total translational kinetic energy (ET) ofa mass of gas with N molecules is, at the macroscopic scale,ET=3·½·kB·T·N.

One of the main problems about this issue is that, very frequently, itis not explicitly stated which one of the previous concepts of df has beenused in a given article, although the interdisciplinary nature of ecology(Naiman, 1999) allows the use of any of them depending on the con-text. Given the foundation of classical ecosystem ecology in thermo-dynamics (Lindeman, 1942; Patten, 1959; Margalef, 1963; Odum,1968), the concept of df linked to statistical mechanics should be themost relevant within the above-explored spectrum of concepts. How-ever, there is a shortcoming in this regard: the foundational approach toecosystem ecology based on conventional physics is outside the main-stream of current thinking in ecology because of the presumptive in-capability of conventional statistical mechanics (devoted to studyclosed equilibrium systems) to study far-from-equilibrium open ecolo-gical systems (Margalef et al., 1991a; Månsson and McGlade, 1993;Ulanowicz, 2004). This conclusion has yield a situation of semi-stag-nation which, despite the criticism by many authors (e.g., Simberloff,1981; Belovsky et al., 2004; Rodríguez et al., 2017a), has kept its rulingepistemological influence for decades. Therefore, the crucial analyticalrole of the concept of ecological df based on statistical mechanics is indanger.

This article is addressed to offer a brief synopsis about how theOrganic Biophysics of Ecosystems (OBEC) has been able to dodge theabove-mentioned paradox, with the goal of establishing what is theminimum and most meaningful set of ecological df, as well as its in-tegration capability from the interdisciplinary point of view. Theminimum and most meaningful combination of ecological df proposedin this paper: (i) contributes to the resolution of the dilemma physics vs.

non-physics in EE; (ii) proposes a plausible and empirically-backedsolution to the meaning, interaction and use of the concept of df in EEby reducing them to an inseparable and all-encompassing triad of in-dicators (species diversity, dispersal intensity, and fresh biomass orbody weight per individual) valid for any kind of ecosystem (non-contingency) and backed by six essential traits (simplicity, universality,evolvability, empirical manageability, inter-model inclusivity in theecological realm, and interdisciplinary scope beyond the ecologicalrealm) and; (iii) explores the aftermaths of the above-mentioned solu-tion to establish the cornerstone of an analytical framework commonlyshared by economics and EE, in order to develop a new way of gettingverifiable results in regard to the interaction between society andnature.

In summary, the results included in the above-mentioned items(from i to iii) are based on previous publications including empiricalfield data from 12 different kinds of taxocenes (marine microalgae;marine interstitial meiofauna of sandy beaches; massive−non-branching− corals; litter invertebrates in laurisilva and pine forest;tropical rocky shore snails; coral reef fishes; ruderal vegetation;Mediterranean shrub vegetation; mixed shrub vegetation; pine forestvegetation; and coastal succulent shrub vegetation) involving a total of1649 plots and 8.874×107 individuals belonging to 1280 species.Besides, this article includes in itself additional data from 638 species ofmammals, 97 samples of ruderal vegetation, and 26 samples of zoo-plankton. The proposal is completed by an example based on humanpopulation data retrieved from official institutions of the U.S. govern-ment (x̄ =2.973× 108±8.657× 106 −S.D.− individuals per year,in combination with data of mean temperature and emission ofgreenhouse gases at the country level over 12 consecutive years) thatoffers a preliminary assessment of the extrapolation of OBEC to theanalysis of society and its interaction with nature. This example is asprout of a non-cosmological TOE whose further development shouldstimulate our thought on a controversial topic, at the same time thatsuggests new ideas for the research areas herein involved (i.e.: eco-system ecology, physics, and economics).

2. Brief summary of OBEC

2.1. Definition and main results

The OBEC, as a denomination to embrace a large set of non-con-tingent models connected to each other, has been defined (Rodríguezet al., 2017b, 2017c, 2017d) as a proposal based on the extrapolation ofthe principles of classical thermodynamics, statistical mechanics andquantum mechanics to the analysis of the living fraction of ecosystemsin accordance with well-known principles of conventional ecology andevolutionary biology. The conventional biophysics of ecosystems dealswith the influence of several typical abiotic factors, e.g.: isotopemovement in food webs, light, temperature, ionizing radiation, all ofthem of physical nature in their origin, even though modified by theirinteraction with biotic factors. In contrast, the OBEC does not take intoaccount any abiotic factor, but analyzes the dynamics of living (or-ganic) creatures assumed as if they were indivisible physical particles inmovement and interacting with each other. In a similar way to the caseof statistical mechanical df, all the essential traits of every organismdepend on transforming solar energy into ecological parameters, thisexplains the necessity of finding the ecological equivalents of statisticalmechanical df in ecosystems. Fig. 1 summarizes in a very simplified waythe main results from OBEC so far. A careful review of these achieve-ments indicates that OBEC seems to be on the right way to rescue theclassical paradigm of ecosystem ecology based on links between physicsand biology in order to put it back in the mainstream of ecologicalthinking.

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