distribution of species diversity values: a link between classical and
TRANSCRIPT
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Ecological Modelling 313 (2015) 162–180
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Ecological Modelling
j ourna l h omepa ge: www.elsev ier .com/ locate /eco lmodel
istribution of species diversity values: A link between classical anduantum mechanics in ecology
icardo A. Rodrígueza,∗, Ada M. Herrerab, Rodrigo Rierac,d, Jacobo Santandere,ezahel V. Mirandae, Ángel Quirós f, María J. Fernández-Rodríguezg,osé M. Fernández-Palaciosh, Rüdiger Ottoh, Carlos G. Escuderoh,ntonia Jiménez-Rodríguezg, Rafael M. Navarro-Cerrillo i, María E. Perdomof,
uan D. Delgadog
Independent Researcher, 6203, Lucy Avenue, Apartment A., Tampa, FL 33619, USAThe Woman’s Group, 13005 S. US Hwy. 301, Riverview, FL 33578, USACentro de Investigaciones Medioambientales del Atlántico C.I.M.A., Av. Los Majuelos, 115, 38107 Santa Cruz de Tenerife, Tenerife, SpainInstituto de Pesquisas, Jardim Bot ´anico do Rio de Janeiro, Rua Jardim Bot ´anico, 1008, CEP 22470-180 Rio de Janeiro, BrazilParque Nacional Sistema Arrecifal Veracruzano, C/Juan de Grijalva no 78, Veracruz, MexicoParque Nacional Los Caimanes, Carretera Central no. 716, CP-50100 Santa Clara, CubaDepartamento de Sistemas Físicos, Químicos y Naturales, Facultad de Ciencias Experimentales, Universidad Pablo de Olavide, E-41013 Sevilla SpainDepartamento de Botánica, Ecología y Fisiología Vegetal, Universidad de La Laguna, E-38071 Santa Cruz de Tenerife, Tenerife, SpainEscuela Técnica Superior de Ingenieros Agrónomos, Universidad de Córdoba. Dpto. Ingeniería Forestal, Avda. Menéndez Pidal s/no., Apdo. 3048, 14080órdoba, Spain
r t i c l e i n f o
rticle history:eceived 20 February 2015eceived in revised form 12 June 2015ccepted 15 June 2015
eywords:iodiversitycological state equationcosystem ecologyaxwell–Boltzmann velocity distributionuantum mechanicstatistical mechanics
a b s t r a c t
Despite the well-known thermodynamic traits of ecosystem functioning, their description by means ofconventional physics should be regarded as incomplete, even if we take into account the most recentadvancements in this field. The analytical difficulties in this field have been especially complex to get areliable modeling of species diversity per plot (Hp) by endowing this indicator with a fully clear theoreticalmeaning. This article contributes to resolve such difficulties starting from (a) the previous proposal of anecological state equation, and (b) the preceding empirical finding of an ecological equivalent of Planck’sconstant at the evolutionary scale. So, in the first instance, this article proposes an equation for densitydistributions of Hp values (EDH) based on a simple transformation of the Maxwell–Boltzmann distributionfor molecular velocity values (M–BDv). Our results indicate that the above-mentioned equation allows anappropriate fit between expected and observed distributions. Besides, the transformation from M–BDv
to EDH establishes connections between species diversity and other indicators that are consistent withwell-known ecological principles. This article, in the second instance, uses EDHs from a wide spectrumof surveys as an analytical framework to explore the nature and meaning of stationary trophic informa-tion waves (STIWs) whose stationary nature depends on the biomass-dispersal trade-off in function ofHp values (B-DTO-H) that characterizes the most of the explored surveys. B-DTO-H makes these surveysbehave as ecological cavity resonators (ECR) by trapping functional oscillations that bounce back andforth between the two opposite edges of the ECR: from r-strategy (at low biomass and diversity, andhigh dispersal) to K-strategy, and vice versa. STIWs were obtained by using the spline-adjusted valuesfrom the arithmetical difference between standardized values of species richness (S) and evenness (J′) infunction of Hp values (i.e., a 2D scalar space Hp, S–J′). Twice the distance on the abscissas (2�Hp) betweensuccessive extreme values on the ordinates (whatever a maximum or a minimum) along the above-
mentioned spline adjustment was taken as the value of ecological wavelength (�e). �e was assessed inorder to obtain the value of the ecological equivalent of Planck’s constant (heec) at the intra-survey scaleec
that was calculated as: he = �e × me × Ie; where me: individual biomass, and Ie: an ad-hoc indicator ofdispersal activity. Our main result is that the observed value of heec’s mantissa is statistically equivalent
to the mantissa of the physical Planck’s constant (h = 6.62606957E − 34 J s) in all of the discontinuous(i.e., with interspersed categories in which n = 0) statistical density distributions of Hp values per survey.
∗ Corresponding author. Tel.: +1 786 479 6354.E-mail address: [email protected] (R.A. Rodríguez).
ttp://dx.doi.org/10.1016/j.ecolmodel.2015.06.021304-3800/© 2015 Elsevier B.V. All rights reserved.
R.A. Rodríguez et al. / Ecological Modelling 313 (2015) 162–180 163
This means that heec = 6.62606957Eϕ Je nat/individual, where ϕ = −xi, . . ., −3, −2, −1, 0,+1, +2, +3, . . .,+xi
depending on the type of taxocenosis explored. That is to say, heec indicates the minimum amount of
energy exchange allowed between two individuals. The exploration of the analytical meaning of thisresult in the final sections of the article explains why quantum mechanics (QM) is a useful tool in orderto explain several key questions in evolutionary biology and ecology, as for example: the physical limitof adaptive radiation; the balance between competitive exclusion and functional redundancy to promotespecies coexistence by avoiding the negative effects of competitive exclusion; the apparent holes in thefossil record; the progression of body size along a wide spectrum of taxa as a general evolutionary trend;the non-continuous nature of net energy flow at the ecosystem level; the way in which the energy levelis stabilized under stationary ecological conditions; the reasons of the higher sensitivity of high diver-sity ecosystems under environmental impact despite their higher stability under natural conditions; thetangible expression of complex concept as ecological inertia and elasticity; as well as the increased riskfrom pushing the biosphere until a rupture limit because of the potential discrete behavior of ecologicalresilience in the large scale due to the quantum nature of ecosystem functioning.
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by step evolutionary gradient (�ke) of exchange of information byeco-kinetic energy at a constant eco-kinetic energy increment rateof he
ev per each unit of �ke = 1 Je nat/individual.
. Introduction
Species diversity is one of the most widespread and interest-ng ecological concepts. Ecologists have considered, for severalecades, that species diversity has multiple and remarkable inter-ctions with other concepts as stability (e.g., McNaughton, 1978;aret, 1982; Kimmerer, 1984; Tilman et al., 1998; Li and Charnov,001; Pfisterer and Schmid, 2002; Ptacnik et al., 2008), habitat per-urbation (e.g., Mackey and Currie, 2001; Solé et al., 2004) androphic energy or “production” (e.g., Tilman et al., 1997, 2001; Evanst al., 2005; Huston and Wolverton, 2009; Witman et al., 2008;odríguez et al., 2013a,b, 2015a,b).
The concept of species diversity was theoretically analyzed in aigorous manner first, by Hurlbert (1971). He pointed out, amongther problems, toward the weak theoretical support for corre-ations between species diversity and other indicators. There areot particularly strong reasons to consider that this situation hasadically changed nowadays (see Rossberg, 2008, p. 21). It is veryrobable that this situation is rooted in the fact that the most fre-uently used measure of species diversity (Eq. (1)) was borrowedrom a foundational external source (Shannon, 1948). This sourcencludes certain degree of relativism in regard to: (1) the namessigned to this concept itself (see Tribus and McIrvine, 1971), and2) the subject taken as a reference point to describe the transmis-ion of signals. That is to say, a net input of information (H) means
reduction of uncertainty that, simultaneously, is equivalent to aocal net output of entropy (Tribus and McIrvine, 1971; Ayres andair, 1984; Ayres, 1994, p. 36).
= −S∑
i=1
[(ni
N
) (ln
ni
N
)](1)
here S: species number, ni: number of individuals of species i, = �ni. H is conventionally expressed in nat/individual when nat-ral logarithms are used.
Several of the current theoretical trends about biodiversityould be seen as attempts to answer those doubts stated by Hurlbertorty-four years ago. Thus, we continue looking for a solid theo-etical meaning (Spellerberg and Fedor, 2003) for a concept whoseain commitment is to reflect a tacit empirical truth: that the unde-
iable successional increase of species diversity is a sort of “centralogma” in ecology (e.g., Margalef, 1963; Odum, 1969).
Nevertheless, is it right to grant so much importance to speciesiversity? What is the “normally” expected statistical behaviorf diversity under a given set of ecological conditions? Is there
ny general way to model diversity starting from other significantcological indicators? Could be used this way to model diversitys an analytical framework to expand our understanding of the© 2015 Elsevier B.V. All rights reserved.
ecosystem functioning? What is the role of other disciplines toassign a solid theoretical meaning to species diversity?
According to recent findings, H can be successfully used as themain state variable to obtain: (1) an ecological state equation (ESE;Rodríguez et al., 2012; see Eq. (6)) that is universally valid provid-ing that the system analyzed is under stationary or quasi-stationaryecological conditions (SEC, hereafter); (2) a consistent explana-tion for the relationship between production and species diversity(Rodríguez et al., 2013a,b); (3) a potential solution to the debatebetween competitive exclusion principle (CEP, one species ↔ oneniche; Hardin, 1960; Darlington, 1972; Gordon, 2000; Wang et al.,2005) and functional redundancy (FR, several species ↔ one niche;Walker, 1992; Lawton and Brown, 1994; Wohl et al., 2004; Petcheyand Gaston, 2006; Mayfield and Levine, 2010) through a stationarywaves model for species coexistence (WMSC) based on the con-currency between transient equilibrium nodes of CEP, and wideantinodes of limited FR; in a similar way to the standing waves ona string (see Rodríguez et al., 2013b).
The attempts to describe the ecosystem via statistical mechan-ics are abundant (e.g., Messer, 1992; Schneider and Kay, 1994;Svirezhev, 2000; Zhang and Wu, 2002; Jorgensen and Svirezhev,2004; Jørgensen and Fath, 2004; Maurer, 2005; Shipley et al., 2006;Dewar and Porté, 2008; Capitán et al., 2009; Banavar et al., 2010;Kelly et al., 2011), and they are based, in general, on the concept ofspecies diversity. Thus, the approach reflected in the items of theprevious paragraph appears to be usual but, in the second instance,it is quite unusual precisely because of its full equivalence to thestandard physical algorithm, in comparison with the proposalsmentioned in this paragraph.
In addition, Rodríguez et al. (2015b) showed, at the inter-surveylevel, that a robust straight line adjustment between Ee and the eco-logical equivalent (ke, see additional explanations in Eq. (6)) of theBoltzmann constant (kB) has a regression constant whose mantissa(he
ev = 6.6260727E − 01 Je·�ke; where �ke = 1 Je nat/individual; ev:inter-taxocenosis or evolutionary scale; Je: ecoJoule, an ad hoc eco-logical measurement unit, see additional explanations in AppendixA, Section 2) coincides with the respective value of physical Planck’sconstant (h = 6.6260727E − 34 J s; see e.g., Halliday et al., 1999;Tipler and Mosca, 20101). The above-mentioned paper concludesthat the evolutionary process can be understood as an initial“micro-clot of life” that has been “ecologically driven” across a step
1 All the following comments about conventional physical principles without cit-ing a particular source are supported by principles, methods and concepts exposedin any of these two books.
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64 R.A. Rodríguez et al. / Ecologi
Rodríguez et al. (2015b, Section 4) also established severalending tasks to develop a dependable theoretical framework tonderstand the ecosystem functioning in terms of QM. This article isimed to resolve some of these pending tasks by combining the tworeviously commented big points (the theoretical meaning of H incology and the empirical emergence of he
ev) starting from a trans-ormation of the Maxwell–Boltzmann distribution for molecularelocities (M–BDv; Eq. (9)) into an expected ecological distributionf diversity values (EDH; Eq. (10)). This article explains how andhy the EDH acts as an analytical link between the classical and
uantum realms in ecology, by clearing doubts about the usefulnessf standard physics to understand the ecosystem functioning.
In the small scale, this article answers three main questions:1) Is it possible to obtain a general theoretical ecological distri-ution of species diversity values (EDH) structurally coherent withhe algorithm of statistical mechanics? (theoretical principles and
ethods in Sections 2.1.2–2.1.4, and results in Section 3.1). (2) Is itossible to find an ecological equivalent of Planck’s constant perDH at the intra-survey scale (he
ec)? (theoretical principles andethods in Sections 2.2.1–2.2.3, and results in Section 3.2). (3) Is it
ossible to get a statistically non-significant difference between thebserved mean values of individual eco-kinetic energy per EDH andhe respective expected values assessed by means of QM? (theoret-cal principles and methods in Sections 2.1.1 and 2.2.3, and resultsn Section 3.3).
All the interspersed comments and notes included in this arti-le are intended to support a subsequent reasoning with the aimf building a self-contained analytical framework that offers: (1) Aunctional summary, as simple as possible, of the essential physicalnowledge to understand our proposal. (2) A review of the basiccological principles that support our proposal (returning to theundamentals is sometimes essential). (3) A step by step explana-ion of an alternative way of thinking ecosystem ecology, including
ethods interlinked with partial results that are necessary to arriveo those final results we are looking for. (4) An eclectic explana-ion of the meaning of our final results in connection with item. In summary, this article shows that there are testable connec-ions between information theory, QM and ecosystem ecology thatannot be easily explained by pure chance, and they also have aotential deep impact on the future development of other sciences,
ncluding physics itself.
. Materials and methods
.1. Procedures to obtain EDH
.1.1. Basic indicators and general theoretical foundationsTotal eco-kinetic energy per plot was calculated as (Rodríguez
t al., 2012, 2013a,b, 2015a,b):
eTp = Np
(Eep
)= Np
(1/2mepI2
e
)= 1/2meTpI2
e (2)
here Np: total number of individuals per plot; Eep: mean eco-inetic energy per individual per plot (p) expressed in Je (asommented above, see Appendix A, section 2); mep: mean indi-idual biomass per plot; meTp: total biomass per plot; Ie: dispersalndicator (expressed in –d unit, see Eq. (3)) with the appropri-te features to replace v (physical velocity) in an interdisciplinarynvariant way in regard to the replacement of T by H from Eqs. (7)o (6), see below. Appendix A, Section 1, includes expanded expla-ations about the characteristics and meaning of Ie according to
odríguez et al. (2013a).e =∑S
i=1
(Iei,j
)S
(3)
delling 313 (2015) 162–180
Iei,j =(
di,j
�i,j
)× 100 (4)
di,j =
∑m
k=1
((√(xj − xk
)2+(
yj − yk
)2
)×((
2ij,k/(
ij + ik))))
m(5)
where di,j is the mean dispersal activity of species i in plot j withcentral geographic coordinates x, y within a total space s0 dividedinto an m number of k plots; ij and ik are the respective abundancesof i in plots j and k; ij,k is the shared number of individuals of iin plots j and k (e.g. if ij = 7 and ik = 12, then ij,k = 7); 2ij,k/(ij + ik)isthe Bray–Curtis similarity index (Washington, 1984); �i,j is thestandard deviation of di,j given that di,j is an arithmetic mean; Iei,j isthe dispersal indicator of species i in plot j; Ie is the dispersal indica-tor of the species group in plot j; and S is the species number in plotj.√
(xj − xk)2 + (yj − yk)2 is the Pythagorean theorem that allows theestimation of a mean Euclidean distance between plot j and all theremaining k elements within the set of m plots.
Our proposal is supported by three main theoretical premises:(1) the analytical equivalence between stationarity in open sys-tems and equilibrium in closed systems (see Montero and Morán,1992; Rodríguez et al., 2012, 2013a,b, 2015a), (2) the anti-kineticand therefore anti-thermic effect of species diversity (Margalef,1963, 1991; Odum, 1972; Rodríguez et al., 2013a,b) and, finally, (3)the local effect of entropy reduction by increasing the informationamount (see Ayres, 1994; Ayres and Nair, 1984; Brillouin, 1951,1953, 1956; Brissaud, 2005; Gabor, 1964; Gallucci, 1973; Jaynes,1957a,b, Lewis, 1930; Odum, 1969; Rothstein, 1951, 1952; Szilard,1929; Tribus and McIrvine, 1971; Volkenshtein, 1985; Rodríguezet al., 2013b, 2015a). Starting from these premises, Rodríguez et al.(2012) combined Eqs. (1)–(3) in order to obtain an ESE (Eq. (6)) ofthe form:
2Np
(1/2mepI2
e
)=
(Npke
)Hp
(6)
2EeTp =(
Npke
)Hp
where Hp: ecological information (Eq. (1)) assessed as speciesdiversity per plot; ke: ecological equivalent of Boltzmann con-stant = mep× Ie2× Hp = 1.3806504Eϕ Je nat/individual, this parame-ter indicates the average rate in which an individual exchanges Hby Ee across the set of plots under SEC; Np: see Eq. (2); Eep: eco-kinetic energy, mean value per individual per plot, see Eq. (2); EeTp:see Eq. (2); mep: see Eq. (2); Ie: see Eq. (3).
2N(
12
mv2)= NkBT (7)
where N: number of molecules; m: molecular mass; v: molecularvelocity; kB: physical Boltzmann constant and T: absolute temper-ature.
Eq. (6) is analytically equivalent to the ideal gas state equation(IGSE, Eq. (7)) and it was proved, later on (Rodríguez et al., 2013a),that it is a generally valid pattern under SEC due to a general trade-off between biomass (me) and dispersal (Ie) in function of diversity(H) values (B-DTO-H). That is to say; me
↓, Ie2↑, Hp↓ or, alternatively,
me↑, Ie2↓, Hp
↑. In such a way: me min↔ Hp min and Ie2min↔ Hp max
(where ↑ means “increase”, and ↓ means “decrease”). As a result:(me× Ie2)max↔ intermediate values of Hp. This implies that Ee
diminishes toward any of the two alternative edges of Hp values. Sothe B-DTO-H supports the constant value of ke (me× Ie2× Hp) along
a gradient from r-strategy (see Pianka, 1970) at low Hp values toK-strategy at high Hp values.The above-mentioned B-DTO-H guarantees that any stationaryecosystem, despite its far-from-equilibrium open nature (exchange
cal Modelling 313 (2015) 162–180 165
owkisaivHecf(abo(bwc
2v
e(tl11aeltsdmtsLe
itteitew
�
tdtdaeauvls
Fig. 1. Wave equivalence between physics and ecology. (a) Graphical summary ofstationary wave (SW) parameters (harmonic number four (4 antinodes)). dt: dis-tance. dp: wave displacement. e: equilibrium line. A: wave amplitude. →: travelingwave from left to right. ←: traveling wave from right to left. dr : total distancebetween both edges (antinode length of the fundamental vibration mode – 2dr = n�,n = 1 – in the case of a physical cavity resonator). os: simple harmonic oscillationof the antinode from crest to troughs, and vice versa, due to interference betweenthe traveling waves and . �: wavelength. The inner panel in the left superiorcorner shows the combination of the two extreme alternative displacement of SWdepending on the change of relative positions of and . (b) A vibration mode(interference pattern) between complementary ecological traveling waves (e→,←e) in an ECR and the resulting stationary wave (STIW) from the EDH with 11Hp classes and cw = 0.19091 nat/individual (fourth overtone, eight macrostates—Ms;data in Appendix B, survey rv1, rows 29–36, EDH# 9). SM(xs): standardized value oftotal species richness per macroestate. J′M(xs): standardized value of total evennessper macroestate. Each bar starting from e indicates the magnitude and sense (y > 0:autotrophic; y = 0: equilibrium node or local climax; y < 0: heterotrophic) of net pro-duction per M. e: equilibrium line (SM(xs) − J′M(xs) = 0). The inner panel shows thecombination of the two extreme alternative displacements of STIW within theirrespective EDH depending on the change of relative positions of e and e . Thescheme shown in this figure is generally valid for all the EDHs under stationaryconditions included in Appendix B (books “cor”–“rv9”); there are only variationsin the number of nodes, antinodes and wavelength values between different EDHsdepending on the number of macrostates and their class width (cw). d1, 2, 3. . . n: halvesof ecological wavelengths (1/2�e) for the set of Ms included by �Hp
′ . �Hp′: total
range (1/2�e o) of Hp′ values per EDH (ecologically equivalent to dr in Fig. 1). Com-
pare Fig. 1b with the original physical pattern (Fig. 1a). (Fig. A.3, Appendix A, includes
R.A. Rodríguez et al. / Ecologi
f energy and substance with other systems), behaves in a similaray to a confined system that contains a gas at the laboratory by
eeping constant state variable values (e.g., like a flask immersednto a thermal bath). That is to say, me× Ie2× Hp = ke “pushes” theet of plots toward a “thermostatistical accretion center” (TAC,
sort of gravity center surrounded by an ecological cavity withntangible but effective limits) that is placed at maximum EeTp
alues per plot and intermediate Hp values. The distribution ofp values, regarded as a single unit, can escape from this TACither by a movement of the system toward a regressive suc-essional position (due to a net leakage of eco-kinetic energyrom the system) or toward a progressive successional positiondue to a net input of eco-kinetic energy into the system). Thisrrangement supports the gradient of energy availability betweenoth edges due to a �EeTp/meTp. In addition, it confines the setf species within mutually compensatory functional boundariesbetween r at me
↓, Ie2↑, Hp↓; and K at me
↑, Ie2↓, Hp↑). Sections
elow explain how the ecosystem avoids falling into the TAChen its total amount of internal eco-kinetic energy remains
onstant.
.1.2. Why we should expect a gamma distribution of diversityalues per plot (Hp)?
According to Margalef (1991, p. 367, Fig. 11–7), any seriesnough representative of H values with regard to time or spaceergodicity) fits a quasi-normal statistical distribution skewed tohe right. The general pattern commented by Margalef is very simi-ar to the M–BDv under equilibrium conditions (see Roller and Blum,986, p. 764, Fig. 24.6). According to several authors (e.g., Stacy,962; Lienhard and Meyer, 1967; Patriarca et al., 2004; Chakrabortind Patriarca, 2008; Khodabin and Ahmadabadi, 2010; Lallouachet al., 2010; Melker et al., 2010), M–BDv can be seen as a particu-ar case of generalized standard gamma distribution. This indicateshat; if it is possible to assign a consistent dual meaning (thermo-tatistical as well as ecological) to any indicator that fits gammaistribution; then it is plausible to model such an indicator byeans of an extension of the M–BDv to a given macroscopic sys-
em (a taxocenosis, in this case). In fact, this procedure has beenuccessfully applied before in other fields (e.g., Lienhard, 1964;ienhard and Davis, 1971; Carruthers, 1991; Ball, 2002; Cockshottt al., 2009).
In regard to “if it is possible to assign a consistent dual mean-ng (thermostatistical as well as ecological) to any indicator that fitshe gamma distribution”: Ie is a variable adjusted to gamma dis-ribution (just as v) and it is negatively correlated with Hp undercological stationarity (see Rodríguez et al., 2012, 2013a). Start-ng from the former relationship and given that the mean (�) andhe variance (�2) in gamma distribution are not independent toach other (i.e., both depend on a parameter a in the followingay):
= 2a ×√(
2�
); and �2 = a2 × (3� − 8)
�(8)
hen � and �2 increase or decrease together in any gamma-istributed variable. Thus, when the distribution of Ie values shiftso the left and its variance as well as its right tail decreases, the Hp
istribution as a whole shifts to the right and its variance as wells its right tail increases, and vice versa. According to Rodríguezt al. (2013b, p. 3), Fig. 1 and respective comments), this oppositend complementary relationship between Hp and Ie reflects the
nderlying reduction of the number of intersections between Ieiectors (see Eq. (4)) that is essential to promote a decreasing over-ap between ecological niches in order to reach a higher degree ofpecies coexistence.a typical empirical example in order to measure local values of 1/2�e).
In summary, the statistical adjustments of Ie and Hp values togamma distribution as well as the opposite correlation betweenthese two indicators are important factors that support speciescoexistence due to the right-skewed character of gamma distri-bution. That is to say, the area under the curve within the largertail placed at the right side (with a higher coexistence level and alower niche overlap) of Hp distribution is relatively larger for thosecommunities of high species diversity. So Rodríguez et al. (2013b,p. 4) concluded that “the right-skewed nature ofHpandIedistributionscan be seen as an evolutionary imperative instead of an accidentalstatistical peculiarity”.
Starting from the above-commented arguments, this article, inthe first instance, explores if the M–BDv in the equilibrium state (Eq.
(9), according to Roller and Blum, 1986, p. 764, Eq. 24.36) can betransformed to an analogue theoretical distribution (EDH, Eq. (10))
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d
w�twtJEvEedC
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(
(
66 R.A. Rodríguez et al. / Ecologi
n order to describe the observed distribution of Hp values (Eq. (1),t the plot level) under SEC.
Nv = 4�N
[√m
2� (kBT)
]3
v2e(−0.5mv2/kBT)dv (9)
here dNv: number of molecular velocity (v) values included fromi to vi + dv; �: 3.14159; e: Euler’s number (2.71828); N: total num-er of v values included in the distribution; m: molecular mass; kB:oltzmann’s constant (1.3806504E − 23 J/K/molecule); T: absoluteemperature; v = vi: minimum of the category of molecular veloc-ty values for which the expected value of dNv was calculated; dv:mplitude of the categories of v values of the distribution.
NHp = 4�N
[√keHT
2� (Ee)
]3
H2p × e
(−0.5keHT H2
p /Ee
)dHp (10)
here dNHp : expected number of Hp values from Hpi to Hpi + dHp;: 3.14159; N: total number of Hp values included in the distribu-
ion (plots with S = 1 and H = 0 were excluded from N, but all speciesere included in the calculation of HT); ke: ecological equivalent of
he Boltzmann constant (1.3806504Eϕ Je nats/individual; Je: 1 eco-oule = 0.5kg –d2; –d: 1 dispersal unit, the unit of expression of Ie, seeq. (3)); HT: Eq. (1) calculated for the survey as a whole; Ee = meanalue of the individual eco-kinetic energy at the survey level (seeq. (2)); Hp = Hpi: minimum of the category of Hp for which thexpected value of dNHp was estimated; e: Euler’s number (2.71828);Hp = amplitude of the categories of Hp values of the distribution.onversion from Eq. (9) to (10) is explained below.
.1.3. Conversion from M–BDv (Eq. (9)) to EDH (Eq. (10))According to Roller and Blum (1986, p. 763), the relationship
/kBT in Eq. (9) is the key point to obtain M–BDv because the vari-nce of v values (�2
v = a2(3� − 8)/�; see Eq. (8), above) is linked tohis relationship in the following way:
2v =
1(m/kBT
) = kBT
m(11)
It is easy to grasp (see Eq. (9)) that kBT/m plays an essentialole to obtain the M–BDv (i.e., m/kBT is the only set of variablesn the central section of the equation that is between bracketsm/(2�kBT)) as well as it is repeated on the exponent of e). As aesult from the meaning of Eq. (11), we would need to find a math-matical expression proportional to (∝) the variance of Hp values�2
Hp) just as kBT/m ∝ �2
v. The reciprocal of this relationship woulde included in Eq. (9) at the same time that physical variables woulde replaced by ecological equivalent parameters in order to obtainhe EDH that we are looking for. Therefore (∴), we performed theollowing analytical sequence:
1) HT (total information amount or species diversity) is the mostgeneral thermo-ecological state variable at the survey level. HT
has an anti-thermic effect (i.e., HT→ 1/T in terms of Eq. (7) incomparison with Eq. (6), see theoretical premises in Section2.1.1). So the relative position of HT in regard to kBT/m (see Eq.(11)) should be just the opposite of T; ∴ �2
Hp∝ 1/HT .
2) The same argument used in item (1) is valid in regard to ke incomparison with kB; ∴ �2
Hp∝ 1/keHT .
3) According to Eq. (11), �2v is inversely proportional to the molec-
ular mass (m). That is to say, because of the level of molecularinertia resulting from a constant m value typical of each gas,
a given energy input (�E) on a volume of gas with heavymolecules produces an M–BDv with a lower dispersion (<�2v)than in the case of a gas composed by molecules lighter thanthe former ones. Besides, the direct influence of �E on v values
delling 313 (2015) 162–180
in Eq. (9) is neglected since this is precisely the physical param-eter whose behavior should be described. Contrastingly, in thecase of SEC, there is a B-DTO-H (see Section 2.1.1). This meansthat there is a crossed or bidirectional connection between Ee
(see Eq. (2)) and species diversity values. In other words, nei-ther mep nor Ie2 have constant values (not even in regard to asingle species) across a given set of Hp-distributed plots, andthe inclusion of both parameters in the calculation of Ee con-tributes to the dispersion of Hp values. In even simpler words,if a unimodal or “hump-backed” distribution (see Rodríguezet al., 2013a) of EeTp (on the y axis) in function of Hp (on thex axis) is transposed, the result is a scatter with the shape ofa lying isosceles triangle with its vertex on the right. With anunlimited supply of increasing energy the triangle vertex wouldgo to infinite on the right in the same measure in which thebase of the triangle would be expanded within certain Hp lim-its that, in practice, seems to go from 0 to ≈5 (see Margalef,1991, p. 371; Margalef, 1995; Kent and Coker, 1992; Jorgensenand Svirezhev, 2004, p. 76). Then it is evident that a given inputof total eco-kinetic energy (�EeT) produces an all-embracingbut differential increase of �2
Hpbecause there is a decreasing
dispersion of Hp values from low to high energy levels: a tinyinitial amount of additional energy produces a stronger effecton species diversity per plot than an equivalent amount addedat already high energy values. This would be especially true ifwe are able to find that the positive influence of the partition ofenergy gradient on species coexistence is limited by quantumeffects (sections below), despite the additional input of energy.In the opposite analytical direction (from Hp to Ee), this is pre-cisely the foundation of the relative increase of entropy perplot (Np× Hp) to promote longer food chains and higher trophicpyramids (see Eq. (10) in Rodríguez et al., 2015a). Finally, dif-ferently from Eq. (9), we do not need to ignore the influencefrom a given �Ee on Ie2 to obtain the EDH, because in thiscase we are trying to predict the distribution of Hp valuesinstead of the distribution of Ie itself; ∴ �2
Hp∝
(1/2meIe
2)
/
keHT = Ee/keHT .(4) Effectively, there is no difference between the mean values of
�2Hp
(�1) and Ee/keHT (�2) for the set of 22 stationary surveysincluded in this study (�1 = 0.2273, �2 = 0.2275; t = −0.0079;p = 0.9937; n = 21 because of the exclusion of 1 survey – css inTable B1 – due to its minimum variance of Hp values in combi-nation with a relatively low number of plots; see original valuesof �2
Hpand Ee/keHT in Table B1, Appendix B).
(5) Finally, just as �2v = kBT/m (compare Eq. (11) with Eq. (9)) has
been included as 1/�2v = m/kBT in Eq. (9), then �2
Hp= Ee/keHT
should be included as 1/�2Hp= keHT /Ee in the EDH (see Eq. (10)
in comparison with Eq. (9)).
2.1.4. General statistical methods applied in regard to Sections2.1.1 and 2.1.3
The sampling methodology per survey is explained in AppendixA, Section 3. The physical requirements for ecological stationarity(i.e., B-DTO-H and absence of statistically significant differences ke(o)vs. ke(e) and 2 Np(1/2me× Ie2) vs. (Np× ke)/Hp) are shown in TableB1, Appendix B.
In order to test our null hypothesis, the observed values ofHp per survey were statistically distributed by using Statistica-6(StatSoft Inc, 2001). It was used a standard relative class width ofcw = range/
√nps± 2 (nps: total number of plots per survey) for all
of the surveys in order to make the distributions comparable toeach another. The adjustment between the observed distributionof Hp values per survey and the expected one according to Eq. (10)was assessed by means of the Chi-square (�2) test by combining
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djacent Hp classes until reaching a number of observations n > 5 inhose Hp categories in which n ≤ 5.
.2. Looking for standing ecological waves within EDH in order tobtain intra-survey assessments of he
ec values
.2.1. Stationary trophic information waves (STIWs) and theirhysical foundation
The conventional division of the ecosystem in many plots ofquivalent size in order to its exploration by means of statisticalethods is arbitrary, similarly to study the anatomical and physio-
ogical links that sustain the biological functioning of a given animalfter its body has been turned into ground meat. On the contrary,he grouping of parcels according to a continuum of Hp values viaDH can be especially fruitful to explore the functioning of ecolog-cal communities. This procedure seems essential due to two maineasons: (a) it is generally suitable to reduce the noise (randomispersion of data) that comes from the arbitrary partition of theystem in many plots, and (b) this procedure is particularly fruit-ul when we are looking for wave indicators whose deploy implies
continuous connection between functional subunits. Each classor category) of diversity values within a given EDH can be seen asn alternative complexion (in Planck’s terms, see Brillouin, 1956, p.20) or internal “macrostate” (henceforth), in the same sense thathis term is used in statistical mechanics (e.g., Eisberg and Lerner,990): as a statistical entity composed by several microstates (plotser Hp class within the respective EDH) whose micro-permutationst the small scale (internal changes between alternative combi-ations of spatial position, mass and velocity under equilibriumonditions) do not affect the value of the gross observable charac-eristics or “state variables” per macrostate (entropy, temperature,ressure, total energy, volume, etc.).
We can suppose that this also occurs in ecology because anyomplex system develops inhomogeneities that enable us to rec-gnize groups of elements that are more similar to one another thanhey are to the background (Levin, 1998). In fact, the configurationf ecosystems is recognizable because they have a particular fractaltructure (Nielsen, 2000; Ulanowicz, 2004; Miller, 2008) composedf different parts arranged in a definite pattern (Margalef, 1963).o, the analytical use of EDH should improve the consistency of ouresults since each one of these EDHs can be assumed as an alterna-ive combination of macrostates (classes of Hp values) with a variableegree of internal connectivity at a smaller scale (microstate). Thisethod was applied to obtain the first empirical assessment of
e (Rodríguez et al., 2012, Fig. 1), and we will use it again heren order to take the EDH as an enclosed functional space (see theoncepts of B-DTO-H and TAC in Section 2.1.1) useful to exploref the ecosystem can be assumed as an ecological cavity resonatorECR). This hypothesis assumes that, within the ECR, there would be
ovements of eco-kinetic energy (Ee) associated to internal “vibra-ions” (the above-mentioned degrees of internal connectivity) thatravel as waves. So these statistical perturbations would travel at anpproximately constant average dispersal rate (Ie, Eq. (3)), bounc-ng back and forth between the two opposite edges of the ECRfrom me
↓, Ie2↑, Hp↓ to me
↑, Ie2↓, Hp↑, and vice versa) interfering
ith each other. This interference creates a pattern of ecologicaltanding waves (SW) or stationary trophic information waves (STIW)hat resonate inside the ECR (Sections 2.2.2 and 2.2.3).
In physics, a SW is a single resulting wave that arises in a sta-ionary medium, and it remains in a constant position as a result ofhe combination between two traveling waves (let’s call wave and
wave) that move in opposite directions (→; ←) and interfere
o each other. If and have equivalent wave parameters (waveelocity (v), wave amplitude (A), wavelength (�) and wave fre-uency (f = v/�)) there is, on average, no net propagation of energyue to the presence of nodes (v = 0 and A = 0) that are “impervious”delling 313 (2015) 162–180 167
to the net flow of energy. But this can change either (a) because oflocal variations of wave parameters in space and time giving placeto a partial SW with net energy transmission capability (see over-tones), or (b) because the fluctuation between alternative modes ofvibration into de ECR produces a “dispersal blending” of Ee acrossthe set of plots as a whole. � is the spatial period of a wave, that is tosay, the distance over which the wave’s shape repeats. � is usuallymeasured by considering the distance between consecutive corre-sponding points of the same phase, such as two crests, two troughs,or two zero crossings of the wave in the same amplitude direction(i.e., 2dn, where dn is the distance between two contiguous nodes orextreme values of A, from +A to −A, in a SW). � is an essential char-acteristic either of traveling waves or SW, as well as other spatialwave patterns. The SI unit of � is m (meter).
If the total distance between both extremes of a resonator is dr,then the length of a round trip of a traveling wave is 2dr and thecondition for resonance within the resonator is that 2dr is equal toan integer number of wavelengths (�) of the wave
2dr = n�, n = 1, 2, 3. . . (12)
and the resonant frequencies (fr) are
fr = nv2dr
, n = 1, 2, 3. . . (13)
So the fr or “normal vibration modes”, are equally spaced multi-ples (or fractions, in terms of wavelength) a.k.a. harmonics, whosenumber is equal to the number of antinodes (overtone number: ot)of a lowest frequency called the fundamental frequency (fo). At fo,ot = 1, that is to say, there is one antinode with a length equal tothe distance (dr) between two edge nodes placed in the oppositeextremes of the resonator (e.g., a single oscillation with only 1 pointof maximum–minimum in between the two fixed extreme points ofa guitar string). This theoretical description assumes waves travel-ing at a constant speed across a rectilinear and isotropic resonator.If any of these three conditions is not fulfilled, then fr may not occurat equally spaced multiples of ot = 1 and they are called “overtones”,instead of “harmonics”, given that the succession of changes ofwave amplitude is associated to a quasi-periodic wave train, insteadoff to a perfectly periodic one. Fig. 1a shows a graphical summary ofthese physical parameters, in the case of a periodic oscillation, that isessential to understand the case at the ecosystem level (explainedand exemplified below).
The inner structure of Eq. (1) is an additional factor to under-stand the physical foundation of STIWs. Eq. (1) combines twosubsidiary variables of a second order: species number or “rich-ness” (S) and the degree of equitability in the relative abundanceof concurrent species or “evenness” (J′, Eq. (14)). Both parametershave certain relative independence to each other, as well as cer-tain mutual compensation capability. In addition, both have, understandard ecological conditions (SEC), a positive correlation with H.
Starting from the observed EDHs selected to be compared withEq. (10) in Section 2.1.4, we continued the obtaining of a continu-ous sequence of EDHs toward a higher number of classes of lower cw
values (i.e., up to an average maximum number of EDHs per surveyof 38.75; an average minimum number of plots per macrostate of2.452 per survey; and an average value of cw expressed as % of themaximum �Hp per EDH of 2.544% per survey; see data in Table B2,Appendix B). This procedure was applied in order to explore a rangeof harmonics as wide as possible (from 1—fundamental harmonicwith fo—to an average maximum value of 16.636 overtones per sur-vey; data in Table B2, Appendix B). The maximum value of �Hp per
EDH has, in the ecological context, an analytical meaning equiva-lent to that of dr in wave’s physics (see Fig. 1a and Section 2.2.3).Along every spectrum of EDHs per survey, we calculated the aggre-gated value of J′ (evenness index of Pielou, 1975) and S (species
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umber) per macrostate (M: class of Hp values per EDH, see com-ents above); where J′:
′M =
HM
ln SM(14)
The standardized values (xs = (xi− �)/�, where xi: observed orig-nal value of variable i, �: mean, and �: standard deviation =
√�2)
f S and J′ per macrostate at the intra-EDH scale (SM(xs) and J′M(xs),espectively) were used in order to obtain a graphical interactionetween these two variables, by dodging the obstacles derived fromhe difference of measurement units between both indicators. SM(xs)nd J′M(xs) were correlated with the class mark (Hp
′: midpoint valueer class) of Hp per macrostate. This procedure is in agreement withhe concept of measurement stated by Niels Bohr in QM (as an inter-ction between instruments that belong to two different scales ofhe physical world in order to obtain information) since SM and J′Mre cumulative indicators at the macrostate level, meanwhile Hp
′ isn indicator at the microstate level. The sequence of values for eachidimensional variable (Hp
′, SM(xs) and Hp′, J′M(xs)) was smoothed by
eans of a piecewise cubic polynomial function (B-spline estima-ion in TableCurve 2D v5.01; SYSTAT Software Inc., 2002) in order toest if the resulting graphical pattern is similar to that of two trav-ling waves ( and , see above, in this same section) that interfereith each other.
.2.2. Connection between STIW and eco-kinetic energy peracrostate
The Pg/R ratio (Pg: gross production; R: respiration) has beenonventionally regarded as an excellent functional index of theelative maturity of ecosystems (see Odum, 1969, pp. 263–265;argalef, 1963, p. 365; Margalef, 1991, pp. 872, 875––876). Inetabolic terms, the Pg/R ratio assesses the aggregate relationship
etween anabolism and catabolism at the gross scale in regard tohe ecosystem as a whole. Those ecosystems where Pg/R > 1 are clas-ified as autotrophic systems (they produce more than their internalonsumption and the excess is “exported” to other systems; seeargalef, 1963). Those ecosystems where Pg/R < 1 are classified
s heterotrophic systems and they are sustained by the importa-ion of energy from autotrophic systems. Finally, those ecosystemsn which Pg/R ≈ 1 are classified as mature or “climax” systemsClements, 1916, 1936; Odum, 1969; Vance, 1988), even takingnto account certain degree of discomfort with the original con-ent of this concept (see Whittaker, 1953; Cook, 1996; Eliot, 2007).ny ecosystem, given its fractal nature, can also be understoods a functional patchwork of these three options repeated at themall scale. The physical foundation of this small-scale structure isnalyzed below, as well as in the following section. Because of thelose thermostatistical relationship between H and Ee (see Section.1.1), the most probable option is that either S or J′, or both, shouldave a significant relationship with the Pg/R ratio (see followingaragraphs).
Given the B-DTO-H (Section 2.1.1), any r-strategy species whosendividuals are predominantly placed on the left side (me
↓, Ie2↑,p↓) of Hp, me, Ie2 space of phases is noticeably more productive
higher EeTp/meTp values) than any K-strategy species whose indi-iduals are preferably placed on the opposite edge (me
↑, Ie2↓, Hp↑),
ee Rodríguez et al. (2013b, Fig. 3b). Besides, the most impor-ant organisms that sustain our food consumption, as well ashose plagues that menace our food supply, are r strategists. Givenhat these kinds of individuals of high productivity are in generalery abundant in comparison with other concurrent species, their
oticeable presence is associated to low evenness values (e.g., com-unity structures of subtropical grasslands and tropical wetlandsn comparison with tropical forests; offshore fishery in comparisonith coral reef fish; agroecosystems in comparison with natural
delling 313 (2015) 162–180
ecosystems – in general – etc.). Thus, it is possible to summarizethe above commented relationship in a simplified way as
Pg ∝ J′−1 (15)
The metabolic counterpart of Eq. (15) depends on the evolu-tionary addition of new species on the microbial root of the lifetree. Taking this origin as a rough benchmark, the general trend istoward the increase of body size either in the evolutionary scale(Cope’s Rule; see Cope, 1896; Kingsolver and Pfennig, 2004; Honeet al., 2005; Marroig, 2007) or in regard to the immediately pre-vious trophic link in food chains (Jonsson and Ebenman, 1998;Neubert et al., 2000; Brose et al., 2006; Akin and Winemiller, 2008;Arim et al., 2010; Nakazawa et al., 2010). This rule is particularlyvalid when food chains are long and very efficient (e.g., communitystructures of tropical forests in comparison with subtropical grass-lands and tropical wetlands; coral reefs fish in comparison withoffshore fishery; natural ecosystems – in general – in comparisonwith agroecosystems, etc).
In such a way, larger and larger body sizes imply the possibilityof more and more species added to the life tree, despite the astro-nomical constancy of solar energy input, thanks to a decreasingvalue of respiration (R) per unit of biomass with the absoluteincrease of size (see Odum, 1972, pp. 84–87; Whittaker, 1999, p.865; Banavar et al., 2002; Enquist, 2002, p. 1053; Makarieva et al.,2003; Li et al., 2004; West and Brown, 2005, p. 1575; Reich et al.,2006, p. 457; Loeuille and Loreau, 2006; Rodríguez et al., 2013b,2015a). In such a way, a lower respiration rate per unit of biomass(lower EeTp/meTp values; i.e., more economical species in terms ofenergy consumption by taking similar biomass values as a commonreference framework of comparison between r and K) provides highvalues of species richness (S), and vice versa. Thus, it is possible tosummarize this relationship in a simplified way as:
R ∝ S−1 (16)
By combining Eqs. (15) and (16), we obtain:
Pg
R∝ J′−1
S−1(17)
Given that 1/3–1/8 = 2.66 = 8/3, then Eq. (17) becomes:
Pg
R∝ S
J′(18)
The Pg/R ratio is proportional to the net production of energy (Pn)that remains available at the ecosystem level. Therefore, if Eq. (18)is consistent with observations, then the richness/evenness ratio(S/J′) at the macrostate level (SM/J′M, see Section 2.1.1) should bepositively correlated with the value of total eco-kinetic energy (Eq.(2)) at the macrostate level (EeTM). Appendix B (books “cor”–“rv9”)includes all the data in regard to test this hypothesis starting from6269 macrostates for all the EDHs and surveys included in thisstudy. Starting from these data, the correlation SM/J′M vs. EeTM (seeAppendix B, Table B3) is high, positive and significant in all of thecases. There is not any significant change in regard to this statisticalassociation if SM/J′M is replaced by SM–J′M.
Therefore, in the first instance, the relationship reflected byEq. (18) coincides with observations as well as with the previ-ously commented theoretical premises. In the second instance, theseresults support the idea about the reliability of taking the com-bined SW (see Section 2.2.1, final paragraph) resulting from theinterference between the ecological traveling waves (e→ = Hp
′,SM(xs); and ←e = Hp
′, J′M(xs)) per sequence of macrostates per surveyas a prototype of STIW. That is to say, from previous explana-
tions in this section, e→ would be equivalent to a gross productionwave (Pg) that travels from me↓, Ie2↑, Hp↓ to me
↑, Ie2↓, Hp↑; mean-
while ← e would be equivalent to a gross consumption wave (R)that travels from me
↑, Ie2↓, Hp↑ to me
↓, Ie2↑, Hp↓. As a consequence,
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e→) − (←e) = resulting STIW or net production wave (Pn) thatervades the ecosystem as a whole.
As a result, depending on a supposed positive graphicalutcome from the previous section (see Section 2.2.1, finalaragraph), the combined wave resulting from the algebraicubtraction of SM(xs)− J′M(xs) would produce a SW of net eco-inetic production (Pn = Pg− R ∝ [(e→) − (←e)]). This means anscillating succession of: (1) (SM(xs)→max) − (J′M(xs)→min) ⇒ SW’srest; (2) (SM(xs)→max) − (J′M(xs)→max) ⇒ SW’s node; (3) (SM(xs)→min)
(J′M(xs)→max) ⇒ SW’s trough, across the total gradient of speciesiversity values (�Hp
′, equivalent to dr in Fig. 1a) belonging to aiven EDH. Or, in even simpler words:
1) (maximum of production) − (minimum of consumption) ⇒autotrophic wave crest of net production.
2) (maximum of production) − (maximum of consumption) ⇒wave node or trophic equilibrium, or local climax.
3) (minimum of production) − (maximum of consumption) ⇒heterotrophic wave trough of net production.
It is explored below if this succession of local ecological statesan be analyzed as a STIW useful to assess ecological wavelength�e) values in order to empirically assess the existence of an eco-ogical equivalent of Planck’s constant.
.2.3. Ecological wavelength (�e) and the intra-surveyssessment of he
ec
Given the alternative analytical framework (in comparison withhysics) that have been used here to obtain STIWs (see Sections.2.1 and 2.2.2), nat/individual will be used as the �e measure-ent unit, instead of m. Fig. 1b shows an empirical example (a
uasi-periodic oscillation) of the graphical output derived from theethod explained in Sections 2.1.1 and 2.2.2.The development of QM began with the study of black body
adiation (BBR, light of low frequency emitted by hot bodies). Andeal black body is a perfect cavity resonator that retains internallectromagnetic energy without losses. Thus, a widely used modelf a black surface is a small hole in a cavity with walls that arepaque to radiation. Under these conditions, there is a constanttationary exchange of energy between the atoms (oscillators) ofhe cavity walls and the trapped radiation inside it. The energy ofhe oscillators is:
=(
n + 12
)× h × fo =
(n + 1
2
)× h ×
( v�o
),
n = 1, 2, 3, . . . (19a)
here v: velocity, h: Planck’s constant, and fo: the lowest or funda-ental frequency associated to the longest wavelength (�o) of therst harmonic (equal to 2 × dr in Fig. 1a and 2 × �Hp
′ in Fig. 1b; seeection 2.2.1 for additional explanations). Since the fundamentalavelength (�o) reduces from the fundamental harmonic of fre-
uency fo to an overtone of frequency f > fo in which � < �o in theame fashion that n (harmonic number; or the number of antinodesn a stationary wave = ot in books “cor”–rv9, Appendix B) increases,hen it is possible to express Eq. (19a) as
= 12
h × f = 12
h ×( v
�
)(19b)
Figuratively, it is possible to say that an atomic oscillator actsust as a tiny musical instrument that spontaneously resonates at aery small scale. In such a way, energy is quantized or “discretelyxchanged” starting from an elemental amount (h), and the prod-
ct h × f defines the variation of energy (�E) allowed between twoontiguous energy levels separated to each other by a gradient.nly during quantum transitions, that is to say, when an oscilla-or “jumps” between two energy levels, those quanta of energy
delling 313 (2015) 162–180 169
are exchanged. If n + 1/2 is reduced to (n + 1/2) – 1, then Eq. (19)indicates that the amount of irradiated energy is
�E = h × f (20)
The initial opinion, at the beginning of QM, was that h was onlyvalid in the instant of interaction atom ↔ cavity radiation. However,Einstein showed later on (1905) that the proposal from Planck wasalso valid for isolated photons, whose kinetic energy is:
E = h × f = h ×(
c
�
)(21)
where c and �: speed and wavelength of light, respectively.Finally, de Broglie (1924) proposed the existence of “mat-
ter waves” by extending the wave-particle duality from light tomaterial particles. This meant that every particle of matter (e.g.,electrons, neutrons, protons, etc.) with a rest mass m and a velocityv was associated to a real “pilot wave” whose � was related to themomentum of the particle (p = m · v) by the equation:
� = h
p= h
(m × v)∴ h = � × m × v (22)
It is evident that in this paper a single individual is taken, on theaverage, as the ecological equivalent of a quantum particle.
Summarizing all the previous sections, there is: (1) a generalcoherence between the results from the ecological application ofconventional physical methods and well-known ecological prin-ciples (Rodríguez et al., 2012, 2013a,b, 2015a); (2) a previousempirical result about the existence of an evolutionary equiva-lent of Planck’s constant at the inter-survey and inter-taxonomiclevel (he
ev; Rodríguez et al., 2015b); (3) an analytical equivalencebetween any ecosystem under SEC and a cavity resonator due to theinfluence from B-DTO-H and TAC, and (4) a noticeable structural sim-ilarity between STIW in Fig. 1b and SW in Fig. 1a. These set of itemsindicates toward the feasibility of applying QM to understand theecosystem functioning, with the essential condition of measuringappropriate �e values.
Each vertical grid (see dashed lines) in Fig. 1b coincides withan extreme value of STIW’s A: either a crest or a trough inwhich the derivative dSTIW/dHp
′ = 0 = first derivative function’s root.So d1, 2, 3. . . n are halves of ecological wavelength (1/2�e) for theset of Ms included by the �Hp
′ between two consecutive grids.This means that �e = 2dn = 2 × �Hp
′ = 2 × (Hp′2M− Hp
′1M), where
Hp′2M− Hp
′1M are the upper and lower wave boundaries (maximum
values either for crests or troughs) per macrostate, respectively.�e has a variable value per macrostate (in a similar way to me
and Ie) since dn is not constant across the full range of Hp′ val-
ues (i.e., STIWs are quasi-periodic waves). In those very rare casesin which a given M position exactly coincided with dSTIW/dH′p = 0
(the sheer vertex of a crest or a trough), then �e was assessed as1�Hp
′ between the two crests or troughs adjacent to the respec-tive M position. The alternative possibility (that yields equivalentresults) in these cases is the measurement of �e as 2�Hp
′ betweenthe two lateral local nodes nearest to a given M. That is to say, twicethe distance between the two maximum and nearest absolute val-ues of dSTIW/dH′p in which dSTIW/dH′p has opposite signs (i.e., �Hp
′
from +dSTIW/dH′p → max to − dSTIW/dH′p → min). In general, this
method (measuring �e values through the curve of first derivativeof the cubic spline adjustment STIW vs. Hp
′) shows the underly-ing pattern in a better way when very partitioned distributions areexplored.
The important point is to be able to measure �e values that
reflect certain local ecological conditions given the functional inter-action between me, Ie and �e, at the same time that the overlapbetween different contiguous combinations of these variables isminimized as much as possible. It was impossible to assess the
1 cal Modelling 313 (2015) 162–180
vdswdfe
e
�
E
tmmep
Eveosd5
paara
h
widc�odt(ctit
cfvsalm
Fig. 2. A conceptual diagram that embraces the essential methodological steps andstructural milestones of this study. v: molecular velocity values. �2
v: variance ofv values. kB: physical Boltzmann’ constant. T: absolute temperature. m: molecularmass. M–BDv: Maxwell–Boltzmann density distribution of molecular velocity val-ues. �2
Hp: variance of Hp (species diversity per plot) values. Ee: eco-kinetic energy
(see Eq. (2). ke: ecological equivalent of Boltzmann’s constant. HT : total value ofspecies diversity at the survey level. B-DTO-H: biomass-dispersal trade-off in func-tion of species diversity values (see Section 2.1.1). ECR: ecological cavity resonator(see Section 2.2.1). Pg : gross production, R: respiration, S: species richness, J′: speciesevenness (see Eq. (14) and Section 2.2.2 about the relationship between these vari-ables). ∝: “proportional to”. EeT: total amount of eco-kinetic energy per macrostate(M) or plot (see Eq. (2)). r: Pearson’s correlation coefficient between S/J′ and EeT
per M (see Section 2.2.2). STIWs: Stationary trophic information waves assessed ina scalar field Hp , S–J′ . s.a.: cubic spline adjustment. he
ev: ecological equivalent ofPlanck’s constant at the evolutionary scale (see Section 1). �2: Chi-square test. p:significance level. meM: mean biomass per individual per M (category of Hp valuesinto a given EDH), xs. IeM: mean dispersal activity (see Eq. (3)) per individual per M,xs. �eM: ecological wavelength (xs) per individual per M: twice the distance on theabscissas (�Hp) between successive extreme values on the ordinates (whatever amaximum or a minimum) along the spline adjustment of S(xs) − J′(xs) in function of Hp
values. xs: standardized value. Hp′: midpoint of the category of Hp values per EDH .
heec: ecological equivalent of Planck’s constant at the ecological scale. C.E.: com-
70 R.A. Rodríguez et al. / Ecologi
alue of �e for some Ms (e.g. the first bar in Fig. 1b by the left)ue to absence of one of the two boundaries required to mea-ure dn. (mainly at the lowest overtones with the longest meanavelength). Local fluctuations with a wavelength outstandinglyifferent to the remaining sections of the STIW were also excludedrom the calculation of �e, mainly when they were near to the rightdge of the full range of Hp
′ values.Starting from the assessment of �e values, the next step was to
xplore if it was possible to prove that
eM =hec
eM
peM= hec
e
(me × IeM)∴ hec
e(o) = �eM × meM × IeM,
or hece(o) = �eM ×
(2
12
meM ×√
I2eM
)∗ (23)
e(e)M =12
hece(e) ×
(IeM
�eM
)= Ee(o) (24)
(according to Eq. (2))where (o) and (e) are observed and expected values, respec-
ively; peM: mean ecological momentum per individual peracrostate; meM: mean individual biomass per macrostate; IeM:ean value of Eq. (3) at the macrostate level; and *: Eq. (23)
xpressed in terms of mean individual eco-kinetic energy (Eq. (2))er macrostate.
With such a goal, the mean value of me and Ie per macrostate perDH per survey were calculated and multiplied by the respectivealue of �e assessed by means of models equivalent to Fig. 1b forvery EDH. This procedure was applied to a representative selectionf EDHs until reaching a minimum value of 1 overtone per EDH perurvey. Fig. A.3 (Appendix A) and Table A1 (Appendix A) includes aetailed example of the assessment of �e and he
ec from EDH number2, rows 317–341, survey ma2, Appendix B.
It was sometimes difficult to differentiate between truly inde-endent local waves and STIW’s insignificant fluctuations. This wasn obstacle to directly measure �e values in 5 surveys (lli, mif1–mif3nd rsn). In those cases, a transformation of Eq. (19) was appliedeplacing physical by ecological variables in order to measure he
ec
s
ece =
EeM[n + 1/2 ×
(IeM/�e0
)] (25)
here EeM: mean value of eco-kinetic energy (see Eq. (2)) perndividual per macrostate; n: number of overtones (ot) per EDH
etected as the number of roots (y = 0) of the spline adjustmenturve of dSTIW/dHp
′; IeM: mean value of Eq. (3) per macrostate; andeo: �Hp = total range of Hp values per EDH (�Hp max− �Hp min = �e
f the fundamental overtone, of). However, this method has threerawbacks: (a) it does not produce direct values of �eM, althoughhis parameter can be indirectly obtained by a transformation of Eq.24) (i.e., �eM = (0.5·he
ec(e)·IeM)/EeM); (b) it depends on the degree of
oincidence between the Hp limits of the sample and the real func-ional limits of the ecosystem, something that is difficult to achieven practice sometimes, and (c) its reliability depends on macrostateshat include several hundreds or even thousands of individuals.
The sequences of values of heec
(o) and Ee(e) were statisticallyompared with he
ec(e) (i.e., a reference value based on a scale trans-
ormation of the physical Planck’s constant mantissa) and the meanalue of Ee(o), respectively, by using the appropriate mean compari-
on tests on Statistica-6 (StatSoft Inc, 2001). Finally, the theoreticalnd empirical consequences derived from these results were ana-yzed. Fig. 2 is a conceptual diagram that embraces the essentialethodological steps and structural milestones of this study.
petitive exclusion. F.R.: functional redundancy. (o) and (e): observed and expectedvalues, respectively.
3. Results
3.1. General statistical results and adjustment between observeddistributions and EDH (Eq. (10))
The whole group of assemblages included a wide range ofvalues (Table B1, Appendix B) with regard to number of plots(from 12 to 193); total number of individuals (from 160 to2.06E+07); bi- or tri-dimensional space (volume) per plot (from0.000001 m3 to 10,000 m2); population density (from 0.51 ind./m2
to 4.68E+11 ind./m3); mean distance between plots (from 4.64 m to2930 m); total species richness per survey (from 11 to 239 species);mean species diversity per plot, and total species diversity persurvey (from 0.652 nat/individual to 2.26 nat/individual, and from1.255 nat/individual to 3.355 nat/individual, respectively); meandispersal capability per plot (from 34.073 –d to 86.896 –d), as well asmean individual biomass per plot (from 5.53E−14 kg to 5.02 kg).So, any regular pattern derived from these data would suggestindependence of scale or taxon, giving place to a pretty generalmodel.
Fig. A.4 (Appendix A, p. 7) shows the graphical results from theset of adjustment tests between observed and expected (Eq. (10))distributions. Firstly, it is appropriate to highlight that the distri-
bution shape derived from Eq. (10) coincides with the expectedshape of a typical gamma distribution in all the panels of Fig. A.4.There are conspicuous cases in this regard due to a larger number of
R.A. Rodríguez et al. / Ecological Modelling 313 (2015) 162–180 171
Fig. 3. Empirical assessment of heec values along a sequence of successive EDHs (ecological distributions of species diversity values) per survey. (v) A typical case from a
stationary survey (rv9). (j) A typical case for a non-stationary survey (mif2). heec
(o): observed value of the ecological equivalent of Planck’s constant (h). heec
(e): expectedvalue of the ecological equivalent of Planck’s constant (h). �eM: ecological wavelength per macrostate (Table A1, Appendix A, includes a typical empirical example in orderto measure he
ec values). meM: individual biomass, mean value per macrostate. IeM: dispersal activity indicator, mean value per macrostate. cw: width of the class of Hp valuesper EDH . #m/Mrg: range of the mean number of microstates per macrostate per EDH per sover each box-whisker for edge EDHs indicates the maximum and minimum values of ot
in which p < 0.05 from the statistical comparison between heec
(o) vs. heec
(e). The respective
Table 1Results from the Chi-square test to assess the statistical similarity between theobserved distribution of species diversity values and the expected one accordingto Eq. (10).*.
Ae NHp �2** df p
cor 66 4.386 4 0.356crf 72 108.109 3 <0.0001css 14 2.067 5 0.839lli 181 9.898 10 0.449ma1 35 3.541 2 0.170ma2 43 5.513 2 0.064ma3 44 2.812 3 0.422ma4 44 2.570 2 0.277mif1 40 5.614 3 0.132mif2 56 25.641 4 <0.0001mif3 51 9.297 2 0.0096Ms 28 1.533 1 0.216mxs’ 19 2.409 4 0.661pli 83 16.129 4 0.003rsn1–3 33 2.611 2 0.271rv1 55 2.172 3 0.537rv2 57 3.079 1 0.079rv3 45 3.712 4 0.446rv4 59 31.964 3 <0.0001rv5 42 7.238 4 0.124rv8 60 6.200 5 0.287rv9 60 4.787 5 0.442
* Ae: type of ecological assembly: cor: massive (non-branching) corals. crf: coralreef fishes. css: coastal succulent shrub vegetation. lli: litter invertebrates in lau-risilva. ma: marine microalgae. mif: marine interstitial meiofauna of sandy beaches.Ms: Mediterranean shrub vegetation. mxs’: mixed shrub’s succulent vegetation. pli:litter invertebrates in pine forest. rsn: tropical rocky shore snails. rv: ruderal vegeta-tion in Tenerife Island. NHp : total number of diversity values distributed. df: degreeso*
e
dTasH
tddoi((
f freedom. p: significance level.* Chi-square test calculated by combining adjacent bins until bin frequency >5 ifxpected bin-frequency is less than or equal to 5.
istributed Hp values (e.g., panels (a), (d), (g), (j), (r), (s), and (v)).his means that the replacement of variables from Eq. (9) to (10) isble to re-create a gamma distribution in ecosystems ecology in aimilar way to statistical mechanics. The observed distributions ofp values also coincide, in general, with this pattern.
Table 1 summarizes specific results from the tests of sta-istical adjustment (�2: Chi-square test) between the observedistribution of Hp values per survey, and the respective expectedistribution according to Eq. (10). There are 3 (crf, mif3 and pli; all
f them with p < 0.05) of 22 surveys in which there was not a fitn agreement with the expected pattern. Two additional surveysmif2 and rv4; also withp < 0.05) were performed under non-SECsee Table B1, Appendix B, rows 17 and 28, columns W and AB),urvey. otrg: range of the number of overtones (ot) per EDH per survey. The numberwithin and almost continuous spectrum of overtones. Daggers indicate those EDHs
figures for the remaining surveys are shown in Fig. A.5, Appendix A.
so the lack of adjustment precisely is the expected result in thesecases. The lack of adjustment in crf and pli could be explained bythe lack of spatial continuity in situ between the whole set of plots(see the sampling description of these surveys in Appendix A, Sec-tion 2). Therefore, the absence of adjustment between observedand expected distribution is unexplainable in only 1 case (mif3).On the opposite edge, survey rv5 should lack adjustment betweenobserved and expected values due to its non-stationary nature (seeTable B1, Appendix B, row 29, columns W and AB); but this contra-diction could be regarded as palliated if we take into account thatthe value of p in rv5 is one of the lower ones in comparison withthe remaining p values.
The above-commented results support a positive answer toquestion number 1 (see Section 1, penultimate paragraph).
3.2. Assessment of the ecological equivalent (heec) of Planck’s
constant (h) at the intra-survey scale
Fig. 3v shows a typical spectrum of observed values of heec cal-
culated from an almost continuous range of EDHs between thenineteenth overtone (ot = 19) and the fundamental one (ot = 1),starting from a stationary survey (rv9). It is possible to see that,despite the continuous reduction of overtones and Hp classes withthe increase of cw from the left to the right, there is a clear “flatness”in which the observed (o) value of he
ec(o)≈ 6.62607E − 01 (being the
value of physical Planck’s constant under intra-atomic-system sta-tionary conditions h = 6.62607E − 34) domains the spectrum of Hp
distributions as a whole from the nineteenth harmonic to the fifthone. When such a pattern seems to abandon the above-mentioned“flatness” (fourth overtone, signaled with a dagger due to p < 0.05for he
ec(o) vs. he
ec(e)), it is only to arrive to a subsequent higher
quantum level in which heec
(o) = 6.62607E + 00 at the fundamen-tal overtone level (ot = 1), or very near to it (ot = 3). That is to say,the system seems to be doubly bounded from the quantum pointof view, with a “transition zone” between two successive values ofhe
ec separated by a unitary increase of the exponent value, but witha constant mantissa. An explanation to this behavior is exposed inthe following section.
Contrastingly, in comparison with the previous case, Fig. 3jshows the spectrum of observed values of he
ec calculated from analmost continuous range of EDHs between the thirtieth overtone
(ot = 30) and the fundamental one (ot = 1), starting from a non-stationary survey (mif2). It is possible to see that there is a clearsequence of continuous increment in the product �eM× meM× IeMfrom the left to the right. Along this sequence, the observed (o) value
172 R.A. Rodríguez et al. / Ecological Mo
Fig. 4. General results (p values per EDH per survey) from the mean comparisonbetween the observed value of individual eco-kinetic energy (Ee , see Eq. (2)) permacrostate (EeM(o)) and the respective expected value (EeM(e)) according to Eq. (24).Tcs
oi
rcsv
q
3ea
b(e(
q
4
tsdoeaTcldc(f(g
tcte
his figure includes only those surveys in which a direct assessment of the ecologi-al wavelength per macrostate (�eM) was possible. Data in the respective book perurvey; columns from “Z” to “AB”, Appendix B.
f heec
(o) coincides with the expected one in only 1 EDH (ot = 6). Thats to say, the system is not bounded from the quantum point of view.
The respective figures equivalent to Fig. 3v and j for theemaining surveys are shown in Fig. A.5 (Appendix A), all of themoincide with some of these two above-commented patterns (i.e.,tationarity – Fig. 3v and panels (a)–(i), (k)–(s), and (u) in Fig. A.5 –s. non-stationarity – Fig. 3j and panel (t) in Fig. A.5).
The above-commented results support a positive answer touestion number 2 (see Section 1, penultimate paragraph).
.3. Equivalence between the observed mean value of individualco-kinetic energy per macrostate per EDH and the expected oneccording to QM
According to Fig. 4, there were no significant differencesetween the observed mean value of individual eco-kinetic energyEe, see Eq. (2)) per macrostate per survey and the respectivexpected value assessed by means of QM (Eq. (24)). Survey rv5under non-SEC) was the exception to this pattern.
The above-commented results support a positive answer touestion number 3 (see Section 1, penultimate paragraph).
. Discussion
Our results support, firstly, that Eq. (10) is useful to modelhe observed distribution of species diversity values by means oftatistical mechanics. In addition, our results sustain that the Hp
istribution is a useful analytical framework to explain the naturef stationary trophic information waves (STIW). STIWs are thessential functional element of a reliable WMSC in order to obtain
significant assessment of heec = 6.62606957Eϕ Je nat/individual.
his constant is the minimum possible value in which an individualan exchange information by eco-kinetic energy at the intra-surveyevel under SEC; with ϕ = −xi,. . ., −3, −2, −1, 0,+1, +2, +3, . . ., +xi,epending on the type of taxocenosis explored. This second out-ome was obtained by using an independent methodological wayi.e., regardless the use of ke as an auxiliary constant) that rein-orces the value of he
ev previously detected at the evolutionary scale4.13566727E − 1 × 1.602176462iB = 6.62606957E − 01 Je per eachradient of �ke = 1 Je nat/individual; see Rodríguez et al., 2015b).
The combination between results from Sections 3.1 to 3.3 and
he importance of the right-skewed nature of Hp distributions (seeitation from Rodríguez et al., 2013b, in Section 2.1.2) indicateshat Eq. (10) must be regarded as a statistical model about how thecosystem works under SEC in order to keep a stable performancedelling 313 (2015) 162–180
in a far-away-from-equilibrium position without a net increase ofenergy input. That is to say, there is not any net flow of trophic energy(i.e., Ee in our analytical context, see Eq. (2)) at the ecosystem levelunder SEC. We can clearly see that a given plant is eaten by a giveninsect and this one is eaten by a given lizard, and so on. However,if there is a thermodynamic open balance (SEC) between energyinput and energy leak at the large scale, then this means that theenergy that has been got by any of the above-mentioned orga-nisms it is either exactly equal to. . . or approximately equal to. . .the energy that has been consumed (i.e., degraded from a higher toa lower metabolic quality) in order to get new trophic energy ofhigh quality.
As a result, by taking SEC as a reference framework, our con-ventional idea about a permanent net flow of trophic energy is anillusion derived from our naïve observation of nature. In fact, theabove-mentioned functional scheme is the only plausible expla-nation that simultaneously matches with (1) the existence of ESE(Rodríguez et al., 2012), with (2) the pro-stationary influence fromB-DTO-H (Rodríguez et al., 2013a), with (3) the ecological fulfill-ment of Boyle–Mariotte’s law (Rodríguez et al., 2013b), with (4)the successful extension of the Maxwell–Boltzmann energy dis-tribution to describe energy pyramids (Rodríguez et al., 2015a),as well as with (5) the previous empirical assessment of he
ec atthe inter-taxocenosis scale (Rodríguez et al., 2015b); because all ofthese 5 patterns are only valid (from the physical point of view)for a steady general condition of random interactions of energyexchange. With a random set of energy exchange any net flow ofenergy is impossible: a net gain in some place and time in favorof any element of the system is immediately compensated by anet energy loss in an alternative space-time point within the samesystem → SEC.
Therefore, the main condition of the ecosystem survival underSEC is to retain its internal energy level, by avoiding any net lossof energy in a way as effective as possible. This goal is attainedby means of internal boundaries (also known as “thermodynamicrestrictions”) that are impervious to the flow of energy (just aswe can be comfortably asleep despite an external temperature of−15 ◦C because we are wearing pajamas – first impervious bound-ary – under a duvet – second impervious boundary – within a closedroom – third impervious boundary – at a house whose doors andwindows are also closed – fourth impervious boundary).
Accordingly, nodes (equilibrium points of maximum potentialcompetition that are physically “impervious” to the flow of energy,see Section 2.2.1 and Fig. 1b) are the internal boundaries to avoidenergy leaks from the ecosystem. However, this seeming solutionposes an even bigger functional problem: a free movement of energyat random in order to supply the metabolic needs of all the ele-ments of the system seems to be impossible at all within a complexsystem full of impervious internal restrictions. That is to say, thefirst solution (impervious internal boundaries: nodes) in order toretain a constant amount of internal energy under SEC requiresa second solution: the dynamic behavior of the ecosystem like aninter-harmonics oscillator or cavity resonator.
For example, the all-encompassing STIW (e− e, according toFig. 1b) from a real survey could be at the vibration mode (over-tone) shown in Fig. 5a, whose original waves (e and e accordingto Fig. 1b) are shown in Fig. 5a . In Fig. 5a (twelve overtone fromthe survey “cor” in Appendix B), there is a particular “ecologicalcomplexion” (see Section 2.2.1, first paragraph) or vibration modecharacterized by certain groups of macrostates (M) isolated to eachother within their respective antinodes. The ecological complexionin Fig. 5a is the following: 1+/2−/3+, 4+/5−/6+, 7+/8−/9+/10−/11+,
12+, 13+, 14+, 15+/16−, 17−, 18−; where + means a wave crest ofecological autotrophy (energy excess) in which e− e > 0; and −means a wave trough of ecological heterotrophy (energy deficit)in which e− e < 0 (see Section 2.2.2, final paragraph). Every slash
R.A. Rodríguez et al. / Ecological Modelling 313 (2015) 162–180 173
(a) (a)
(b) (b)
(c) (c)
Fig. 5. Examples of fluctuations between alternative overtones (or vibration modes) in a survey under SEC. (a) Overtone 12, EDH 20, macrostates 229–246, rows230–247, survey “cor” (massive coral reef) in Appendix B; he
ec(o) = 6.3704E − 04 Je nat/individual, p = 0.7814 in comparison with he
ec(e) = 6.62607E − 04 Je nat/individual.
(b) Overtone 9, EDH 12, macrostates 101–114, rows 102–115, survey “cor” in Appendix B; heec
(o) = 7.6276E − 04 Je nat/individual, p = 0.2429 in comparison withhe
ec(e) = 6.62607E − 04 Je nat/individual. (c) Overtone 17, EDH 22, macrostates 266–284, rows 267–285, survey “cor” in Appendix B; he
ec(o) = 5.3990E − 04 Je nat/individual,
p = 0.1739 in comparison with heec
(e) = 6.62607E − 04 Je nat/individual. The basic trophic waves for (a)–(c) are shown in the respective panel on the right: (a) , (b) , and(c) . M: macrostate. Hp
′: species diversity value, class mark per M. SM(xs): total species richness per M, standardized value (wave e→). J′M(xs): total evenness (Eq. (14)) perM, standardized value (wave ←e). EDH : distribution of species diversity values. STIW: resulting stationary trophic information wave (e − e). ↑↓: alternative changes ofS EDH 2“
imibpttoopi
TIW’s polarity (from STIW+: e − e to STIW−: e − e). The respective positions ofcor” are labeled in panel (a), Fig. A5, Appendix A.
n the previous complexion means a node that hampers the freeovement of energy at random between contiguous antinodes. It
s obvious that this overtone cannot last forever given that its sta-ility is precarious. For example, those individuals at M5
− coulderish due to starvation (ever taking STIW↑ = e− e, or STIW+, ashe reference wave polarity) since they are at a heterotrophic waverough of net production (e− e < 0). Meanwhile, there is an excess
f energy (autotrophic wave crest of net production) at that antin-de on the left frontier in regard to M5−, in which M3+ and M4
+ arelaced. This situation is equivalent for M2
−, M8−, M10
−, M16–18−
n comparison with M1+, M6
+, M7+, M9
+, M11–15+. As a result, the
0, EDH 12, and EDH 22 within the whole spectrum of EDHs explored for the survey
ecosystem “needs” to transfer the “excess” of energy from crest totroughs in order to compensate the disadvantageous trophic situa-tion in M2
−, M5−, M8
−, M10− and M16–18
−. The periodic inversion ofpolarity from STIW↑ to STIW↓ (e− e, or STIW−), and vice versa,is not enough for such a goal because, due to the fixed positionof nodes, a polarity inversion creates an alternative situation thatis perfectly equivalent to the former one, but just the other way
around. That is to say, with a simple change of the STIW’s polar-ity the previously heterotrophic nodes change to autotrophic onesand vice versa, but there is not any longitudinal transfer of energybetween antinodes at the local scale.
1 cal Mo
ooi1acclwr(stti
vtn2(olt
caiAdmHapm
E
wbuhtovckBwp
(
74 R.A. Rodríguez et al. / Ecologi
There is a unique possible solution: an all-encompassing changef the vibration mode, for example, from overtone 12 (in Fig. 5a) tovertone 9 (in Fig. 5b). Now there is a different ecological complex-on in regard to STIW↑: 1−/2+, 3+,4+/5+, 6+, 7+/8−, 9−,10−/11+, 12+,3+, 14+, 15+,16+/17−, 18−; and the macrostate previously taken as
reference point (M5) has been transferred from a heterotrophicondition to an autotrophic one in a similar way to M2 and M16, inomparison with the previous condition in overtone 12. Neverthe-ess, this new complexion also has certain drawbacks: (a) there is a
ide open area (an antinode without any node on the right) at theight edge of the distribution that could mean an energy leaking,b) the first two nodes on the left are about to fail, and (c) M17 istill under heterotrophic conditions in regard to STIW↑. Therefore,his situation is not either permanent because, in order to feed cer-ain macrostates (e.g., M2, M5 and M16), it is necessary to relax thenternal set of energy boundaries.
So the ecosystem can perform a new leap between alternativeibration modes: from overtone 9 to overtone 17 (i.e., from Fig. 5bo c) by getting a more “conservative” position (a higher number ofodes) in order to restrict the probability of an energy leak: 1−,−/3+, 4+/5−/6−/7+/8−/9+/10−/11+/12−/13+/14−/15+/16−/17+/18−
taking STIW↑ as the reference wave polarity). When this lattervertone is too limiting in regard to energy movement due to itsarge number of nodes, then the ecosystem leaps again to an over-one similar to those in Fig. 5a or b, and so on.
Obviously, the former example of a succession of ecologi-al overtones implies a noticeable oversimplification, since thoserrows included in Fig. 5b and c are only indicative of fixed pos-tions in comparison with Fig. 5a, taken as a reference condition.ctually, every change of overtone also mean fragmentation (whene number of Hp categories is higher and the respective EDH isore partitioned) and fusion of macrostates (when de number of
p categories is lower and the respective EDH is less partitioned),s well as changes of their relative trophic positions, species com-osition, number of individuals, and mean total trophic energy peracrostate:
eTM =EeTs
M#(26)
here EeTs: total eco-kinetic energy per survey, and M#: total num-er of macrostates per EDH. The value of EeTs remains constantnder SEC regardless inter-overtones fluctuations. Additionally, aigher EDH’s partition level (higher overtones) implies the simul-aneous mean reduction of NM and EeTM (where NM: mean numberf individuals per macrostate). So, it is obvious that increasing thealue of �e is the only way to keep the relationship shown by Eq. (23)onstant in front of any of the two alternatives for individual eco-inetic energy fluctuations (either due to me or Ie fluctuations—see-DTO-H in Section 2.1.1). Nevertheless, the former example (Fig. 5)ould have been impossible to understand without such oversim-lification.
Some corollaries in order to profile the issue so far:
1) Ecological steadiness (SEC) at the large scale is paradoxicallykept by constant fluctuations or internal thermodynamic adjust-ments at the small scale. A funambulist that successfully keepshis position on the tightrope could be a good quotidian exampleabout the essential dynamics of this model. The system of locksof the Panama Canal also mimics the above-commented pat-tern, since each antinode restricted by two movable nodes atthe ecosystem level is a sort of double floodgate for energy in a
similar way to the locks system of the Panama Canal transientlyretains water: “real ecosystems are near steady-state in long-termmean characteristics, but are dynamic in short-term responses”(Schramski et al., 2007).delling 313 (2015) 162–180
(2) The “blending of energy” across the ecosystem due to leapswithin the spectrum of overtones “shakes up” the metabolic andbehavior performance of living things by getting a more effi-cient management of eco-kinetic energy. For example, energywill be consumed in a more exhaustive way if two or moremacrostates that are complementary to each other (partialfunctional redundancy) remain transiently included within acommonly shared antinode.
(3) Every inter-overtone oscillation means a switching betweenalternative competition/redundancy intensities due to varia-tions of the inter-node distance, with the significant advantagethat there is no macrostate that perfectly coincides with animpervious node of competitive exclusion under stable natu-ral conditions. That is to say, in the classic lab experiment aboutcompetitive exclusion (Gause, 1934) two species were compet-ing to each other and one of them was eliminated at the longrun. But, under normal natural conditions, the position of eachequilibrium node of competitive exclusion is determined by theset of positions of several surrounding macrostates, and each ofthem embraces a complex internal assemblage of species. Sincecompetition is a mutually inhibitory relationship (−,−; seeOdum, 1972), the whole set of species of the surrounding sys-tem of macrostates produces local states of mutual and generalcompetitive inhibition among species that lead to a sequence ofempty virtual spaces along the distribution of Hp values: a chainof nodes. In such a way, competition performs an underlyingorganizational influence that promotes a coexistence parti-tioned into antinodes by avoiding the spontaneous eliminationof species at random: “I treat competitive exclusion as a centralorganizing concept in community ecology, though the relaxation,disruption, or prevention of interspecific competition is probablythe status quo in most communities” (Palmer, 1994, p. 552).
(4) In a similar way to a pilot wave that keeps the electronfar from the atomic nucleus, a dynamic ecological structurebased on an ever-changing sequence of many local equilib-rium points (nodes) is able to keep the ecosystem far awayfrom a single aggregate equilibrium point that would meana position of maximum internal entropy (i.e., a maximumlevel of ecological degradation (i.e., ��H′p → 0) if the ecosys-
tem shrinks toward its TAC; see Section 2.1.1). In such asense, higher values of total species diversity (HT) producewider distributions of Hp values (see relationship between� and � in gamma distribution, Eq. (8), Section 2.1.2) withmore available “space” (�Hp
′) to insert a larger number ofnodes in order to avoid the spontaneous drift (second law ofthermodynamics) of the set of macrostates toward the TACunder SEC. This means a better control of the internal envi-ronment, just as it has been pointed out before (e.g., Margalef,1991; p. 368).
(5) In spite of the inter-overtone permutations under SEC,the observed value of he
ec per macrostate (M) per EDH
(heec
(o) = meM(o)× IeM(o)× �eM(o)) tends to keep a mean valuewith p > 0.05 in comparison with the expected value ofhe
ec(e) = 6.62606957Eϕ Je nat/individual, with ϕ = −xi,. . ., −3,
−2, −1, 0,+1, +2, +3, . . ., +xi (Fig. 3 and A5, Appendix A). Thatis to say, there is an all-encompassing pattern in the oscil-lating functioning of the ecosystem that yields an ecologicalquantum of action: the minimum and discrete (non-continuous)mean value of eco-kinetic energy per individual that can beexchanged per each information unit across a species diversitygradient within a given ecosystem.
(6) If items from (1) to (5) are reliable inferences there should be a
significant change of the basic wave pattern (Hp′, SM(xs) vs. Hp′,
J′M(xs), or e vs. e waves) from surveys under SEC to surveysunder non-SEC.
R.A. Rodríguez et al. / Ecological Modelling 313 (2015) 162–180 175
Fig. 6. Miscellany of inferences in regard to the functional meaning of heec taken the survey of massive corals (“cor”) as a reference example under SEC. (a) B-DTO-H. meM: mean
value of individual biomass per macrostate (M) per EDH ’s spectrum. IeM mean dispersal value per M per EDH ’s spectrum. xs: standardized values. (b) Distribution of meanvalues of ecological wavelength per M (�eM) per EDH across the B-DTO-H shown in (a). (c) Correlation between the general mean value of �eM per EDH per survey (�e(�EDH,s))and total species diversity per survey (HT ). Two outlier observations (surveys rsn1 and rsn2) were excluded. Meaning of labels in footnote of Table 1. (d) Stationary trophicinformation waves (STIW: e (photosynthesis, production, anabolism; dark gray curve) & e (respiration, consumption, catabolism; pale gray curve) waves from an EDH (#20)in a survey (“mif2”, rows 94–11, Appendix B) under non-SEC. The same non-SEC wave pattern can be obtained from rv7, rv6 and rv5. (e) e and e waves of the fundamentalovertone (of ); each small square dot is a macrostate. (f) Combination of waves in Fig. 5 and waves in the of . TAC: thermostatistical accretion center (see Section 2.1.1, lastparagraph) given by the point of maximum concentration of nodes (Hp
′ ≈ 0.77 nat/individual). (g) Propagation of the “ecological message” in function of time (t) as a processof wave modulation (wm) based on a sequence of the wave packet of Fig. 6f. cw: carrier wave. mwe: message wave envelope, a.k.a. “enveloping wave”. White area: currentmoment of observation. Gray spectrum: flow from pass to future observations. (h) Total EDH ’s spectrum of the survey in terms of Hp
′ , EeTM values. Hp′: class mark (midpoint
per Hp category) per M. EeTM: total eco-kinetic energy per M. ot : overtone number. The panel in the right inferior corner shows the typical waves pattern from a drop in thewater surface. (i) Discontinuous EDH (#26, rows 347–367 in “cor”, Appendix B). K–S and �2: Results from the Kolmogorov–Smirnov and Chi-square goodness of fit tests,respectively, from the comparison between the observed distribution of Hp values and the standard gamma distribution. (j) Continuous EDH (Fig. A4a, Appendix A; EDH#3,rows 9–16 in “cor”, Appendix B). (k) Example of STIW in a simple graphical model that includes three trees. Arrow 1: local alternation of e & e between morning and eveninga n (of ,
raphie
FUe
nd vice versa, tree#1. Arrow 2: alternation of e & e between midnight and nooe & e between morning and evening and vice versa, tree#3. (l) Scale-dependent gcosystem composed of 5 taxocenosis with different he
ec values.
In regard to item (5): heec emerges because, on the B-DTO-H (see
ig. 6a) that, in turn, sustains the hump-backed curve (inverted-shape) of production (EeTp) vs. species diversity (Rodríguezt al., 2013a), it is functioning a secondary U-shaped trade-off
see above) and vice versa, for the system as a whole. Arrow 3: local alternation ofcal model of the “ecological message” in space (s) and time (t) from a hypothetical
(i.e., inverted in regard to the former one) between �e and Hp′
within every EDH under SEC (see Fig. 6b). This connection betweenthe classical realm and the quantum realm (∩\∪) is clearer inthe same measure in which the ecosystem is better adjusted
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76 R.A. Rodríguez et al. / Ecologi
o SEC. That is to say, in the analytical context of statisticalechanics a gradient of species diversity per plot (Hp) can be dis-
ggregated (⇔) in three main functional sections, differentiated byhe interrelationships of increase (↑) or decrease (↓) between three
ain variables:
↓ep, I2↑
e , H↑p ↔(
mep × I2e
)↑, Hp (intermediate values) ↔
m↑ep, I2↓e , H↑p ∴ mep × I2
e × Hp = ke (27)
Equivalently in the analytical context of quantum mechanics inegard to an intra-EDH gradient of Hp values fragmented in its threeain functional sections, it is fulfilled that:
↓eM, I↑eM, �↑eM ↔ (meM × IeM)↑p (intermediate values), �↓eM ↔
m↑eM, I↓eM, �↑eM ∴ meM × IeM × �eM = hece (28)
In even simpler words, �e compensates the variation from anyf its two complementary variables, or from the highest product ofntermediate values of both (meM(�)× IeM(�)), in a certain way thateeps a constant mean value of the product meM× IeM× �eM.
If we want to grasp the above-commented pattern in a betteray, it is possible to use a simple mechanical model: let us suppose
system of many balls (i.e., macrostates), all of them connected toach other by many springs (i.e., STIWs) glued around and placednto a deep and wide gravitational field (i.e., an inverted gamma dis-ribution of Hp values) whose center is the respective TAC in whichhere is a maximum local value of total eco-kinetic energy per plotEeTp = Np× 1/2 mep× Iep
2). The force of attraction inter-balls will betronger in the same measure in which certain balls are closer tohe TAC, and so the springs connecting these balls will be propor-ionally compressed (shorter �eM values) and exerting a strongerpposite force, by avoiding in such a way the collapse of the sys-em as a whole by shrinking toward the TAC. So there is a trade-offetween ecological momentum (pe = me× Ie) and �e in function ofhe distributed gradient of Hp.
Alternatively, if we prefer an ecological simile equivalent to theormer one, then we can say that those species and individualslosely associated with each others in the form of a given set ofacrostates tend to move toward the intra-EDH point of intermedi-
te Hp′ values (TAC), in which there is a maximum concentration of
igh values of total eco-kinetic energy per plot. Therefore, it is nec-ssary to increase the number of nodes (i.e., shorter wavelengths)nterposed between those macrostates near to TAC by regulatinghe free movement of energy in order to avoid a too high andntoward level of competition. Or, by inverting the cause–effectelationship, it is also possible to assume that nodes of competi-ive exclusion can be closer to each other at the TAC level preciselyue to the higher amount of total eco-kinetic energy per plot thatvoids a harmful increase of competition. Anyway, from the pos-tivist point of view, a theoretical disagreement in this regard isonsense because these three models are similar to each otheriven that their observable results are perfectly equivalent.
In the first place, these examples highlight the pro-structuralole of species diversity. In the second place, the net increase of �e
i.e., the U-shaped distribution of �e in Fig. 6b is not perfectly sym-etrical; it is biased to the right because of the right-skewed nature
f gamma distribution) from the left to the right edge of �Hp′ in
DH indicates that: (1) Those macrostates of high species diversityend to be placed into wider antinodes (more spaced nodes of com-
etitive exclusion) with a lower frequency of replacement betweenvertones (e.g., the positions of M16, M17 and M18 across the inter-vertones permutations in Fig. 5 are more stable or “conservative”han those positions either in the middle or in the opposite edge ofdelling 313 (2015) 162–180
the gradient). (2) From the previous item, all of the macrostates ofhigh species diversity fulfills some typical traits:
a) They require a cheaper and more efficient metabolism in termsof energy consumption per unit of biomass (EeTp/meTp), becausethe probability of a “trophic subsidy” due to a random changeof overtone (as in regard to M5 from Figs. 4a to 5b) is lower forthem. This is in agreement with previous statements by sev-eral authors (e.g., Margalef, 1963; Odum, 1969; Rodríguez et al.,2013a) about the existence of a general trade-off between totalproduction (EeTp) and trophic efficiency (availability of energyper unit of biomass: EeTp/meTp) with the increase of speciesdiversity.
b) If �e is longer (wider antinodes) at highest Hp′ values, then this
means higher coexistence and lower competition levels.(c) Finally, from (a) and (b): there is a structure of differential selec-
tive pressures from low to high Hp′ values.
The U-shaped distribution (Fig. 6b) of Hp′, �e at the intra-survey
scale does not indicate that the same pattern is valid for inter-surveys and inter-taxocenosis comparison in regard to total valuesof species diversity per survey (HT). Quite on the contrary, accordingto Rodríguez et al. (2015b), it would be very probable that ecosys-tems of high species diversity have a higher capability to magnifythe human impact on the environment (i.e., a higher resolvingpower from the point of view of ecological monitoring; in a simi-lar way in which an electronic microscope has a higher resolvingpower than an optical one because of the shorter wavelength usedfor image explorations in the first case) due to their trend to reducethe mean value of �e in the large scale. Now this hypothesis is sup-ported by our results (see Fig. 6c; based on original data in TableB4, Appendix B).
In regard to item (6): there is a noticeable degree of steadiness(flatness of y values at the large scale either in regard to e or e)under SEC (see Fig. 5a–c). On the contrary, there is a noticeableincrease (y values with slope either for e or e, mainly regardingthe second one) under non-stationary ecological conditions (NEC;see Fig. 6d in comparison with Fig. 5a–c). This conspicuous incre-ment of catabolic activity (e, see theoretical foundations in Section2.2.2) from low to high Hp
′ values cannot be steadily sustainedfrom inside the ecosystem itself, it should be getting a net input ofenergy from the outside that explains its NEC (i.e., 2EeTp /= Npke/Hp;ke(o) /= ke(e); he
ec(o) /= he
ec(e)). As a result, it is justified to expect a
net flow of eco-kinetic across trophic chains only under NEC. Thus,under NEC, there would be a weakening of the internal set of nodesused to keep species apart from each other in conditions of highintensity of competition under SEC. This is perfectly understand-able if we think that conserving internal energy in a so strict manneris not necessary, because there is a net input of energy from theoutside.
This explains why the observed distribution of Hp values is morecompressed than the expected one (Eq. (10)) under NEC (see, e.g.,Fig. A4j; Appendix A). That is to say, from the mechanical pointof view, we could say that Hp distributions deploy certain degreeof combination between elasticity and inertia. These concepts havebeen profusely used in ecology before (e.g., Holling, 1973; Orians,1975; Westman, 1978; Koons et al., 2007; Bodin and Wiman, 2007),but with drawbacks derived from recurrent terminological misun-derstandings, as well as from modeling obstacles due to difficultiesregarding their empirical measurement (e.g., Lewin, 1983; Grimmand Wissel, 1997; Odenbaugh, 2001).
However, inertia and elasticity can be tangibly seen by using
our model: Hp distributions are more compressed to the left edgeof Hp values than the theoretically expected pattern when theyare accelerating toward a higher spectrum of diversity values(see Fig. A4j, Appendix A). Contrastingly, they should be more
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R.A. Rodríguez et al. / Ecologi
ecompressed (expanded) to the right edge of Hp values than theheoretically expected pattern when they are in motion toward aower spectrum of diversity values (see, e.g., Fig. A4s and Fig. A4t;ppendix A). This is the expected behavior from the B-DTO-H: on
he one hand, those macrostates at the right edge of Hp distribu-ion are heavier and slower. Hence they react more slowly (with
higher inertia) to net energy inputs, jamming the translationf the distribution as a whole by obstructing the movement ofhose macrostates with the opposite set of features toward higheriversity values. This results in compressed distributions of Hp
alues under NEC, in comparison with the stationary expected pat-ern (Eq. (10)).
On the other hand, those macrostates at the left edge of Hp dis-ributions are lighter and faster. Thence they react more rapidlylower inertia) to net energy leaks, pulling the distribution as ahole toward lower diversity values. But this movement is also
bstructed by those macrostates of higher inertia that react withelay in the opposite edge of Hp gradient. This results in decom-ressed or lengthened distributions of Hp values under NEC, inomparison with the stationary expected pattern (Fig. A4s and Fig.4t; Appendix A). Finally, the elasticity of the all system is sup-orted by the internal set of nodes connected with each other byrophic waves. As a result, those macrostates of high diversity val-es have an important conservative influence in favor of ecologicaltability, either when HT increases or when it decreases, becausehey act as a ballast system that avoids abrupt variations of HT. This
eans that high diversity ecosystems are more stable under naturalonditions but, paradoxically, more sensitive to the environmentalmpact due to their shorter �e (see previous explanations in regardo Fig. 6c).
Coherently, the perfect coincidence between the observed pat-ern and the expected one according to Eq. (10) indicates that theistribution is stopped from the successional point of view (i.e., it isnder SEC). As a result, the stability of the set of interspecific rela-ionships of trophic chains under SEC is not kept by the net flowf trophic energy, but by the conservative influence of the set ofechanisms of reproductive isolation between species (hybridiza-
ion barriers) which act as nodes from the genetic point of viewhen the system is “waiting” for the next net input, either positive
r negative, of energy from the outside.There are three additional points that require clarifications: (a)
hy there is a leap from heec
(e) = 6.62606957Eϕ Je nat/individualo he
ec(e) = 6.62606957Eϕ + 1 Je nat/individual from those EDHs with
t� 1 to those EDHs in which ot→ 1 in almost all the surveys? (b)hy there is a transition in which he
ec(o) /= he
ec(e) (see † in Fig. 3v
nd Fig. A4) across the succession of Hp distributions, even in thoseurveys under SEC? (c) Regardless the explanatory capability of thisodel; is it possible to offer a simple example about the nature of
nformation trophic waves (STIW) in practice?About point (a): the fundamental overtone (of) means total
bsence of internal boundaries to control the free movement ofnergy (see Fig. 6e) since, at the fundamental frequency, there arenly two edge nodes which coincide with the minimum and max-mum values of Hp per survey (always excluding those plots in
hich Hp = 0). So, taking into account previous arguments in thisection, this overtone has a very low level of efficiency to retainnergy. Therefore, the only effective solution to keep the inter-al stability of of, despite its almost full open nature, is that theurveyed subsystem must be enveloped in a super-system with aigher level of functional hierarchy. So the value of he
ec of thisuper-system (6.62606957Eϕ + 1 Je nat/individual) does not coin-ide with the internal he
ec value (6.62606957Eϕ Je nat/individual)
f the surveyed subsystem itself. This explains the leap from Eϕ toϕ + 1 in the exponent of heec from the wide spectrum of EDHs inhich ot� 1 to the narrow spectrum of EDHs in which ot→ 1 in all
f the surveys (see Fig. 3 as well as Fig. A5, Appendix A).
delling 313 (2015) 162–180 177
That is to say, the surveyed system is able to keep its level of ther-modynamic isolation from the environment in the same measure inwhich ot� 1 because the larger the number of nodes, the strongerthe level of internal control on the free movement of eco-kineticenergy (Ee). But, in the same measure in which the system is nearto ot→ 1, the system shares more and more of its internal controlwith the super-system in which it is enveloped. I fact, according toour data, the of means a “lethargic internal state” characterized bya minimum exchange individuals-environment: despite the oppo-site correlation between the mean value of total eco-kinetic energyper macrostate (EeTM) and overtone number (ot) per EDH per survey(see example in Fig. A6a, Appendix A); there are direct and signifi-cant correlations between the mean values of dispersal activity (Ie)and individual eco-kinetic energy (Ee) per macrostate per EDH andot (see Fig. A6b and A6c, respectively). That is to say, individualsmove faster and more frequently near to the TAC in the same mea-sure in which ot� 1, hence the greater number of nodes – shorter �e
values – to avoid drifting to maximum entropy levels (at the TAC’spoint). Thus, when ot→ 1, there is a general state of ecological lax-ity and lack of internal control: the system needs to be supportedby the super-system to which it belongs.
If this explanation is plausible it should be possible to observe anecological structure that mimics wave modulation (wm) in physics.Wm is based on a low hierarchy carrier wave (cw) that changes itsamplitude and frequency by means of groups of oscillation (inter-nal wave packets with either different frequency or amplitude,see Fig. 6f and g) in order to transmit a secondary or envelopingwave (a.k.a. message wave envelope: mwe) with a higher hierar-chical level. Graphically, mwe seems to envelope the carrier wavein space and time (Fig. 6g). If the internal oscillations of Fig. 6f andg are shown in terms of Hp
′, EeTM fluctuations, then it is possible tosee in an even clearer way (Fig. 6h) that all of the internal STIWswhen ot� 1 are enveloped by the of (ot→ 1), which acts as an all-encompassing resonance cavity. It is quite difficult to overvaluethe implicit importance of this fractal structure. For example, itindicates that:
(1) The connection, theoretically assumed in the models so far,between all the ecological systems and taxocenosis on theplanet it is an empirical necessity given the leap from heEϕto heEϕ + 1 across the change from ot� 1 to ot→ 1 in all theanalyzed surveys.
(2) The pattern shown in Fig. 6g and h could be analytically under-stood as a perturbation in a preceding field (in a similar way tothe fall of a drop in water surface; see inset in the left inferiorcorner of Fig. 6h). This would permit to apply modifications ofthe basic algorithms of the physics of fields in order to modelthe eco-evolutionary process: that is to say, the evolutionaryemergence of reptiles as a perturbation in a “field” of amphib-ians that, in turn, would be equivalent to a physical perturbationin a field of fish, and so on, until a point in which there wouldbe an analytical bifurcation between the emergence of life onthe Earth as either a pro-negentropy perturbation in a molec-ular field (see, e.g., England, 2013) or as a perturbation in anall-encompassing universal field of microbial life (panspermia,e.g., Nicholson, 2009; Wickramasinghe, 2011).
(3) All the ecosystems are super-systems that include many sub-systems interwoven with each others. Each of these subsystemshas its particular quantum vibration mode coupled in an all-encompassing arrangement characterized by a state of quantumdecoherence (inequality of A, �e, and wave phase between theinternal elements of the super-system). For example, the tax-
ocenosis of nocturnal animals can be regarded as in its of inthe morning, when diurnal animals are not in their of, and viceversa. In a similar way, the peaks of activity of several kindsof animals living in a coral reef are not exactly tuned to each
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78 R.A. Rodríguez et al. / Ecologi
other. Then the processes of destructive interference and reso-nance (see, e.g., Rodríguez et al., 2013b) among all these set ofconnected STIWs should have an important regulation role.
4) From items (1) and (3), a multilevel ecological failure is very dif-ficult. However, it is possible to suppose that several systemsbecome in a process of inter-systems resonance (a periodic andmutual reinforcement of the wave amplitude due to a generalcoincidence between wave phases, i.e., linkage crests ↔ crestsand troughs ↔ troughs between different STIWs) because of apervasive negative influence on nature. In this case, the mostrational expectation would be to observe a discrete (discontinu-ous) catastrophic change (instead off a gradual change) towardworse ecological conditions. In fact, the most probable optionis that the biosphere as a whole (the enveloping STIW of max-imum level) has its particular and all-encompassing he
ec valuethat could change in a discrete fashion, in a similar way to asingle taxocenosis, but in an overwhelmingly higher value. Thispossibility ought to be regarded as a very disturbing alternative,mainly if we take into account that high diversity ecosystems(the stabilizer ballast at the biosphere level, according to pre-vious comments) are the most affected systems by humaninfluence all over the world.
As a general result from this point, the ecosystem is triplyestricted from the quantum point of view: First restriction: a gen-ral value of he
ev = 6.62606957E − 1 Je nat/individual (Rodríguezt al., 2015b) valid for any migratory or evolutionary transi-ion between taxocenosis. Second restriction: a local value athe intra-taxocenosis level of he
ec = 6.62606957Eϕ Je nat/individual.hird restriction: a general value at the intra-ecosystem level ofe
ec = 6.62606957Eϕ + 1 Je nat/individual (of).About point (b): every analytical algorithm has certain draw-
acks. The main methodological hypothesis that sustains thisroposal is the following: if the emergence of he
ec implies a non-ontinuous (discrete) behavior regarding the exchange of energy,hen it would be expected that such a behavior can only be seen inhose cases of distributions of Hp values with certain level of sta-istical discontinuity (e.g., see Fig. 6i – discontinuous distribution –n comparison with Fig. 6j—continuous distribution). The analyticalruitfulness of this hypothesis has been supported by its explana-ory capability in regard to several well-known functional patternsf ecological dynamics (remaining content in this section). That iso say, all those cases in which he
ec(o) /= he
ec(e), even under SEC,
re cases of Hp distributions of low statistical resolution (that cane easily modeled by Eq. (10)), which are very similar to Fig. 6j, butissimilar in comparison with Fig. 6i (EDH # 26, book “cor”, rows47–367, Appendix B).
In other words, the distribution of species diversity values, atow levels of statistical resolution (i.e., with a small number of
ide categories of Hp values ordered in a continuous way), can benderstood as a classical system. Alternatively, the same analyticalontext becomes a quantum system at high levels of statistical reso-ution (i.e., with a larger number of narrow categories of Hp valuesrdered in a discontinuous way). This is the key point that explainshe title of this article. After all, this dual behavior indicates thathe statistical distribution of Hp values acts right as a very potent
agnifier that mimics the physical reality itself: a naked eye visi-le physical object can be macroscopically described (i.e., at a lowesolution level) by means of classical mechanics but, if this objects observed at a very high resolution level (atomic level), then itsnternal quantum properties become clear in the smallest scale. Inddition, those traits in the small scale have a strong influence on
raits at the large scale, and vice versa, because the point (TAC) ofaximum accumulation of nodes in Fig. 6f–h coincides with theoint of maximum probability (distribution mode) in Fig. 6i and j.his pattern is valid for all of the surveys under SEC.
delling 313 (2015) 162–180
From the point of view of wave modulation (see commentsabove) the transition in which he
ec(o) /= he
ec(e) (between ot� 1 and
ot = 1) is an expression of the dual nature (wave and particle) of theecosystem as a whole. That is to say, the above-mentioned discon-tinuity in regard to he
ec(o) = he
ec(e) is recovered when ot→ 1 given
that beyond this transition zone the taxocenosis as a whole behaveseither as a single particle, or as a compact and indivisible packet ofwaves (both structures are equivalent to each other, as it is possi-ble to infer from Fig. 6g) that externally vibrates in its fundamentalfrequency.
About point (c): it is evident for any ecologist with fieldexperience that animals can change their physical positions, inter-specific relationships and consumption strategies depending on themoment of the day as well as on the season of the year. This fluc-tuating behavior is enough to produce STIWs. However, STIWs canbe evident by themselves even in the case of completely sessile orga-nisms like algae, biological crusts, coral reefs and trees. For example,let us suppose the simplest case of three interacting rain forest trees(in addition that any tree is a dual organism that performs photo-synthesis – e wave – and respiration – e wave – a forest tree is socomplex that can be regarded as an ecosystem in itself) aligned withthe daily movement of the sun across the sky (see Fig. 6k). Depend-ing on the alternation of night and day as well as on the size anddirection of the shadows of the trees due to sun’s position changes,there are alternative local fluctuations between the relative inten-sities of e and e that are summarized in the inferior section ofthe figure. This pattern can be extended to millions of trees andtheir associated fauna in all the ecosystems on the planet. In fact,the wave structure shown in the inferior section of Fig. 6k wouldremain approximately constant if we replace the tree line from eastto west as well as the three sun positions, by the successions ofnorthern hemisphere ↔ equator ↔ southern hemisphere, and win-ter ↔ spring ↔ summer, respectively, at the planetary scale. Then itis pretty obvious that the resulting wave lattice can produce verycomplex ecological patterns.
In addition, our model is based on a scale-independent analyticalframework in regard to Hp
′ gradients in order to grasp a patternthat takes place in space (s) and time (t) under heterogeneous eco-logical conditions in the small scale. However, these conditionsare scale-dependent in regard to biomass (hence the variations ofEϕ in he
ec, since the ecological succession is a biomass-dependentprocess from the point of view of increase of total energy per taxo-cenosis; see Rodríguez et al., 2015a, Fig. 5a). That is to say, a givenvalue of �Hp
′ = 2.70261 nat/individual can be measured either inregard to a space unit of 1.0E − 06 m3 per plot (e.g., in the case ofmarine microalgae), or in regard to a space unit of 100 m2 per plot(i.e., in the case of coastal succulent vegetation; data in Table B1,Appendix B). Thus, if we combine waves of e and e from severalconcurrent taxa in function of s and t in a scale-dependent chart,then we are able to glimpse a minimum hypothetical vision of theastonishing complexity of the “ecosystem’s message” (see Fig. 6l).
As a result, besides the association �e vs. HT summarized inFig. 6c (see comments above), those ecosystems of high speciesdiversity are capable to reflect the human influence on nature in amore sensitive way just because they embrace the widest spectrumof wave’s amplitudes and frequencies. Therefore, the interferenceprobability of their ecological waves with a given type of humaninfluence is always higher than in the case of those ecosystems oflow species diversity.
The relatively easy and holistic way in which the model devel-oped in this article can describe a variety of underlying basicpatterns of the ecosystem functioning is, perhaps, the key point
of this proposal. This simplicity seems to be a plausible answer tothe essential question posed by Goldenfeld and Kadanoff (1999,p. 87: “everything is simple and neat—except, of course, the world(. . .) Each situation is highly organized and distinctive, with biological
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R.A. Rodríguez et al. / Ecologi
ystems forming a limiting case of exceptional complexity. So why,f the laws are so simple, is the world so complicated?”), as well as a
ell-defined positive response to a previous sprout in favor of link-ng ecology and quantum mechanics (Jørgensen et al., 2007, pp.7–49; Jørgensen, 2012, pp. 61–64). Our results seem to indicatehat, despite the external complexity that seems to emerge fromur hiper-specialized current scientific view about reality, therere simple underlying patterns that are valid for several sciences,s well as for a wide variety of functional scales all over the universe.
cknowledgments
The final work on this paper would not have been possible with-ut the professional and business direct support provided by Theoman’s Group (Tampa, FL), in general, and Miriam Martínez, in
articular, in favor of Ada Ma. Herrera.
ppendix A. Supplementary data
Supplementary data associated with this article can be found, inhe online version, at http://dx.doi.org/10.1016/j.ecolmodel.2015.6.021
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