Biomass-dispersal trade-off and the functional meaning of species diversity

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Ecological Modelling 261– 262 (2013) 8– 18

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Ecological Modelling

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Biomass-dispersal trade-off and the functional meaning of speciesdiversity

Ricardo A. Rodrígueza,∗, Ada Ma Herreraa, Juan D. Delgadob, Rüdiger Ottoc,Ángel Quirósd, Jacobo Santandere, Jezahel V. Mirandae, María J. Fernándezb,Antonia Jiménez-Rodríguezb, Rodrigo Riera f, Rafael Ma Navarrog, Ma Elena Perdomod,José Ma Fernández-Palaciosc, Carlos G. Escuderoc, José R. Arévaloc, Lorenzo Diéguezh

a Fundación Canaria Rafael Clavijo, E-38320 La Laguna, Tenerife, Spainb Departamento de Sistemas Físicos, Químicos y Naturales, Facultad de Ciencias Experimentales, Universidad Pablo de Olavide, E-41013 Sevilla, Spainc Departamento de Parasitología, Ecología y Genética, Universidad de La Laguna, E-38071 Tenerife, Spaind Parque Nacional Los Caimanes, Carretera Central n◦ 716, CP-50100 Santa Clara, Cubae Parque Nacional Sistema Arrecifal Veracruzano, C/ Juan de Grijalva n◦ 78, Veracruz, Mexicof Centro de Investigaciones Medioambientales del Atlántico (C.I.M.A.), C/ Arzobispo Elías Yanes n◦ 44, E-38206 La Laguna, Tenerife, Spaing Escuela Técnica Superior de Ingenieros Agrónomos y de Montes, Universidad de Córdoba, Dpto. Ingeniería Forestal, Avda. Menéndez Pidal s/n◦ , Apdo.3048, 14080 Córdoba, Spainh Centro de Higiene y Epidemiología, Departamento de Control de Vectores, C/ República n◦ 217, Camagüey, Cuba

a r t i c l e i n f o

Article history:Received 20 December 2012Received in revised form 29 March 2013Accepted 31 March 2013

Keywords:Biomass-dispersal trade-offProduction–diversity patternsr–K selection theorySpecies diversityEcological Boltzmann’s constantEcological state equation

a b s t r a c t

Production–diversity patterns lack a single explanation fully integrated in theoretical ecology. An ecologi-cal state equation has recently been found for ruderal vegetation. We studied 1649 plots from twenty-nineecological assemblages and analyzed the relationship between diversity, biomass and dispersal look-ing for a pattern across these ecosystems. We found that high biomass and low dispersal values weresignificantly associated with high diversity plots under stationary conditions, and vice versa, involv-ing a biomass-dispersal trade-off that is coherent with well-established ecological principles. Therefore,energy per plot, estimated as one half of the product of mean individual biomass and mean square dis-persal multiplied by the number of individuals per plot, only reaches its maximum at intermediate levelsof diversity. This explains the well-known humped relationship between production and diversity. Wealso explore why the rest of the diversity–production patterns can be explained starting from disruptionsof this basic pattern. Simultaneously, the product of diversity, biomass and square dispersal is statisticallyequal to the ecological equivalent of the Boltzmann’s constant included in the ecological state equationthat remains valid for all the assemblages explored due to scale variations in the value of the above-mentioned constant. Biomass-dispersal trade-off resembles the principle of equipartition of energy fromthe kinetic theory of gases but in a characteristic way, because the alternative micro-associations ofdispersal-biomass in function of species diversity are not randomly distributed as it happens with thecombinations of molecular mass and velocity in a mixture of gases. Therefore, this distinctive ecologicalfeature should be assumed as one of the main pro-functional gradients or thermodynamic constraintsto avoid chaos and ecological degradation under stationary conditions. Hence, biomass-dispersal trade-off explains production–diversity patterns and the ecological state equation in simultaneous agreementwith conventional ecology and physics.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

The relationship between species diversity and ecosystemfunctioning is one of the most important points for ecosystems

∗ Corresponding author at: Fundación Canaria Rafael Clavijo, Calle Ofra s/n, E-38320 La Laguna, Tenerife, Islas Canarias, Spain. Tel.: +34 618 034 208.

E-mail addresses: [email protected], [email protected] (R.A. Rodríguez).

conservation (Tilman, 1999; Hector et al., 2001; Willig,2011). For instance, diversity and trophic production can beassociated according to the following patterns: (a) a uni-modal hump-backed curve (e.g. Chase and Ryberg, 2004;Graham and Duda, 2011); (b) direct linear (e.g. Marquardet al., 2009; Li et al., 2012); (c) inverse linear (e.g. Moserand Hansen, 2009); there may also be (d) no evidenceof any significant relationship (Huston et al., 2000; Long andShaw, 2010), or even more than one of these patterns can emerge

0304-3800/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.ecolmodel.2013.03.023

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R.A. Rodríguez et al. / Ecological Modelling 261– 262 (2013) 8– 18 9

for a single taxocene depending on the observation scale (Zhangand Wang, 2012). The proportions of these four patterns accord-ing to a survey of the ecological literature including a totalof 200 cases (Waide et al., 1999) are 30%, 26%, 12% and 32%,respectively.

Such a variety of patterns may be due to: (i) the absence ofsimple laws at the biological community level (Lawton, 1999;Simberloff, 2004); (ii) the spurious influence of variables notexplicitly included in the analysis (Mouquet et al., 2002; Hawkinset al., 2003; Mittelbach et al., 2003); or (iii) methodological issues(Huston, 1997; Naeem and Wright, 2003; Gamfeldt and Hillebrand,2008).

Alternatives (ii) and (iii) may share a common cause due to theuse of surrogate indicators instead of direct measurements. Forexample, the number of species (richness) is frequently assumedto be a good indicator of diversity, on the one hand, and biomass agood measure of production on the other. However, this procedureto evaluate production–diversity patterns has been questioned(Whittaker and Heegaard, 2003; Dangles and Malmqvist, 2004;Wilsey et al., 2005).

According to the conventional concept of biological diversity(Magurran, 2004, p. 17), the assumption that richness can replacediversity leaves evenness (i.e. the relative abundance betweenspecies) out of the analysis. Similarly, the assumption that biomasscan replace production neglects the fact that production is theflow of energy that sustains standing biomass (Odum, 1968,1969).

On the other hand, an ecological state equation (ESE, Eq. (7),below), similar to the ideal gas state equation, has recently beenfound for ruderal vegetation (Rodríguez et al., 2012). In comparisonwith the well-consolidated theory regarding the thermodynamicnature of the ecosystem (Odum, 1968, 1969), the ESE is, for themoment, an empirical result restricted to a particular kind of eco-logical assemblage. Thus, three main issues should be clarified: Canit be assumed that the ESE is a general pattern under stationary con-ditions (SC)? What is the ecological foundation of this equation? Isthere any reliable reason to consider this equation as unconnectedfrom the conventional ecological theory?

In this article, we develop an all-encompassing explanationof the above-mentioned issues that becomes a general explo-ration of the functional meaning of species diversity includingseveral new findings. With such a goal, we examine the rela-tionship between diversity and a proxy for trophic productioncalculated in a similar way to that of kinetic energy in physics.As a result, we found a biomass-dispersal trade-off (B-DT) thatexplains production–diversity patterns in agreement with conven-tional ecology and physics. We also obtained a general validationof the ESE for all the stationary systems sampled, confirming at thesame time the congruence of this equation with well-establishedecological principles. Finally, we discuss the general meaning ofthese findings.

2. Materials and methods

2.1. Indicators

(a) Species diversity per plot (Shannon, 1948; Magurran, 2004):

Hp = −S∑

i=1

ni

Npln

ni

Np(1)

where S is the species number per plot; ni is the number ofindividuals or colony elements (e.g. corallites) of species i; Np

is the total number of individuals per plot =∑

ni.(b) Total biomass per plot (meTp) and mean individual biomass per

plot (me) in kg: biomass has been recognized as one the most

important indicators to describe ecological dynamics at anyhierarchical level (e.g. Margalef, 1963; Odum, 1969; Bombelliet al., 2009).

(c) Ergodic (transitivity or meaning equivalence between spaceand time from the analytical point of view, see Appendix A,pp. 1–3) indicator of dispersal activity (Rodríguez et al., 2012):

Ie =∑S

i=1(Iei,j)

S(2)

Iei,j =(

di,j

si,j

)× 100 (3)

di,j =∑m

k=1((√

(xj − xk)2 + (yj − yk)2) × (2ij,k/ij + ik))

m(4)

where di,j is the mean dispersal activity of species i in plot jwith central geographic coordinates (x, y) within a total spaces0 divided into an m number of k plots; ij and ik are the respectiveabundances of i in plots j and k; ij,k is the shared number of indi-viduals of i in plots j and k (e.g. if ij = 7 and ik = 12, then ij,k= 7);(2ij,k)/(ij + ik) is the Bray-Curtis similarity index (Washington,1984); si,j is the standard deviation of di,j (given that di,j is anarithmetic mean); Iei,j is the ergodic indicator of dispersal activ-ity of species i in plot j; Ie is the ergodic indicator of dispersalactivity of the species group in plot j; and S is the species num-ber in plot j.

√((xj − xk)2 + (yj − yk)2) is the Pythagorean theorem

that allows the estimation (see Eq. (4)) of a mean Euclidean dis-tance between plot j and all the remaining k elements withinthe set of m plots. The calculation of Ie regarding 3D space(coordinates x, y, z per plot) only requires the replacement of√

((xj − xk)2 + (yj − yk)2) by√

((xj − xk)2 + (yj − yk)2) + (zj − zk)2).The calculation of Ie, heuristically explained in seven steps, isas follow: (Step 1) Can we consider the distance reached by aspecies i starting from an arbitrary x, y or x, y, z point withina stationary system as a good indicator of dispersal capability?Yes ⇒ Euclidian distance calculation (EDC; the left element onthe numerator of Eq. (4)). (Step 2) EDC, assumed as a singlevector-parameter for dispersal, has a drawback: its modularvalue for all the valid cases (ij /= 0 and ik /= 0) is the sameeither if ij = ik or if ij /= ik. Nevertheless, it is obvious that, forinstance, when ij = 100 and ik = 1 our certainty about the realdispersal capability of species i would be considerably lesserthan when ij = 100 and ik = 100. This limitation should be solved⇒ see step (3). (Step 3) The Bray-Curtis Similarity Index (BCSI)can function in this case in a similar way to a relative prob-ability (since it is ranged from 0 to 1 with 1 value only ifij/ik = ik/ij = 1) that either maintains or discounts a given amountfrom the modular value of EDC depending on the degree ofquantitative similarity between ij and ik: if (2ij,k/ij + ik) = 1 theresult from EDC×BCSI = EDC, but it is proportionally shorter as(2ij,k/ij + ik) < 1. Therefore, the value of BCSI can be used as aprobabilistic assessment of our certainty regarding EDC ⇒ cal-culation of EDC×BCSI (numerator of Eq. (4) as a whole). (Step 4)The mean value of EDC×BCSI for species i in plot j in regard toall the remaining k plots included in the total set of m plots willindicate the mean dispersal capability of i taking j as a referencepoint ⇒ di,j calculation (Eq. (4)). (Step 5) However, two equalmeans can be obtained from two very different set of data ⇒ seestep (6). (Step 6) What species would have a higher dispersalcapability according to the distribution of its di,j values? Thatspecies (generally, a dominant species) whose distribution ofdi,j values is as homogeneous as possible regarding all the setof m plots. The coefficient of variation (CV = s/�×100; where s:

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standard deviation and �: mean) is a normalized or dimension-less indicator of dispersion of a given probability distribution.Contrarily, a reduction of the statistical dispersion of di,j willindicate a more homogeneous dispersal capability of species i⇒ calculation of 1/CV (�/s × 100) in regard to Eq. (4) = Iei,j (Eq.(3)). This is the final indicator of dispersal for species i taking j asa reference point which also indicates the degree of isotropismof s0 “from the point of view” of species i. The mean value fromIei,j1, Iei,j2, Iei,j3, Iei,j4, . . . Iei,jm (which is not analytically impliedin this article) is an indicator of the realized niche width of igiven the environmental conditions in s0 as a whole. (Step 7)Each one of the i1, i2, i3. . .iS elements belonging to the total setof S species that coexist in plot j makes its respective contri-bution to the resulting dispersal activity of the biota from j ⇒calculation of the mean value of Iei,j = Eq. (2) = Ie. In summary,using a didactic example and given that time is not explicitlyincluded in any of the former calculations, the meaning of Ie isequivalent to the information we can obtain from a photo fin-ish. In such a case, our notion about the winner of the race doesnot depend on our point-by-point knowledge about the perfor-mance of each runner along the whole race; i.e. in photo finish:a larger distance of a given runner (“frozen in the time” by thepicture) from the starting point = a more intense physical effortinvested in the race. Similarly, in the set of plots: a higher Ievalue= a higher investment of trophic energy in dispersal start-ing from the center x, y of the reference plot given that, understationary condition, there is no any specially privileged posi-tion within the system. The influence of time can be neglected tocalculate Ie due to the time-independent character of stationarystates (see additional comments in the first paragraph of section5). In fact, ergodicity is a pure statistical property in all thosesystems under the influence of random interactions rather thana physical or ecological property. For instance, to clarify thenature of these two types of averages (space vs. time), let usassume that we want to determine the probability that the tossof a coin results in “heads”. We can do a time average by takingone coin, tossing it in the air many times in one place (�t � 0and �s → 0) and counting the fraction of heads. In contrast, anensemble average can also be found by obtaining many similarcoins and tossing them into the air at one time in a scatteredway (�t → 0 and �s � 0). It is reasonable to assume that thetwo ways of averaging are equivalent. This equivalence is calledthe quasi-ergodic hypothesis (see Gould et al., 1996, p. 20) andit is coherent with the expected results obtained from the cal-culation of Ie for typical stages of the taxon cycle (see AppendixA, Fig. A2). Ie must have several features similar to those of themolecular velocity (see Appendix A), with the goal of facilitat-ing the extrapolation of mechanic kinetic energy equation (seebelow) from physics to ecology but, as its own name indicates,Ie is not only a velocity. Any value of physical speed (v) is an indi-cator of motion intensity due to an input of energy in regard toisolated elements with a constant mass (in the non-relativisticrange) whose speed values can be assessed one by one in thecase of macroscopic elements. On the other hand, (i) Ie alsoreflects a given quota of effort devoted to reproduction (sincereproduction is the main population source for dispersal), (ii)the reference system to calculate Ie is the plot at least insteadof the individual and, therefore, (iii) it is also a measure with alarger statistical load than that of v, in reference to a group ofindividuals with a non-constant mass per type of element (seecomments referred to Eq. (10), section 4). For additional detailsabout Eq. (2) see Appendix A. The original software (IeCalc-2.1;free of viruses and including clear instructions for use) to cal-culate Iei,j and Ie values can be requested, free of any charge, tothe main author of this article; only the name, e-mail addressand institution of the applicant should be supplied.

(d) Total eco-kinetic energy per plot:

EeTp = NpEe = Np

(12

me · I2e

)= 1

2meTp · I2

e (5)

where Ee is the mean eco-kinetic energy per individual per plot.

Eq. (5) reflects that part of the energy captured by organismsis converted into biomass (me), and another part is invested indispersal (Ie) via reproduction coupled with individual movement(non-sessile animals), or emission of propagules and vegeta-tive reproduction (in sessile organisms). Eq. (5) assumes thattrophic energy can be estimated in a similar way as mechanicalkinetic energy (E = 1/2mv2, where m = physical mass, v = velocityand kg m2/s2 = Joule, J) is calculated in the case of any other body,from atoms to galaxies (Resnick et al., 2001). Therefore, there is noreason to suppose that an equation similar to E cannot be equallyuseful in the case of ecosystems, mainly if we accept the equiv-alence and free transformation of all kinds of energy (First Lawof Thermodynamics). The quantitative proportionality between E,EeTp and trophic energy is an ulterior goal that depends on find-ing, previously, a reliable ecological meaning for the relationshipbetween EeTp and Hp (see below).

The orthodox way to evaluate the amount of trophic energy inecosystem depends on measuring the primary production (the bal-ance of photosynthetic CO2 assimilation with photorespiratory CO2release) expressed in g C m−2 d−1 (e.g. Gilpin et al., 2002). The exactmeasurement of this parameter has become very complex (e.g.Corno et al., 2005; Beardall et al., 2009), almost as a little sciencein itself whose resulting data are sometimes difficult to interpret(e.g. Chapin et al., 2006) and cannot inform us about the amountof energy at a small-scale or disaggregated level as in the case ofeco-kinetic energy assessments (e.g. total energy/species/plot ormean energy/individual/species/plot). On the contrary, one of themain methodological advantages of calculating Eq. (5) is that itdoes not require any special device or adjustment, but the datacan be directly obtained from the conventional sampling procedureapplied in the standard biotic community studies.

Eq. (5) does not indicate the energetic balance of a given organ-ism, evaluating this is neither a goal nor an element of our proposal.The understanding of the meaning of this decision requires sev-eral physical and statistical details. In the first place, if the systemis under stationary conditions (balance between inputs and out-puts or null net exchange with its surroundings; the ecological stateequation – Eq. (7), below – should yield an equality only under suchcircumstances), then the individuals of the system neither gain norlose energy at the large scale; consequently, the study of the ener-getic balance of a given organism means nothing by definition fromthe statistical viewpoint applied in our approach.

Secondly, the thermostatistical degrees of freedom (t.d.f. or eco-logical degrees of freedom -e.d.f.-, in ecology) are three generalways in which a given molecule can fix energy: rotational motion;internal vibration motion between the atoms of the molecule and;finally, the translational linear motion of the molecule as a whole.Only the last t.d.f. (which also encloses three additional t.d.f., oneper each spatial dimension x, y, z) contributes to our physical notionof temperature (Roller and Blum, 1986; Aguilar, 2001; Resnicket al., 2001). Our proposal assumes an approximated equivalencebetween individuals and molecules to extend the classical formal-ism from physics to ecosystem ecology. Consequently, only thetranslational motion at the plot level statistically assessed by meansof Ie matters within our model.

The internal energy embodied in the living tissues of any organ-ism is neglected since it is assumed to be equivalent to thoseinternal degrees of freedom of a molecule that do not contributeto temperature and, therefore, their implication to perform a givenecological work can be ignored for our purpose. Is it approximately

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R.A. Rodríguez et al. / Ecological Modelling 261– 262 (2013) 8– 18 11

in agreement with reality this decision? Whales, elephants, wal-ruses, crocodiles, whale sharks, moon fishes, large felines andsequoia trees are, the most of them, very large organisms with ahuge amount of internal energy and a strong top-down ecologicalcontrol, but none of them is in the bottom of a significant food chain.Contrarily, several of these species are in danger of extinction due tohuman consumption and, in contrast, our farms are full of animalsand plants that, even in nature, are smaller than the former speciesin individual size but more abundant and with a higher capability ofdispersal and population replacement. This latter kind of organismsis also in the bottom of the most important natural food chains. Thisindicates something that is well-known in ecology: abundance anddispersal matter.

(e) Ecological equivalent (ke, Rodríguez et al., 2012) of the Boltz-mann constant (kB = 1.3806504E−23 J/mol/K):

ke = 2Np(1/2meI2e ) · Hp

Np

→ statistically constant for all the plots under SC (6)

In the case of ruderal vegetation ke ≈ 1.3806504E+02 Je nat/individual; where Je is an ecoJoule (kg –d2); –d is thedispersal unit (the unit of expression of Eq. (2)); nat/individual isthe unit of expression of Eq. (1) when natural logarithms are used.

Eq. (6) indicates the amount by which the individual eco-kinetic energy fluctuates with a change of species diversity of onenat/individual at the plot level. If we take Ie as a one-dimensionalaction then ½ke = �Ee/e.d.f.·nat/individual, since Eq. (7), below,can be expressed in terms of individual energy values dividingboth sides by 2Np (i.e. (2Np(½meIe2))/2Np = (Npke/Hp)/2Np) and thenEq. (7) becomes ½meIe2 = ½ke/Hp = Ee (see Eq. (5)). According toRodríguez et al. (2012) the key factor for the detection of stationar-ity is that ke maintains, in the mean, the same value for all the plotsincluded in the ecological assemblage explored. In other words, ke

is the conversion factor between total biomass (meTp) and squaredispersal (I2

e ), placed on the left of the ESE (Eq. (7)), and individualabundance (Np) and diversity per plot (Hp), placed on the right ofthe equation:

2Np

(12

meI2e

)= Npke

Hpor 2

(12

meTpI2e

)= Npke

Hpor 2EeTp = Npke

Hp(7)

Eq. (7) is very similar to the ideal gas state equation PV = nRT =n(R/NA)T = NkBT = Nmv2

x = 2N(1/2mv2x ), Tipler and Mosca, 2010,

pp. 570–575, Eqs. (17.7)–(17.12) and (17.16)–(17.18), where P isthe pressure, V is the volume, n is the number of moles, R is theuniversal constant of gases (the increment of kinetic energy permole per Kelvin), T is an absolute temperature value, NA is theAvogadro’s number (the number of molecules in a mole), kB is theBoltzmann’s constant (1/2kB the increment of kinetic energy pert.d.f. per molecule per Kelvin), N is the number of molecules, mis the molecular mass, v2

x is the mean square molecular velocityalong the x axis = 1/3v2 and v2 is the mean square molecular velocityregarding 3D space. Appendix A includes further comments aboutthe underlying assumptions in Eq. (7).

2.2. Hypothesis

Under SC, biomass (me or meTp) and Hp are positively correlated,whereas square dispersal (I2

e ) and Hp are negatively correlated. Thiscombination of correlations sets plots with high biomass and lowdispersal at high diversity values and plots with an inverse combi-nation of biomass, dispersal and Hp values at the opposite end ofthe diversity gradient (B-DT).

2.3. Consequences of the hypothesis

Consequence 1: EeTp (Eq. (5)) decreases at very low diversityvalues due to the reduced values of biomass (me or meTp). EeTp isalso depleted at very high diversity values due to the low values ofdispersal (I2

e ). Therefore, EeTp = 1/2meTpI2e (total eco-kinetic energy

per plot assumed as a proxy for trophic energy) only can reachits maximum value at intermediate diversity levels (hump-backeddiversity–production curve). The rest of the diversity–productionpatterns can be explained starting from disruptions of this basicpattern.

Consequence 2: B-DT is coherent with three well-known eco-logical regularities: (a) Cope’s rule (Cope, 1896; Kingsolver andPfennig, 2004; Hone et al., 2005): there is a positive relationshipbetween increased individual size of new species added to ecosys-tems during times of stability which promotes diversity (edgeof high diversity and high biomass within B-DT). (b) Rapoport’srule: at the hemisphere level, species dispersal (Ie, in this article)is reduced, promoting coexistence, as latitude descends towardcommunities of greater diversity (Rapoport, 1975; Stevens, 1989).This rule can act locally (Rohde, 1996; Hernández et al., 2005),which is coherent with the opposite correlation between I2

e andHp embraced by a BD-T. (c) r–K selection theory (MacArthur andWilson, 1967; Pianka, 1970; Reznick et al., 2002): r-species tend tobe small in individual size and with a high rate of reproduction anddispersal in micro and macro-environments of low diversity values.K-species are typical of the opposite ecological condition. Conse-quently, r–K selection theory is a sort of combination in parallelbetween Rapoport’s rule (reduction of dispersal from r- to K-edge)and Cope’s rule (increase of size from r- to K-edge) and it is alsoequivalent to the hypothetical B-DT as a whole.

Consequence 3: Simplifying Eq. (6) with a didactic intention, weobtain:

ke = 2Np(1/2meI2e ) · Hp

Np= Np · me · I2

e · Hp

Np

= meTp · I2e · Hp

Np= me · I2

e · Hp (8)

Thus, deriving a rational inference (∴ = therefore) symbolicallyrepresented in terms of alternatives (∨ = or) and fluctuations direc-tion (↓↑) in reference to a given ecological assemblage:

(biomass ↑, I2e ↓, Hp ↑) ∨ (biomass ↓, I2

e ↑, Hp ↓) ∴ me · I2e · Hp = ke

→ constant (9)

Eq. (9) is the main ecological foundation of Eq. (7) given that ke is asort of “hinge” or switch factor between the right and the left sideof the equality in the ESE.

2.4. Statistical procedures

I2e , me and meTp were correlated with Hp (diversity per plot) using

Pearson’s correlation coefficient (r) and the correlative relationshipbetween these indicators was graphically analyzed by means ofthe spline-fit (piecewise cubic polynomial interpolation) curves ofboth sets of correlations with regard to the whole group of eco-logical assemblages ordinally ranked according to their taxonomicresemblance and temporal sequence.

Cover percentage is considered a good proxy for biomass (e.g.Röttgermann et al., 2000; Muukkonen et al., 2006). Therefore, thecorrelation between cover (mean cover per species per plot, ce, andtotal cover per plot, ceT) and Hp replaced the correlation betweenbiomass and Hp due to the absence of biomass values in one (code:pf, see below) of the twelve kinds of assemblages included in ourstudy.

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The value of ke (Eq. (6)) was calculated for all the plots perassemblage. The difference between the observed mean value ofthe ecological equivalent of the Boltzmann’s constant (ke(o)) andthe respective expected value (ke(e)) was tested by means of thet-test for a single sample to verify if the ESE (Eq. (7)) was alsovalid for other cases besides ruderal vegetation. Collaterally, wealso assessed the difference between the mean values of the twomembers of the ESE by using either the t-test for dependent sam-ples or the t-test for unequal samples sizes in those cases in whichNpke(e)/Hp was undefined due to Hp = 0.

The statistical standardization of i data (i − mean/standard devi-ation) allows comparison of the slopes of correlations calculatedfrom variables expressed in different units. Thus, the absolute dif-ference between the values of (regression coefficient) of thestandardized values of I2

e and biomass with regard to Hp was corre-lated with the p-value (significance level) of the difference betweenke(o) and ke(e) to explore the influence of the balance between thedispersal gradient and the biomass gradient on ecological station-arity.

The total eco-kinetic energy per plot (EeTp, Eq. (5)) wasgraphically related with Hp and its maximum values for each�Hp = 0.1 nat/individual were adjusted by means of polyno-mial equations to highlight the presumptive hump-backeddiversity–production curve in those assemblages under SC. Thisanalysis also supported our interpretation about the rest ofthe patterns as contingent disruptions of the hump-backeddiversity–production curve. Data from comparable assemblageswith a similar range of Hp values were included in the same graph.Statistica-6 (StatSoft Inc., 2001) was used for all the statistical tests.

All the values of the main indicators for the statistical proceduresapplied to each ecological assemblage are included in Appendix B.All the graphical results from correlations and hypothesis tests perecological assemblage are also included in Appendix B.

2.5. Samplings

The statistical procedures described above were applied to datafrom the following kinds of ecological assemblages:

Marine microalgae; marine interstitial meiofauna of sandybeaches; massive (non-branching) corals; litter invertebrates insubtropical laurel and Canarian pine forest; tropical rocky shoresnails; coral reef fishes; ruderal vegetation; Mediterranean shrubvegetation; mixed shrub vegetation; coastal succulent shrub veg-etation and pine forest vegetation. The sampling method and dataprocessing for each one of the se assemblages are explained inAppendix A.

3. Results

The whole group of assemblages included a wide range ofvalues (Table 1) with regard to number of plots (from 12 to219); total number of individuals (from 160 to 2.06E+07); bi- ortri-dimensional space (volume) per plot (from 0.000001 m3

to 10,000 m2); population density (from 0.51 ind. m−2 to4.68E+11 ind. m−3); mean distance between plots (from 4.64 mto 29,259 m); species richness (from 11 to 239); mean speciesdiversity per plot (from 0.47 nat/individual to 2.26 nat/individualas well as mean individual biomass per plot (from 5.53E−14 kg to5.02 kg). Therefore, any regular pattern derived from these datawould suggest independence of scale or taxon.

The main general pattern that we obtained deserves particularattention (Table 2): one hundred percent of negative correlations(and 82.76%, 24 of 29, were significant) between square dispersal(I2

e ) and species diversity (Hp) matched with positive correlationsbetween biomass (either me or meTp) and Hp (85.18%, 23 of 27, weresignificant, or almost significant: ma3).

ma1

ma2

ma3

ma4

mif1

mif2

mif3 co

r lli pli

rsn1

rsn2

rsn3 cr

frv

1rv

2rv

3rv

4rv

5rv

6rv

7rv

8rv

9M

sm

xsm

xs'

pf1

pf2

css

Ecological assemblages

-1.0

-0.6

-0.2

0.2

0.6

1.0

Pea

rson

's c

orre

latio

n (r)

r : Hp, Ie2

r : Hp, bio mass (m ax. values for me or meTp)

** * * **

Fig. 1. Relationship between the correlations I2e vs. Hp , and biomass (me or meTp) vs.

Hp , for all the ecological assemblages. Equivalent graphs for the rest of indicators inTable 2 are unsuitable due to large scale differences even for a single indicator. max.:maximum. *Non-significant p-value of r. Ecological assemblage codes in Table 1.

An auxiliary graph (Fig. 1) helps to illustrate the pattern: thespline fits of r-values for dispersal vs. diversity and biomass vs.diversity are mirror images of each other (r = −0.644, p = 0.0003between the set of r-values of I2

e vs. Hp and the respective r-valuesof biomass vs. Hp). Any displacement of the correlation between I2

eand Hp toward negative and significant r-values matched with anopposite movement of the respective correlation between biomass(either me or meTp) and Hp. There were exceptions to this pattern:(i) mif2, in which there was a negative and significant correlationbetween meTp and Hp (r = −0.31, p = 0.019) combined with a low andnon-significant correlation between I2

e and Hp (r = −0.08, p = 0.56)and (ii) the segment from rv5 to rv7 whose low and non-significantcorrelations between I2

e and Hp were combined with a trend to pos-itive correlations between biomass and Hp that were estimatedfor rv6 and rv7 by means of the interpolation of the sequence ofdata from the same place. These exceptions were cases withoutstationarity (see comments below).

In all of the cases in which the above-mentioned pattern wasfound, there was no significant difference (Table 2) between theobserved and the expected value of ke (p) nor between the left andthe right side of the ESE (p′).

Rsn1 was the most typical isolated example of this previ-ous combination of conditions (r: Hp, I2

e = −0.91, p < 0.001; r: Hp,meTp = +0.64, p = 0.013; ke(o) = 1.37 vs. ke(e) = 1.38, p = 0.977; 2EeTp vs.Npke(e)/Hp, p′ = 0.91).

Additionally, all the assemblages adjusted to this combinationshowed a clear trend to a humped relationship between the totaleco-kinetic energy per plot (EeTp, Eq. (5)) and species diversity(Fig. 2, panels a–j).

Panels k and l (Fig. 2) show the most remarkable differencesin comparison with the stationary hump-backed pattern. Coinci-dently, these two cases also showed the most significant differencesbetween ke(o) and ke(e) (see the p values for mif2 and rv5, Table 2)as well as between the two sides of Eq. (7) (see the respective p′

values for mif2 and rv5, Table 2). On the one hand, the key factor inmif2 seems to be the space without EeTp observations in the lowerleft corner of panel k in Fig. 2 due to the lack of biomass limitationdespite the reduction of Hp (r = −0.31, p = 0.019). This explains theleft shift in maximum value of total eco-kinetic energy.

On the other hand, the key factor in rv5 is the space withoutobservations in the lower right corner of panel l in Fig. 2, due tothe lack of significant limitation for dispersal despite the increaseof Hp (r = −0.17, p = 0.223) combined with the expected increaseof biomass (r = 0.50, p < 0.001). This explains the trend of the

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R.A. Rodríguez et al. / Ecological Modelling 261– 262 (2013) 8– 18 13

Table 1General ecological indicators of each ecological assemblage (Ae).a

Ae np Na s0p d = Na/(s0pnp) dmp S Hp(m) me

ma1 35 1.241E+07 0.000001(3) 3.55E+11 5.04 55 1.097 5.53E−14ma2 43 1.833E+07 0.000001(3) 4.26E+11 5.11 57 1.012 5.82E−14ma3 45 1.710E+07 0.000001(3) 3.80E+11 5.32 54 0.892 6.50E−14ma4 44 2.059E+07 0.000001(3) 4.68E+11 5.11 62 0.966 6.48E−14mif1 40 4360 0.00032(3) 3.41E+05 13.33 79 1.758 2.81E−08mif2 56 19,034 0.00032(3) 1.06E+06 19.99 60 1.064 1.24E−08mif3 51 34,143 0.00032(3) 2.09E+06 16.99 137 2.256 1.58E−08cor 71 1.951E+07 10 2.75E+04 29.38 19 0.841 3.21E−05lli 193 11,854 0.25 245.68 35.02 239 2.085 1.49E−04pli 115 608 0.25 21.15 34.06 98 0.762 1.34E−04rsn1 14 1531 1 109.36 4.64 23 0.899 1.54E−03rsn2 15 949 1 63.27 4.64 21 0.897 9.71E−04rsn3 12 1094 1 91.17 4.64 17 0.698 5.95E−04crf 72 7062 40(3) 4.98 29.38 58 1.134 4.10E−02rv1 59 319 4 1.35 30.05 24 1.045 4.01E−02rv2 59 323 4 1.37 30.05 26 1.055 4.48E−02rv3 56 192 4 0.86 30.05 19 0.748 8.81E−02rv4 59 412 4 1.75 30.05 28 1.248 4.81E−02rv5 54 160 4 0.74 30.05 20 0.652 6.07E−02rv6 60 23,055 4 96.06 30.05 41 1.404rv7 60 19,805 4 82.52 30.05 83 2.184rv8 60 2618 4 10.91 30.05 57 1.122 2.97E−02rv9 60 2117 4 8.82 30.05 29 0.823 3.94E−02Ms 28 664,726 10,000 2.37 5656.51 37 1.654 5.16E−01mxs 19 2725 100 1.43 29,259.23 77 1.821mxs’ 19 962 100 0.51 29,259.23 27 1.257 5.02E+00pf1 17 1293 100 0.76 18,413.55 53 1.355pf2 219 100 8897.14 63 0.469css 14 1110 100 0.79 267.596 11 1.013 2.58E+00

a np: number of plots. N: total number of individuals per assemblage. s0p: space per plot in m2 or m3(3). d: population density per space unit per plot. dmp: mean distancebetween plots in meters. S: total species richness. Hp(m): mean species diversity (Eq. (1)) per plot in nat/individual. me: mean individual biomass per plot in kg. ma: marinemicroalgae. mif: marine interstitial meiofauna of sandy beaches. cor: massive (non-branching) corals. lli and pli: litter invertebrates in laurisilva and pine forest, respectively.rsn: tropical rocky shore snails. crf: coral reef fishes. rv: ruderal vegetation. Ms: Mediterranean shrub vegetation. mxs: mixed shrub vegetation. pf: pine forest vegetation.css: coastal succulent shrub vegetation.

Table 2Combinations of correlations (r) that support biomass-dispersal trade-off and significance level (p) of the t-test between the observed (o) and the expected (e) value of ke aswell as (p′) between the left (2EeTp) and the right (Npke(e)/Hp) side of the ecological state equation (Eq. (7)).a

Ae r: Hp , I2e r: Hp , me r: Hp , meTp ke(o) ke(e) p p′

ma1 −0.38, 0.023 0.37, 0.027 0.18, 0.299(1) 1.72E−10 1.38E−10 0.106 0.672ma2 −0.44, 0.003 0.64, <0.001 0.25, 0.107 1.74E−10 1.38E−10 0.099 0.142ma3 −0.31, 0.036 0.29, 0.058 0.01, 0.966 1.78E−10 1.38E−10 0.136 0.188ma4 −0.41, 0.006 0.35, 0.021 0.19, 0.226(1) 1.65E−10 1.38E−10 0.288 0.261mif1 −0.49, 0.001 0.44, 0.005(2) 0.16, 0.311 1.56E−04 1.38E−04 0.417 0.855mif2 −0.08, 0.561 −0.03, 0.832 −0.31, 0.019 3.93E−05 1.38E−05 3E−15 2E−10mif3 −0.45, <0.001 0.35, 0.012 −0.03, 0.861 1.65E−04 1.38E−04 0.066 0.508cor −0.31, 0.009(1) 0.04, 0.738 0.54, <0.001(2) 1.67E−01 1.38E−01 0.228 0.658lli −0.23, 0.002(1) −0.02, 0.837 0.23, 0.002 1.36E+00 1.38E+00 0.834 0.496pli −0.33, <0.001 −0.01, 0.912 0.56, <0.001(1) 1.35E−01 1.38E−01 0.825 0.164rsn1 −0.91, <0.001 0.33, 0.251 0.64, 0.013 1.37E+00 1.38E+00 0.977 0.910rsn2 −0.71, 0.003 0.58, 0.023 0.49, 0.0609 1.44E+00 1.38E+00 0.863 0.099rsn3 −0.76, 0.004 −0.64, 0.026 0.72, 0.0080 1.40E+00 1.38E+00 0.945 0.416crf −0.39, <0.001 0.38, 0.001(1) −0.02, 0.874 1.50E+02 1.38E+02 0.671 0.125rv1 −0.39, 0.002 0.32, 0.014 0.67, <0.001 1.36E+02 1.38E+02 0.848 0.822rv2 −0.45, <0.001 0.22, 0.097 0.63, <0.001 1.39E+02 1.38E+02 0.969 0.652rv3 −0.30, 0.024 −0.19, 0.152(1) 0.32, 0.016(1) 1.45E+02 1.38E+02 0.557 0.744rv4 −0.44, <0.001 0.06, 0.675 0.22, 0.096 2.73E+02 1.38E+02 0.004 0.028rv5 −0.17, 0.223 −0.09, 0.514 0.52, <0.001 6.94E+01 1.38E+02 2E−13 9E−11rv6 −0.05, 0.692(1)

rv7 −0.09, 0.519rv8 −0.52, <0.001 0.48, <0.001 0.40, 0.002 1.55E+02 1.38E+02 0.113 0.880rv9 −0.51, <0.001 0.35, 0.007(1) 0.24, 0.062 2.00E+02 1.38E+02 0.173 0.323Ms −0.11, 0.582 −0.30, 0.115 0.10, 0.598 1.58E+03 1.38E+03 0.611 0.420mxs −0.47, 0.042 0.62, 0.005 0.73, <0.001mxs’ −0.47, 0.049(1) 0.56, 0.013 0.60, 0.006 1.48E+04 1.38E+04 0.844 0.827pf1 −0.84, <0.001 −0.55, 0.024 0.63, 0.007pf2 −0.43, <0.001 −0.32, <0.001(1) 0.48, <0.001css −0.71, 0.005 0.55, 0.041 0.47, 0.091 1.23E+04 1.38E+04 0.633 0.416

a The first number in each correlations column is the value of r and the second (on the right of the comma) is the respective p value. Sub-indexes between parenthesesindicate the number of outliers excluded from the respective correlation. Biomass data were replaced by a proxy (cover) in mxs, pf1 and pf2. rv6 and rv7 lack biomass valuesdue to adverse weather conditions. The r-values used in Fig. 1 are highlighted in boldface. Appendix B includes the graphical analysis that support the main results includedin Table 2 for each ecological assemblage.

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14 R.A. Rodríguez et al. / Ecological Modelling 261– 262 (2013) 8– 18To

tal e

co-k

inet

ic e

nerg

ype

r plo

t(E eTp

, Eq.

(5))

0.0 0.4 0.8 1.2 1.6 2.0 2.40E-01

4E-05

8E-05

1E-04

2E-04 (a)

! 1.0 1.4 1.8 2.2 2.6 3.0 3.40.00

0.02

0.04

0.06

0.08 (b)

0.0 0.4 0.8 1.2 1.6

0E-01

2E+04

4E+04

6E+04

8E+04 (c)

0.4 0.8 1.2 1.6 2.0 2.4 2.80

40

80

120

160

200 (d)

! 0.0 0.4 0.8 1.2 1.6 2.00.0

0.4

0.8

1.2

1.6

2.0

2.4 (e)

0.0 0.4 0.8 1.2 1.6 2.00

40

80

120

160

200

240 (f)

0.2 0.6 1.0 1.4 1.80E-01

2E+04

4E+04

6E+04

8E+04 (g)

0.0 0.4 0.8 1.2 1.6 2.00E-01

4E+03

8E+03

1E+04

2E+04(h)

1.0 1.2 1.4 1.6 1.8 2.0 2.20E-01

1E+07

2E+07

3E+07

4E+07(i)

0.4 0.6 0.8 1.0 1.2 1.4 1.60E-01

4E+05

8E+05

1E+06

2E+06

2E+06 (j)

0.0 0.4 0.8 1.2 1.6

1E-052E-05

3E-054E-05

5E-056E-05

k e(o

)

-0.002

0.002

0.006

0.010

0.014

EeT

p

(k)

0.0 0.4 0.8 1.2 1.60

100

200

300(l)

Species diversity per plot ( Hp, Eq. (1))

Fig. 2. EeTp vs. Hp . (a) ma1–ma4(2). (b) mif1 and mif3. (c) cor(1). (d) lli. (e) pli(2). (f) rsn1–rsn3. (g) crf. (h) rv1–rv4 and rv8–rv9. (i) Ms. (j) mxs’ and css(1). (k) mif2. (l) rv5(1).Gray lines represent polynomial equations that fit the maximum values of EeTp per interval of 0.1 nat/individual in Hp (equations list in Appendix A). EeTp is on the right axisin panel k and left axis for ke(o) (r = 0.452, p < 0.001, regression equation – black fit line: ke(o) = 1.406E−05 + 2.371E−05 Hp; intercept indicated by an arrow) has been added tosupport an analysis below. Sub-indexes between parentheses indicate the number of outliers excluded from the fit.

maximum value of total eco-kinetic energy to the right extremeof Hp values.

The correlation between the absolute difference of the slopes(AbsˇIe − ˇme) of the standardized values of dispersal and biomasswith regard to species diversity and the p-value of the compari-son between ke(o) and ke(e) was negative and significant (r = −0.409,p = 0.048). Thus, less difference between slopes tends to be associ-ated to higher equality between the observed and the expectedvalue of ke even if such a result depends only on three observationsout of stationarity (mif2, rv5 and rv4 to a lesser extent) given thatunder SC the absence of correlation (r = −0.091, p = 0.694) betweenAbsˇIe − ˇme and pke(o) vs. ke(e) was the expected.

4. Discussion

In summary, a gradient of increasing species diversity per plotunder SC was associated with: (i) reduction of dispersal activity

and (ii) increase in biomass either at the individual (me) or theplot (meTp) level. Therefore, any stationary gradient of speciesdiversity can also be assumed to be a gradient of r–K selectionwhose internal changes are quantified in terms of a given scalevalue of ke. Such a B-DT leads directly to a humped pattern ofthe total energy per plot, because at both ends of H there is ashortage of one of the two parameters needed for the growth ofEeTp (EeTp = 1/2meTp · I2

e ). Recent evidence suggests that this patterncould even apply on a global scale: the gradient of productiv-ity associated with the latitudinal gradient of species diversitypeaks between 30◦ and 50◦ latitude (Huston and Wolverton,2009). In other words, the most productive ecosystems in eachhemisphere are not located equatorially (usually very diverse) ornear the poles (usually with low diversity), but rather at mid-latitudes.

The above-discussed results indicate that our hypothesis as wellas its three main consequences is difficult to reject.

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R.A. Rodríguez et al. / Ecological Modelling 261– 262 (2013) 8– 18 15

The opposite correlation between AbsˇIe − ˇme and p from ke(o)vs. ke(e) is an additional factor that supports the influence of thebalance between dispersal and biomass to sustain the humpbackedpattern between EeTp and Hp. For instance, there were no significantp values either for Ie vs. Hp or me vs. Hp in Ms (Table 2), but the r-values for Ie and me were very similar; this seems to be sufficientto produce a typical stationary state (see the hump-backed shapeof the scatter in panel i, Fig. 2).

The maximum power principle states that “during self-organization, system designs develop and prevail that maximizespower intake, energy transformation, and those uses that reinforceproduction and efficiency” (Odum, 1995, p. 311; see also Hall, 2004).Therefore, the most parsimonious option is to assume that mif2(panel k, Fig. 2) is drifting to the right edge of the species diver-sity spectrum; on the contrary, rv5 (panel l, Fig. 2) is drifting to theopposite position.

Summarizing, stationarity is supported by a maximum valueof energy per plot placed at intermediate diversity values (hump-backed pattern); if this maximum is displaced to the left (panel k inFig. 2: negative linear correlation pattern between total energy andspecies diversity) it pulls the whole system toward lower diversityvalues where energy maximum is more probable, and vice versa(panel l in Fig. 2: positive linear correlation trend between the totalenergy per plot and species diversity).

Stationarity means constancy of internal energy (gains bal-anced by losses) at the system level. So, if we establish a netconsumption relationship with any ecosystem that spontaneouslytends to stationarity, it will eventually be extinguished, since a netenergy gain is only possible out of stationarity. Correspondingly,non-stationarity is a transition stage in which the system eithergains or loses energy in its movement from one stationary state toanother.

Two additional elements support this approach:

a) In regard to mif2 (panel k, Fig. 2), the regressionequation between ke(o) and Hp is ke(o) = 1.406E−5 + 2.371E−05 Hp (r = 0.452, p < 0.001) where the intercept1.406E−5 → 1.380650E−5 Je nat/individual. This equationseems to indicate that the non-stationary assemblage mif2 ismoving toward the left edge of the diversity gradient, startingfrom a previously stationary state in which ke(o) ≈ 1.3806504E−4(as in mif1 and mif3, see Table 2), to reach an ulterior sta-tionary state with a different maximum power positionat a lower Hp value in which ke(o) ≈ 1.3806504E−5 Je nat/individual.

b) With regard to rv5 (panel l, Fig. 2) the monthlysequence of the total number of individuals per assem-blage (Na) in the same place from rv1 to rv9 (308 →319 → 193 → 406 → 160 → 23,055 → 19,805 → 2618 → 1995)indicates that in rv6 (Na = 23,055) and rv7 (Na = 19,805) theremust have been a substantial increase in biomass per plot(meTp). However, the correlation between I2

e and Hp for rv6 andrv7 does not correspond to what would be expected accordingto dispersal-biomass trade-off (see r: Hp, I2

e and its p-value forrv6 and rv7, Table 2). Thus, in rv6 and rv7, biomass per plotincreased with no reduction of I2

e , so positive displacementfrom stationarity toward greater Hp values of the ecosystemas a whole in these two assemblages indicates a breach ofthe dispersal-biomass trade-off due to increased net energyinput to the ecosystem (see the fit segments for dispersal andbiomass from rv5 to rv7, Fig. 1). In such circumstances theobserved pattern of total energy per plot is not unimodal andhumped for intermediate values of Hp, but direct and lineardue to a net increase of EeTp in the right edge of Hp, similarlyto the right half of panel l in Fig. 2. Consequently, rv5 canbe assumed to be a “sprout” of successional change toward

new positions of higher diversity (rv6 and rv7) in which thenew energy maximum of a subsequent stationary state will belocated.

This does not mean that there is a pre-eminence of maximumenergy per plot over species diversity or vice versa. On the con-trary, it is a question of “tuning” between the biological communityand its environment. When the supply of energy from the envi-ronment is reduced, the maximum value of internal total energyper plot moves toward lower diversity values, in which r-selectionprevails, and vice versa (increased net supply of energy from out-side → “tuning” of the ecosystem at high diversity per plot values,where K-selection prevails). As a result, there should be a direct cor-relation between the mean value of species diversity per plot andthe value of species diversity where maximum energy is reached(Fig. 3a).

On the other hand, the movement of maximum energy towardhigher diversity values does not mean that a stationary ecosys-tem of high diversity has a higher availability of energy per unit ofbiomass. A physical body at rest (v = 0) has physical mass (m) andpotential energy, but it has no available kinetic energy (1/2m·02 = 0)to perform any physical work. Similarly, a species i present only inone (Iei,j = 0) or very few plots (Iei,j → 0) has biomass and potentialenergy (the energy chemically embodied in its living tissues froman initial photosynthetic transduction of light energy into chemicalenergy) but, due to its low level of replication in space and time,has insufficient eco-kinetic energy (ni(1/2)meI2

ei,j→ 0) to support

any kind of intense trophic exploitation due to its low capability forpopulation replacement.

Such species (with large biomass and low Iei,j) are frequentin high-diversity communities. Therefore, there should be anopposite correlation between the availability of energy per unitof biomass per plot (EeTp/meTp = 1/2NpmeI2

e /Npme → 1/2I2e ) and

species diversity (Fig. 3b). This is in agreement with a very oldclassical statement on the reduction of productivity expressed interms of the energy/biomass ratio or turnover rate (Margalef, 1963)or, conversely, the gain of efficiency expressed in terms of thebiomass/energy ratio (Odum, 1969) in those communities in whichspecies diversity is high.

In a mixture of two gases (1 and 2) in equilibrium (i.e.two species in our analytical context), the molecules witha higher mass (e.g. m1 > m2) move at a lower mean squarespeed (v1

2 < v22) and those with a lower mass move faster (i.e.

½m1v12 = ½m2v2

2 = 3kBT/2, in the mean, per molecule per cell ofphase space for all the concurrent gases). This is the simplestexpression of the principle of equipartition of energy betweent.d.f. (Roller and Blum, 1986, p. 699; Aguilar, 2001, p. 567; Tiplerand Mosca, 2010, p. 577). Similarly, in the sampled ecosystems,“heavier” plots (which accumulate more meTp) tend to have lowervalues of I2

e ; conversely, “lighter” plots tend to have higher values ofI2e . Thus solar energy, for the whole ecosystem and under stationary

conditions, tends to be equally invested at the “macroscopic” levelbetween alternative e.d.f. (internal energy embodied in biomassvs. dispersal), in an approximately consistent way with a sort ofprinciple of trophodynamic equipartition on a large scale.

However, there is an internal gradient on a “microscopic” levelbecause the micro-associations of species are not randomly dis-tributed (as it happens with molecules in the gas phase space):those plots with a combination of heavier species and less dispersalare concentrated in a stable way at the high end of species diversity,meanwhile the plots with lighter species combinations and greaterdispersal tend to be at the low end. This is perhaps the main pro-functional gradient or thermodynamic constraint to avoid chaosand ecological degradation under far from equilibrium stationaryconditions.

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16 R.A. Rodríguez et al. / Ecological Modelling 261– 262 (2013) 8– 18

ma1ma2

ma3 ma4

mif1

mif2

mif3

cor

lli

plirsn1 rsn2

rsn3 crf

rv1

rv2rv3

rv4

rv5

rv8

rv9

Ms

mxs'

.

0.4 0.8 1.2 1.6 2.0 2.4

Hp (mean value)

0.2

0.6

1.0

1.4

1.8

2.2

Hp,

(EeT

p m

ax.)

r = 0.803, p < 0.0001 Hp, (EeTp max.) = 0.06 + 0.92 Hp

css

(a)

.. ma3.

mif1mif 2

mif 3

cor

lli

pli

.rsn2

rsn3

crf ..

rv3

rv4

rv5

rv8

.

. m xs'

css

1.0 1.4 1.8 2.2 2.6 3.0 3.4

HT

500

1500

2500

3500

4500

EeT

p/m

eTp

r = -0.52, p = 0.009EeTp /meTp = 3570 .94 - 774.09 HT

(b)

Ms

rv9

rv2

rv1

ma2ma4

ma1

Fig. 3. Relationship between energy indicators at the plot level and species diversity. (a) Correlation between the value of diversity per plot in which the maximum value ofthe total energy (EeTp max.) is reached and the general mean value of species diversity per plot. (b) Correlation between the mean value of available energy per unit of biomassper plot (EeTp/meTp) and the total species diversity of each assemblage (HT).

Therefore, Eq. (9) indicates a peculiarity of the ecosystem incomparison with gases: in the latter case molecular mass is con-stant for each kind of gas even at the individual level regardless ofkinetic energy; but in the case of ecosystem this is not even true forindividuals that belong to the same species. Therefore, if we wouldrewrite Eq. (9) in a comparative way for two masses of the sametype of substance in equilibrium at different temperatures (T), thenwe would establish the following didactic expression (mc: constantmolecular mass of the substance; vx: mean molecular velocity alongthe x axis):

(mc, v2x ↑, T ↑) ∨ (mc, v2

x ↓, T ↓) ∴ mc · v2x

T= kB → constant (10)

The noticeable difference between Eq. (9) and Eq. (10) indicatesthat the slow down process (Ie reduction) from low to high diver-sity values is an essential feature that distinguishes the ecosystemfrom the non-living physical systems and it can be assumed as the“source” of biomass increment either from the ontogenetic or phy-logenetic point of view. This trend toward larger sizes can be relatedto the influence of selective pressures in favor of the less expensiveenergetic maintenance per gram of biomass under high diversityconditions (see Fig. 3b).

Thus, maximum productivity depends on those non-stationarysystems (with quasi-continuous growth) of low diversity thatcan support intense trophic exploitation, as in the case of agroe-cosystems. Alternatively, high diversity communities have lowerproductivity but higher efficiency (a greater amount of biomasssupported per unit of energy, i.e. a higher meTp/EeTp ratio). Thiscompensatory relationship between productivity and efficiencydepends on the correlative position of dispersal and biomass in theabove-commented ratios (dispersal is part of the numerator andbiomass part of the denominator to estimate productivity, and viceversa to assess efficiency). As a result, a primary B-DT implies asecondary trade-off between productivity and efficiency.

Additionally, items (a) and (b) indicate that the time trend of anecological assemblage can be glimpsed by means of a momentaryobservation of the spatial pattern of an ecosystem (ergodicity, seecomments about Eq. (2) in section 2 as well as in Appendix A, pp.1–3).

In summary, our proposal is congruent with the opinion ofWillig (2011, p. 1710): “the absence of a modal [hump-backed]pattern suggests the absence of defining trade-offs. Finally, fora modal pattern to emerge, the trade-offs must be strong andpervasive.”

5. Concluding remarks

Our proposal opens a new branch or family of models for eco-logical modeling based on an affirmative answer to a question

that is fundamental to understand ecosystem functioning from thephysics perspective. Can we describe ecosystem functioning understationary conditions by means of the orthodox models developedby the conventional thermodynamics and thermostatistics?; ourproposal responds affirmatively to this questions since the sta-tionary ecological state is consistent with the formal definitionof equilibium state as the pivotal concept of thermodynamics andthermostatistics: “By definition (. . .) thermodynamics describes onlystatic states of macroscopic systems” (Callen, 1985, p. 6), (. . .) “inall systems there is a tendency to evolve toward states in which theproperties are determined by intrinsic factors and not by previouslyapplied external influences. Such simple terminal states are, by defi-nition, time independent. They are called equilibrium states” (Callen,1985, p. 13). The values of state variables do not change with timeneither under equilibrium nor under stationarity. That is to say,the stationary state is for the analysis of open systems what equi-librium state is for the analysis of isolated systems (Montero andMorán, 1992).

Additionally, such classical models were conceived for systemsof weakly interacting elements and, accordingly, a reduction in thestrength of interactions with the increase of diversity is an essentialrequirement to maintain community stability (Margalef, 1974, p.366; McCann et al., 1998; Neutel et al., 2002; Banasek-Richter et al.,2009). In agreement with the former, the general ecological trendsalong non-stationary transitions could be deduced by comparisonbetween different stationary states or by means of a harmoniousarrangement between the outcomes derived from our proposal andthe results obtained from those approaches that, implicitly, arebased on a negative response to the question above outlined orare independent of this question (e.g. those models intended todescribe non-stationary ecological states).

According to our results, Eq. (7) is more than a particular caserestricted to ruderal vegetation. An additional key element in com-parison with Rodríguez et al. (2012) is that ke(e) = 1.3806504Eϕ withϕ = −xi, . . ., 0, . . ., + xi, where xi is an integer number that tends to betypical of each particular kind of ecological assemblage, e.g. xi = −10from ma1 to ma4 (marine microalgae); xi = 00 from rsn1 to rsn3(tropical rocky shore snails) or xi = +02 from rv1 to rv9 (ruderalvegetation). This implies that the ESE is valid (as a pattern) regard-less of environment, body size or taxon because the mantissa ofke(e) (1.3806504) is the same for all ecological assemblages underSC but ϕ varies following a suitable scale progression according tovariation of ecological conditions.

Three main general positions can be assumed about this subject:

1) Very pessimistic and conservative position: these findings arecircumstantial. This is the typical case of a fluke in whichan improbable combination of coincidences emerges whenexploring one scientific field by using tools from another field.

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R.A. Rodríguez et al. / Ecological Modelling 261– 262 (2013) 8– 18 17

There are two main facts to take into account regarding thisposition: (a) all our foundational knowledge about ecosystemfunctioning is based on thermodynamics rather than on “classi-cal” ecology itself. (b) “. . .in the history of human thinking themost fruitful development frequently take place at those pointswhere two different lines of thought meet” (Heisenberg, 2000, p.129).

2) Moderately optimistic position: This approach is simplya mathematical artifact whose main instrumental contri-bution is a better understanding of the underlying pat-terns in ecosystem functioning. In this case our pro-posal is congruent with the ultimate purpose of scienceitself: to gain understanding of the natural world (Oreskes,2003).

3) Optimistic position: This model shows internal consistencyand reflects a natural reality. In this case, this article sug-gests some answers but, at the same time, it poses many newquestions (e.g. What is the cause of the discrete or discon-tinuous scale variation in the value of ke between successivestationary states? Are there underlying principles connectedwith B-DT that can be useful to explain species coexistenceissues, for instance, the old controversy between competi-tive exclusion and functional redundancy (see Lewin, 1983)?Is it possible to complete the understanding of ecologicalstationary states in terms of conventional thermostatistics?What are the advantages, disadvantages and epistemologicallimits of the models based on physics to explain ecosys-tem functioning?). This would be the best alternative, sinceany reliable answers to these new questions hold promiseof probable new findings, and this is an essential goal ofscience.

Nevertheless, one of the main values of this approach lies inlogical transitivity: our results indicate that A (ecological stateequation) is consistent with B (biomass-dispersal trade-off) andB is consistent with C (conventional ecology; i.e. Cope’s rule,Rapoport’s rule, r–K selection theory, production–diversity pat-terns and productivity-efficiency trade-off). As a consequence, Ais also consistent with C.

Acknowledgements

The authors wish to thank Drs. Manuel Antonio González dela Rosa and Marta González Hernández, as well as the “FundaciónCanaria Rafael Clavijo”, for personal, institutional and financial sup-port which allowed us to carry out this research. R.Ma. Navarrocontribution was supported through DIVERBOS (CGL2011-30285-C02-02). We also thank Dr. José Juan Cáceres Hernández for hisguidance on the statistical treatment of data. Our special thanksare due to Michael Lee McLean who undertook the dauntingtask of translating the text into English. The manuscript wassignificantly improved from the comments of two anonymousreviewers.

Appendix A. Supplementary data

Supplementary data associated with this article can be found,in the online version, at http://dx.doi.org/10.1016/j.ecolmodel.2013.03.023.

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