J. theor. Biol. (1999) 197, 371–392Article No. jtbi.1998.0881, available online at http://www.idealibrary.com on

0022–5193/99/007371+22 $30.00/0 7 1999 Academic Press

The Application of Mass and Energy Conservation Lawsin Physiologically Structured Population Models of

Heterotrophic Organisms

S. A. L. M. K*†, B. W. K* T. G. H‡

*Department of Theoretical Biology, Vrije Universiteit, de Boelelaan 1087, 1081 HVAmsterdam, The Netherlands and ‡Department of Ecology & Evolutionary Biology,

University of Tennessee, Knoxville, TN 37996, U.S.A.

(Received on 5 March 1998, Accepted in revised form on 26 November 1998)

Rules for energy uptake, and subsequent utilization, form the basis of population dynamicsand, therefore, explain the dynamics of the ecosystem structure in terms of changes instanding crops and size distributions of individuals. Mass fluxes are concomitant with energyflows and delineate functional aspects of ecosystems by defining the roles of individuals andpopulations. The assumption of homeostasis of body components, and an assumption aboutthe general structure of energy budgets, imply that mass fluxes can be written as weightedsums of three organizing energy fluxes with the weight coefficients determined by theconservation law of mass. These energy fluxes are assimilation, maintenance and growth, andprovide a theoretical underpinning of the widely applied empirical method of indirectcalorimetry, which relates dissipating heat linearly to three mass fluxes: carbon dioxideproduction, oxygen consumption and N-waste production. A generic approach to thestoichiometry of population energetics from the perspective of the individual organism isproposed and illustrated for heterotrophic organisms. This approach indicates that masstransformations can be identified by accounting for maintenance requirements and overheadcosts for the various metabolic processes at the population level. The theoretical backgroundfor coupling the dynamics of the structure of communities to nutrient cycles, including thewater balance, as well as explicit expressions for the dissipating heat at the population levelare obtained based on the conservation law of energy. Specifications of the general theoryemploy the Dynamic Energy Budget model for individuals.

7 1999 Academic Press

1. Introduction

The importance of ecological energetics, aconfluence of ecology and thermodynamics,has been widely recognized since Lindeman’spioneering efforts (Lindeman, 1941, 1942) butthese concepts go back at least to Lotka (1924),

who was interested in a ‘‘law of maximumenergy’’ for biological systems. Wiegert’s synop-sis (Wiegert, 1976) covers a trophic structureenergetics perspective of ecological systemsalong with some of its inherent difficulties.

Because energy limitations frequently controlpopulations, the coupling between mass andenergy fluxes is fundamental to ecologicalenergetics. The application of both mass and

†Author to whom correspondence should be addressed.E-mail: bas.bio.vu.nl

. . . . .372

energy conservation laws to community dynam-ics has been considered too complicated to bepractical. For example, Berryman et al. (1995)state

‘‘However, it can be argued that strict adherence to thelaws of conservation may unnecessarily constrainpredator–prey theory because predators do not alwayskill their prey and sometimes kill without eating.’’

We think, however, that this does not hamperthe application of conservation laws; kills noteaten combine with deaths via ageing or othercauses, in an explicit flux. In our opinion,population dynamical theories and modelscould benefit considerably by making explicit useof conservation laws. The discipline of ecologyhas few basic principles it can rely on, so it isimportant to not dismiss the valid ones tooreadily.

This paper presents a generic framework toaccount for both conservation of mass andenergy in heterotrophic organisms (animals andmicro-organisms), a discussion of autotrophicorganisms (plants) is beyond the scope of thispaper. The theory can be extended to includeautotrophs (Kooijman, 1998; Kooijman &Nisbet, 1999), however, their metabolic versatil-ity involves more state variables. The interactionbetween the individuals being restricted to simplecompetition, our treatment can be judged as justa step towards a more realistic (and complex)modelling framework (Grover, 1997). Wedemonstrate that energy uptake and utilizationgovern the dynamics of the structure of theecosystem in terms of changes in standing cropsand size distributions of individuals and weindicate that mass fluxes are the basis offunctional aspects of ecosystems, explaining theroles of populations in the ecosystem. Thestoichiometry of metabolic transformations atthe population level is developed from anenergetics perspective; this indicates that massand energy fluxes are intimately linked to eachother through the concept of homeostasis ofbody components. Consequently, the rules forenergy uptake and use do not allow supplemen-tary assumptions on mass fluxes, such asrespiration or nitrogen waste, without creatinginconsistencies. In Reiners’ opinion (Reiners,1986) models for energy and mass fluxes in

ecosystems are complementary, and can bedeveloped independently. He does recognize thatthese models have many points of intersection;we will show here that the intersection iscomplete for heterotrophic organisms; mass andenergy fluxes should be considered as two aspectsof the same concept.

We believe that ecological energetics has awider applicability than its name suggests,because of the coupling between energy andmass. Many populations are limited by theavailability of nitrogen rather than energy(White, 1993). In an energetic framework, thissituation translates to a reduced efficiency of theconversion of food into assimilation energy,because of the nutritional inadequacy of thefood. A more appropriate name for ecologicalenergetics could refer to limitations by a singlecomponent of the resource.

Energy fluxes are generally difficult to measuredirectly. Attempts to measure energy fluxes viarespiration rates (i.e. carbon dioxide or oxygenfluxes, also called ‘‘metabolic rates’’) havecreated problems, because a careless mapping ofrespiration rates to energy fluxes can easily yieldincorrect budgets. Although actual growth inbiomass during the short period of the measure-ment of respiration rates can be negligibly small,this does not imply that energy investment intothe growth process is also negligibly small.Respiration is frequently, but incorrectly,identified with routine metabolic costs, con-ceived as maintenance costs. The interpretationof respiration rates in terms of energy invest-ments has been the source of a long standingproblem in animal physiology: why is the slopeof the regression line of the log metabolic rateagainst the log body weight about 0.75? Withers(1992) calls this ‘‘one of the most perplexingquestions in biology’’. A sound approach to therelationship between respiration and energyfluxes provides a solution for this problem(Kooijman, 1986, 1993).

Although the specification of energy fluxes(known as ‘‘powers’’) is dependent upon theindividual model employed [such as the DynamicEnergy Budget (DEB) model (Kooijman, 1993)],the coupling between energy and mass fluxesholds for a broad class of models representinguptake and utilization of resources. We here

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show how the coupling follows from consistencyarguments given a short list of seemingly simpleand ‘‘harmless’’ assumptions about the generalstructure of the budget model. This powerfulresult can be rephrased by pointing to the farreaching implications of these assumptions; ifthese implications do not apply, the generalassumptions should be reformulated, and theirharmlessness is deceptive.

As an example of the application of the theoryfor mass fluxes in ecosystems we consider amyxamoeba–bacteria–glucose food chain in thechemostat. Experimental data by Dent et al.(1976) have been used for parameter estimation,and the mass fluxes have been analysed (Kooi &Kooijman, 1994b; Kooijman & Kooi, 1996). Wehere present the transfer of mineral compounds,carbon dioxide, water, oxygen and nitrogenwaste that are implied by the organic fluxes. Thepredator as well as the prey propagate by binaryfission and this justifies the use of a simplifiedversion of the (DEB) model (Kooi & Kooijman,1994a). With this model for the individuals, thesteps from the physiology of the individual tofluxes at the population level and further tomulti-species systems can be made on theassumption that conspecific individuals onlyinteract by competition (Kooi & Kooijman,1995). Although this simple assumption aboutinteractions seems to apply in the artificialenvironment of a well-mixed chemostat in thiscase, most less artificial environments require themodelling of elaborate forms of interactions,spatial structure, and many other effects of thelocal environment. Many of these complicatingphenomena affect population dynamics viafeeding (while effects on reproduction, develop-ment, growth and survival follow from effects onfeeding), which means that the equationsdescribing population dynamics should beadjusted. Most of the theory that is presentedhere about mass-energy coupling, however, stillapplies, given the feeding flux. The formulationof realistic models for population dynamics inthese more complex situations is beyond thescope of this paper.

We first summarize the mass fluxes anddissipating heat flux for an individual. Ratiosbetween these mass fluxes represent stoichio-metric coefficients. Theory on the stoichiometry

of metabolic transformations has been developedin the microbiological literature, which selectsthe food uptake flux as a reference. Becauseembryos do not eat and grow at the expense ofreserves, these ratios are not practical foranimals with an embryonic life stage. Astoichiometry on the basis of the reserve flux hasbeen worked out (Kooijman, 1995), but in thispaper a substantial simplification is obtained byworking with fluxes directly and avoiding the useof stoichiometric coefficients. Following fluxesthrough individuals, we consider fluxes for apopulation based on these individuals which areassumed to interact only via competition for thesame resource. We then discuss mass fluxes infood chains as a step towards ecosystems. Wedemonstrate that this approach is extremelyconvenient for analysing the processes ofnutrient cycling at the ecosystem level, providedonly that a model for the energetics of anindividual is prescribed.

2. Mass Fluxes for Individuals

Table 1 gives the man symbols and notation;Table 2 lists all assumptions about generalaspects of mass and energy fluxes.

2.1.

Our approach is to represent and follow thetransfer of chemical elements, because elementsobey conservation laws; compounds generally donot. For illustration, we follow the four mostabundant elements in living systems, C, H, O andN, but this list can be extended readily becauseeach new element comes with a correspondingbalance equation. We delineate two sets ofchemical compounds:

‘‘Mineral’’ (M) ‘‘Organic’’ (O)

C carbon dioxide X substrate (food)H water V structural body massO oxygen E reservesN nitrogen waste P product (faeces)

The structural body mass (V) and reserves (E)constitute the individual (Assumption 1 inTable 2), the other organic compounds and theminerals define the chemical environment ofthe individual. For simplicity’s sake, water in thenitrogen waste (urine) is included in its chemical

. . . . .374

T 1List of main symbols. The symbols in the dimension-column stand for t time, e energy, (number (i.e. C-mole). Vectors and matrices are denoted in bold face; the notation nT meansthe transpose of n. All rates are designated with dots. The symbol * is used as a placeholder,

for which another symbol can be substitutedSymbol Dimension Interpretation

X, V, E, PC, H, O, N 7 Indices for

compounds 6 Food, body mass, reserves, product ($O)Carbon dioxide, water, oxygen, nitrogen waste ($M)

Mm , M, Mh Indices for Maturity maintenance, somatic —, endothermic heating —,Gm , G, R energies maturity growth, somatic —, reproduction,7 6A, D, C, T assimilation, dissipative, catabolic, heat

n*1*2 — Number of atoms of element *1 in compound *2 per C-atomnM, nO — Matrix of chemical indices of minerals, organic compoundsJ� *, J� M, J� O (t−1 Flux of compound *, —of minerals, organic compoundsM*, M*0, M*m ( Mass of compound * (*= E,V), initial mass, maximum massMX, MK (l−3 Density of mass of food, —as saturation constantpt*, pt et−1 Power *, the three basic powers ptA, ptD, ptGm*, mM, mO e(−1 Chemical potential of compound *, —of minerals, organic compoundsm*1*2 e(−1 Power *2 per flux of mass *1

h (e−1 Matrix of coefficient that weigh powers to obtain mass fluxest, a t Time, agef — Ingestion as fraction of its maximum, given l: scaled function responsee — Reserve density as fraction of its maximuml, lb, lp, la — Boby length as fraction of its maximum, —at birth, puberty, divisionht, hta, hte t−1 Hazard rate, —for ageing, —for embryosR� (t−1 Reproduction ratef, fe t−1 Stable age distribution of juveniles+adults, —of embryosF, Fe (l−1t−1 Relative frequency density of juveniles+adults, —of embryosN, Ne (l−3 Density of number of juveniles+adults, —of embryos

‘‘composition’’, as is done for methane in faeces(which is relevant for mammals). Faeces includesbile and enzymes that are excreted in the gut,since these excretions are tightly coupled to thefeeding process. In contrast to the ‘‘static’’energy budget tradition, urine production (thenitrogen waste) is not tightly coupled to thefeeding process, because maintenance processescontribute via protein turnover.

Food for micro-organisms is usually called‘‘substrate’’, and faeces ‘‘metabolic products’’.

These products generally do not originate fromsubstrate directly, but indirectly through themetabolic machinery of the organism. Thisproblem is addressed by including such productsinto the overheads of the three basic energyfluxes (Assumption 3 in Table 2). The number ofdifferent products can be extended in astraightforward manner. Not only bacteria andfungi produce compounds that are excreted intothe environment, many animals do this as well(e.g. mucus, moults, milk). The (energy/carbon)

T 2Assumptions about general aspects of energy and mass fluxes through an individual

1. The amounts of structural body mass and reserves are the state variables of the individual; body mass and reserves areinvariant in composition (strong homeostasis assumption).

2. Food is converted into faeces; food and faeces are invariant in composition.3. Assimilates derived from food are added to reserves, which fuel all other metabolic processes. These processes are

classified into three categaories: synthesis of structural body mass, of (embryonic) reserves (i.e. reproduction), andprocesses not associated with net synthesis. Products that leave the organism may be formed in direct association withany or all of these three categories of processes, and with the assimilation process.

4. If the individual propagates via reproduction (rather than via division), it starts in an embryonic stage that initially hasa negligibly small structural body mass, but a substantial amount of reserves.

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substrate for micro-organisms can be poor innitrogen, such that nitrogen must be taken upfrom the environment, rather than excreted. Thecompound ‘‘nitrogen waste’’ (this terminology isappropriate for metazoans living on protein-richfood) should be identified by ‘‘nitrogen source’’;the sign of the flux defines uptake or excretion.Bacteria that live on glucose as energy source willhave a negative nitrogen waste flux. Nitrogen(and/or oxygen for aerobic organisms, and/orcabon dioxide for bacteria feeding on methane)is assumed to be available in sufficient quantity.The theory can be extended, however, to includesimultaneous limitations, (cf. Zonneveld, 1996,1997; Zonneveld et al., 1997; Kooijman, 1998).

Assumptions 2 and 3 in Table 2, when pushedinto the extreme, imply that chemical potentialsof the organic compounds per C-mole areconstant, which is consistent with the biochemi-cal literature (Westerhoff et al., 1983). Themotivation of these assumptions can be based onthe idea that structural body mass and reservesmainly consist of polymers (polysaccharides,lipids and proteins), which do not take part inthe metabolism directly, while the concentrationof monomers, which are directly involved inmetabolism, is low and constant. Food (sub-strate) is digested intracellularly, or in the gut,which also represents a constant chemicalenvironment. Products are also formed intra-cellularly; variations in concentrations in theenvironment are assumed not to affect energyconsiderations for individuals.

2.2. fi

Assumption 3 implies that the relationshipsbetween powers and mass fluxes involve threegroups of basic powers:

pA

p0gG

G

F

f

pDhG

G

J

jpG

assimilation power (coupled to food intake)

=dissipating power (no net synthesis of biomass). (1)

anobolic power (somatic growth)

A variety of metabolic processes contributes todissipating power; it is sufficient at this point toassume that the dissipating power is a function

of the two state variables, V and E, of theindividual and not delineate the representation.Most of the dissipating power leaves thethermodynamic system, consisting of the individ-ual and relevant organic and mineral com-pounds, as heat, while a portion leaves thesystem as nitrogen waste or (other) products.Part of the growth and assimilation power willalso contribute to dissipating heat because of theoverhead costs; growth and assimilation do notoccur with 100% efficiency (see below).

Reproduction power ptR has a special statusbecause reserves of the adult female areconverted into reserves of the embryo, each ofwhich have the same composition by virtue ofAssumption 1 in Table 2. The efficiency of thisconversion is denoted by kR, which means that(1− kR)ptR is dissipating and kRptR returns to thecompound class ‘‘reserve’’, but now of theembryo. The amount of reserves allocated toreproduction during a very small time incrementis very small, not nearly enough to make oneembryo. This property, shared by all time-continuous models, necessitates the existence ofa buffer of reserves with destination reproduc-tion. Reproduction itself, i.e. the conversion ofthe reserves in this buffer to embryos, is treatedas an instantaneous event. The overhead costs ofthe reproduction event are taken into account inthe allocation to reproduction through theparameter kR.

In the considerations below, the fluxes ofreserves and reserves in the reproduction bufferare added. This makes sense biologically,because the buffer is still in the individual; thereason for the addition is that the assumptions inTable 2 imply that the sum of both fluxes will beshown to be a weighed sum of the three basicpowers, however, this does not necessarily holdfor each of the fluxes separately. (It does not holdin the DEB model, for instance.) Mineral fluxesdepend only on the sum, so there is no need totreat them separately. From a chemical point ofview, reproduction does not represent a trans-formation, because reserves are converted intoreserves with an identical composition, while theoverhead costs for reproduction contribute todissipating power. The inclusion of reproductiveinvestment into the reserve flux allows generaliz-ation to organisms that propagate via division

Food Faeces

Heat

Somaticwork

Maturitywork

Volume Maturity

Storage Gametes

X

G GmM Mm

P

A

C RMh

. . . . .376

T 3Assumptions of the DEB model, in addition to the ones listed in Table 2

5. The transition from embryo to juvenile initiates feeding. The transition from juvenile to adult initiates reproduction andceases maturation. Transitions occur when the cumulated energy invested in maturation exceeds a threshold value.

6. Somatic and maturity maintenance are proportional to body volume, but maturity maintenance does not increase aftera given cumulated investment in maturation. Heating costs for endotherms are proportional to surface area.

7. The feeding rate is proportional to surface area and depends hyperbolically on food density.8. The reserves must be partitionable, such that the dynamics are not affected, and the energy density at steady state does

not depend on structural body mass (weak homeostasis assumption).9. A fixed fraction of energy, utilized from the reserves, is spent on somatic maintenance plus growth, the rest on maturity

maintenance plus maturation or reproduction (the k-rule).

but do not allocate to reproduction. Division istreated here as an instantaneous event, whichoccurs when the individual reaches a thresholdsize. Details about the division only play a roleat the population level because individuals arefollowed up to the division event. Division isassumed to produce two identical daughterindividuals, but this restriction can be relaxedin several ways, without affecting the mainargument.

2.2.1. Specific budget models

Table 3 gives the assumptions of the DEBmodel, in addition to the ones listed in Table 2.These assumptions have been underpinnedmechanistically and tested against empirical data(Kooijman, 1993, Hanegraaf 1997). The threegroups of powers should be specified as functionsof the state of the individual. Figure 1 displaysthe energy fluxes. As an example, Table 4

delineates these relationships for the DEB modelfor a three-Stage–three-Dimensional (3S–3D)isomorph, i.e. an individual that does not changein shape during growth, and reproduces via eggs,which implies the three stages embryo, juvenileand adult. Since the DEB model requires thatfood uptake is proportional to surface area(Assumption 7 in Table 3), volume/massconversions should be considered; we presentthem in Appendix A.

Table 5 presents the DEB model for aone-Stage–one-Dimensional (1S–1D) isomorph,i.e. an individual that grows only in length, sothat surface area is proportional to volume, anddivides into two identical daughter individuals.This simplifies the DEB model considerably.Bacteria and fungi are interesting examples of1S–1D isomorphs. Table 5 also specifies the threebasic powers for the well-known Marr–Pirt(Marr et al., 1969; Painter & Marr, 1968; Pirt,

F. 1. Energy fluxes through a heterotroph. The rounded boxes indicate sources or sinks. All powers contribute todissipating heat, but this is not indicated in order to simplify the diagram. The powers ptX = JtXmX and ptP = JtPmP for ingestionand defecation ‘‘connect’’ the individual with the environment. The DEB model assumes that JtXAJtPAptA. The dissipatingpower is ptD = ptMm + ptM + ptMh + ptGm +(1− kR)ptR.

/ 377

T 4The powers as specified by the DEB model for a 3S–3D isomorph of scaled length l and scaled reservedensity e at scaled functional response f0X/XK +X, where X denotes the food density and XK thesaturation constant. Relationships are given in the diagram 1. The table presents scaled powers, wheremE denotes the chemical potential of the reserves. Parameters: g investment ratio, kM maintenance ratecoefficient, k partitioning parameter for catabolic power, lh scaled ‘‘heating length’’. Ectotherms do notheat, i.e. lh =0. Implied dynamics for eq lq lb: d/dt e= f−e/lktM g and d/dt l= e− l− lh/e/g+1

ktM/3Embryo Juvenile AdultPower

mEMEmktMg 0Q lE lb lb Q lE lp lp Q lQ 1

Assimilation, ptA 0 fl2 fl2

Catabolic, ptC el2g+ lg+ e el2g+ l+ lh

g+ e el2g+ l+ lhg+ e

Somatic maintenance, ptM kl3 kl3 kl3

Maturity maintenance, ptMm (1− k)l3 (1− k)l3 (1− k)l3p

Endothermic heating, ptMm 0 kl2lh kl2lh

Somatic growth, ptG kl2 e− le/g+1 kl2e− l− lh

e/g+1 kl2e− l− lhe/g+1

Maturity growth, ptGm (1− k)l2 e− le/g+1 (1− k)l2e− l+ lhe/g

e/g+1 0

Reproduction, ptR 0 0 (1− k)(l2e− l+ lhe/ge/g+1 + l3 − l3p)

1965) and the Monod model (Monod, 1942) formicrobial growth.

2.3.

Let n*1*2 denote the chemical index ofcompound *2 for element *1. We will choose the

chemical indices of the organic compounds forcarbon equal to 1 and refer to their amounts asC-moles. The homeostasis Assumptions 1 and 2in Table 2 are equivalent to the condition thatthe chemical indices do not change.

Let J� * denote the rate of change of thecompound * as a result of utilization (J� * Q 0) or

T 5The powers as specified by the DEB model for a 1S–1Disomorph of scaled length l and scaled reserve densitye at scaled functional response f. We take ptMh =ptR =0,so that ptD =ptM +ptMm +ptGm, and 2−1/3ld Q lE ld. Anindividual of structural volume V0MV/[MV] takes upsubstrate at rate [J� Xm]fV. The implied dynamics for eand l: d/dt e= f−e/ldktMg and d/dt l=ktMl/3 e/ld −1/e/g+1. We also present the Marr–Pirt and the Monod

models in the same notationPower

mEMEmktMg DEB Marr Monod

Assimilation, ptA l3f/ld l3f/ld l3f/ld

Dissipating, ptD l3 l3 0

Somatic growth, ptG l3e/ld −1e/g+1 l3f/ld − l3 l3f/ld

. . . . .378

production (J� * q 0) by the individual. Theconservation of mass states

0=GG

G

F

f

1 0 0 nCN

0 2 0 nHN

2 1 2 nON

0 0 0 nNN

GG

G

J

jGG

G

F

f

J� CJ� HJ� OJ� N

GG

G

J

j

+GG

G

F

f

1nHX

nOX

nNX

1nHV

nOV

nNV

1nHE

nOE

nNE

1nHP

nOP

nNP

GG

G

J

jGG

G

F

f

J� XJ� V

J� E + J� RJ� P

GG

G

J

j(2)

This can be summarized in matrix form asO= n−1

M J� M + nOJ� O. Thus the fluxes for the‘‘mineral’’ compounds can be written explicitlyas

J� M =−n−1M nOJ� O (3)

with

n−1M =

1

0

−1

0

0

2−1

−4−1

0

0

0

2−1

0

−nCN

nNN

−nHN

2nNN

n4nNN

1nNN

(4)GG

G

G

G

G

G

F

f

GG

G

G

G

G

G

J

j

and n0 4nCN + nHN −2nON.We will now explain why the organic fluxes J� O

follow from the basic powers pt via

J� O 0GG

G

F

f

J� XJ� V

J� E + J� RJ� P

GG

G

J

j=G

G

G

F

f

−m−1AX

0m−1

E

m−1AP

00

−m−1E

m−1DP

0m−1

GV

−m−1E

m−1GP

GG

G

J

j

pA

×gG

G

F

f

pDhG

G

J

j

0 hp (5)

pG

where mE is the chemical potential of the reserves,and m*1*2 denotes the power *2 per flux of mass *1,i.e. the coupling between mass and energy fluxes.The m*1*2s serve as model parameters.

The substrate flux J� X =−ptA/mAX follows fromAssumptions 1 to 3 in Table 2, which imply aconstant conversion coefficient from food to

assimilation energy. We quantify assimilationenergy by its fixation into reserves, so reservesare formed at a rate ptA/mE, where mE stands forthe chemical potential of the reserves. The ratiomAX/mE equals the C-moles of food ingested perC-mole of reserves formed. The amount of workthat can be done by ingested food is mXJ� X; a part,ptA, is fixed into reserves, a part, ptAmP/mAP, is fixedinto product, and the rest dissipates as heat andmineral fluxes associated with this conversion.The ratio mAX/mAP equals the C-mole of foodingested per C-mole of product that is deriveddirectly from food; products can also be formedindirectly from assimilates, via the processes ofgrowth and maintenance, explaining the productflux J� P.

If the individual is a metazoan and the productis interpreted as faeces, we take m−1

DP = m−1GP =0.

Faeces production is then coupled to food intakeonly. Alcohol production by yeasts that live onglucose is an example of product formationwhere m−1

DP $ 0 and m−1GP $ 0. Carbon dioxide and

water partially serve as faeces here; this is takeninto account by (3) and (5), via the coefficientsfor the assimilation flux. At this point, moleculardetails about the process of digestion being intra-or extra-cellular are not needed. This knowledgeonly affects the precise interpretation of thecoefficients in h.

The body mass flux J� V = ptG/mGV follows fromAssumptions 1 and 2 in Table 2, which implythat a constant amount of energy, mGV, is investedper C-mole of structural body mass. Note that mV

is actually fixed in a C-mole of structural bodymass, so that mGV − mV dissipates (as heat or viaproducts that are coupled to growth) per C-mole.

The flux of (parent) reserves is given byJ� E = m−1

E (ptA − ptC), because reserve energy isgenerated by assimilation and used bycatabolism, i.e. the sum of all other metabolicpowers (Assumption 3 in Table 2). The catabolicpower can be written as ptC = ptD + ptG + kRptR.The flux of embryonic reserves (reproduction),J� R = m−1

E kRptR, appears as a return flux to thereserves because embryonic reserves have thesame composition as that of the parent as aconsequence of the homeostasis Assumption 1.The sum of the (parent) reserve and embryonicreserve fluxes amounts to J� E + J� R = m−1

E

(ptA − ptG). This completes the derivation of (5).

Food

–1 JX

.

.Structure biomass

40 JV

Faeces

1 JP

.

Reverses

10 ( JE + JR). ..

Carbon dioxide

2 JC

.

Water

2 JH –2 JO

Ammonia

10 JN

.

Oxygen

..

l

flu

x J

*

.

Scaled length l

J*

/ 379

F. 2. The organic fluxes JtO (top) and the mineral fluxes JtM (bottom) for the DEB model as functions of the scaledlength l at abundant food (e=1 for lq lb; 0Q lQ 1). The various fluxes are multiplied by the indicated scaling factorsfor graphical purposes. The stippled curve separates adult from embryonic reserves (reproduction). The parameters: scaledlength at birth lb =0.16,—at puberty lp =0.5 (both indicated on the abscissa), scaled heating length lh =0 (ectotherm),energy investment ratio g=1, partition coefficient k=0.8, reproduction efficiency kR =0.8. The coefficient matrices are:

mEh=GG

G

F

f

−1.501

0.5

00

−10

00.5−10

GG

G

J

j, nM =G

G

G

F

f

1020

0210

0020

0301

GG

G

J

j, nO =G

G

G

F

f

11.80.50.2

11.80.50.2

11.80.50.2

11.80.50.2

GG

G

J

j.

Figure 2 illustrates the fluxes of organic andmineral compounds, J� O and J� M, of the DEBmodel as a function of the structural body mass(i.e. scaled length, see next section), at abundantfood. The embryonic reserve flux is negativebecause embryos do not eat. The growth justprior to birth is reduced because the reservesbecome depleted. The switch from juvenile toadult implies a discontinuity in the mineralfluxes as functions of size (not age), but thisdiscontinuity is negligibly small. The reason ofthe discontinuity is that energy invested indevelopment dissipates because it is not fixed inmass, while energy invested in reproduction is(partly) fixed in (embryonic) reserves.

2.4. .

The reserves mass and the structural bodymass relate to the fluxes as ME(a)=ME0 + fa

0

J� E(t) dt and MV(a)=MV0 + fa0 J� V(t) dt. Assump-

tion 4 in Table 2 states that the initial value of

the structural body mass is negligibly small, i.e.MV0 =0. The mass of reserves of an embryo inC-moles at age 0, ME0, might be introduced as amodel parameter, but the DEB model obtainsthe value from the constraint that the reservedensity of the embryo at birth equals that of themother, i.e. e(ab)= f.

The changes in structural body mass, MV, andreserve mass, ME, relate to the powers as d/dtMV = J� V = ptG/mGV and d/dt ME = J� E =(ptA − ptC)/mE. If the model for these powers implies theexistence of a maximum for the structural bodymass, MVm, and for the reserve mass, MEm, itproves convenient to replace the state of theindividual, MV and ME, by the scaled lengthl0 (MV/MVm)1/3 and the scaled energy reservedensity e0MEMVm/(MVMEm). The change of thescaled state then becomes

ddt

l=ptG/mGV

3M2/3V M1/3

Vm

. . . . .380

and

ddt

e=MVm

MVMEm 0ptA − ptCmE

−ME

MV

ptGmGV1.

The reproduction rate, in terms of number ofoffspring per time, is given by Rt = J� R/ME0. Thethree basic powers, supplemented with the re-productive power, therefore, fully specify theindividual as a dynamic system.

2.5.

We introduced the dissipating power ptD, whichis an element of pt, to quantify a group of powers,such as maintenance, that is not allocated tobiomass production. In addition to this energyloss, heat dissipates in association with theprocesses of assimilation and growth becausethese processes are less than 100% efficient. Thetotal dissipating metabolic heat ptT follows fromthe energy balance equation

0= ptT + mTOJ� O + mT

MJ� M

= ptT +(mTO − mT

Mn−1M nO)mpt (6)

where

mTM 0 (mC mH mO mN) and mT

O 0 (mX mV mE mP)

designate the chemical potentials of the variousmineral and organic compounds, respectively.The thrust of the formulation is that the energyallocated to reserves and structural body massappears as parameter values, while the energyfixed in these masses is given by the chemicalpotentials, the differences appearing as dissipat-ing heat, i.e. overhead costs.

The dissipating metabolic heat contributes tothe thermal fluxes to and from the individual.The individual loses heat via convection andradiation at a rate ptTT = 4vtT5 (Tb −Te)V2/

3+ 4vtR5 (T4b −T4

e )V2/3. Here Te denotes theabsolute temperature in the environment, includ-ing a relatively large sphere that encloses theindividual. For radiation considerations, thesphere and individual are assumed to have gray,opaque diffuse surfaces. Tb is the absolutetemperature of the body; V2/3 is the body surfacearea; 4vtT5 is the thermal conductance and4vtR5= os is the emissivity times the Stefan–Boltzmann constant s=5.67×10−8 Jm−2 s−1

K−4; see, for instance (Kreit & Black, 1980). Thebody temperature does not change if ptT = ptTT.This relationship can be used to obtain the bodytemperature, given knowledge about the othercomponents. Most animals, especially theaquatic ones, have a high thermal conductance,which gives body temperatures only slightlyabove the environmental ones. Endotherms,however, heat their body to a fixed target value,usually some Tb =312 K, and have a thermalconductance as small as 4vtT5=5.43 J cm−2

hr−1 K−1 in birds and 7.4–9.86 J cm−2 hr−1 K−1

in mammals, as calculated from Herreid II &Kessel (1967), (see Kooijman, 1993).

Most endotherms are terrestrial and lose heatvia evaporation of water at a rate ptTH. Here wecan use the relationship ptT = ptTH + ptTT to obtainthe thermoneutral zone: the minimum environ-mental temperature at which no endothermicheating is required (ptMh =0). Alternatively, wecan deduce the heating requirements at a givenenvironmental temperature. To this end, we firstconsider the water balance in more detail, toquantify the heat ptTH that goes into theevaporation of water. The individual loses watervia respiration at a rate proportional to the useof oxygen, i.e. J� HO = dHOJ� O, (see Kooijman, 1995;Verboven & Piersma, 1995), and via transpira-tion, i.e. cutaneous losses. The latter routeamounts to 2–84% of the total water loss in birdsdespite the lack of sweat glands (Dawson, 1982).Water loss J� HH via transpiration is proportionalto surface area of the body, to the differencebetween vapour pressure of water in the skin andthe ambient air, to the square root of the windspeed, and depends on behavioural components.To maintain homeostasis, the animal has todrink at a rate J� H − J� HO − J� HH. The heat loss byevaporation amounts to ptTH = mTH(J� HO + J� HH),with mTH =6 kJ mol−1. Within the thermo-neu-tral zone, endotherms often control their bodytemperature by evaporation, through panting orsweating, which affects the water balance, andconsequently, requires enhanced drinking. Thepresent reasoning can be used to work outthe quantitative details.

2.5.1. Thermodynamic constraints

Given the assumption of constant partialmolar chemical potentials, and, therefore, con-

/ 381

stant partial molar entropies, for the organiccompounds, the second law of thermodynamicsimplies that the processes of assimilation,dissipation and growth are exothermic. Hence,we can decompose the dissipating heat into threepositive contributions ptTT 0 (ptTA ptTD ptTG), withptTT1= ptT and 1T =(1 1 1), which follow from thebalance equations for these three processes

OT = ptTT +(mTO − mT

Mn−1MMnO)h diag(pt) (7)

where diag(pt) is the diagonal matrix with theelements of pt on the diagonal. The sum of thethree equations (7) returns the overall balanceequations (6), because diag(pt)1= pt. We see thatthe heat that dissipates in connection with a basicpower, is proportional to that power. We also seethat the three factors that multiply the basicpowers in these three balance equations, shouldall be negative. This implies a constraint for eachcolumn of h. In a combustion frame of reference,where mM= 0, these constraints translate tomT

OhQOT. This constraint leaves us with theproblem to obtain the chemical potentials mO,which we need for the chemical environmentwithin the organism. The method of indirectcalorimetry can be used for this purpose(Kooijman, 1995). This method relates thedissipating heat to the mineral fluxes asptT = mT

TJ� M with regression coefficients

mTT 0 (mCT mAT mOT mNT) (8)

2 (60 0 −350 −590) kJ mol−1 (9)

for aquatic animals that use ammonia asN-waste (Brafield & Llewellyn, 1982). Smallcorrections have been proposed for birds(Blaxter, 1989) and mammals (Brouwer, 1958).The energy balance (6) and the mineral fluxes (3)lead to the desired result

mTO = mT

Tn−1M nO (10)

in a combustion frame of reference. Thisrelationship is sufficient, and perhaps necessary,to obtain the chemical potentials of complexorganic compounds in a complex environment.If knowledge of these potentials is available(from previous experiments, for instance), (10)

can be used as a constraint on the chemicalcoefficients nM and nO.

2.6.

The three basic powers form a basis for avector space of mass fluxes. The reserves play theimportant role of rendering the three powersindependent. Without reserves, these threepowers only span a two-dimensional vectorspace. The Marr–Pirt model for microbialgrowth is an example of such a model. Table 5shows that ptA = ptD + ptG holds for this model.Within this framework, product formationshould be taken as a weighted sum of twopowers, such as maintenance (and, therefore,biomass) and growth, as has been proposed byLeudeking & Piret (1959). The Monod model(see Table 5) has neither reserves nor mainten-ance. It is an example of a one-dimensionalvector space for mass fluxes, with ptA = ptG andptD =0. Within this framework, product for-mation should be taken proportional to growth(or assimilation) only. The Marr–Pirt modelrepresents a special case of the DEB model, andthe Monod model represents a special case of theMarr–Pirt model. We can generalize on theassumptions of Table 2 in several ways, onebeing to add the reproduction flux as a fourthbasic flux to the vector space of mass fluxes. Thisincrease in flexibility is bought at a cost ofadditional parameters.

Mineral and organic fluxes can be partitionedinto the contributions by assimilation, dissipa-tion power and growth in a simple way. When wewrite J� * = J� *A + J� *D +J� *G for *$4M, O5, andcollect these fluxes in two matrices, we have

J� M* =−n−1M nOJ� O* and J� O* = h diag(pt)

(11)

where diag(pt) represents a diagonal matrix withthe elements of pt on the diagonal, so thatdiag(pt)1= pt, and JtM*1= J� M, J� O*1= J� O.

If food input is too small for too long a period,the reserve flux J� E will eventually become toosmall to ‘‘pay’’ the maintenance costs. The modelshould specify what happens in those situations.One possibility is that the individual dies bystarvation, another is that it shrinks, i.e. J� V Q 0.In the latter case, we might assume that the

. . . . .382

structural biomass mineralizes instantaneously,which contributes to a mineral flux.

We conclude that all mass fluxes, mineral andorganic, as well as the dissipating heat, areweighted sums of the powers ptA, ptD and ptG. It isthe task of the model for resource uptake and useto specify these three basic powers as functionsof the state variables of the individual. Becauselinear functions of linear functions are linear aswell, dissipating heat is a weighted sum of threemineral fluxes, which is the basis of the widelyapplied method of indirect calorimetry. Itsempirical success can be conceived as support forthe general assumptions about the basic struc-ture of budgets.

3. Mass Fluxes in Populations

The step from individual to populationrequires a specification of the interactionsbetween the individuals. The simplest specifica-tion is feeding on the same resource, whichimplies competition. The sections below discussthe mass and energy fluxes in transient states;steady states are discussed in Appendix B.

3.1.

We now consider a population of partheno-genetically reproducing individuals that developthrough embryonic, juvenile and adult stages.Sexually reproducing animals can be included ina simple way, as long as the sex ratio is fixed. Thepopulation structure, derived from the collectionof individuals that compose the population, isbased upon individual characteristics. We willassume the existence of a maximum amount forstructural body mass and reserves for individ-uals, and use the scaled length l, the scaledreserve density e and the age a to specify the stateof the individual, but the techniques to model thedynamics are readily available for an arbitrarynumber of state variables (Hallam et al., 1990,1992).

Suppose that a population of individuals livesin a ‘‘black box’’ and that the individuals onlyinteract through competition for the same foodresource. Food is supplied to the black box at aconstant rate htXMX, where htX has dimension

time−1 and MX is the amount of food (in C-molesper black box volume). Eggs are removedfrom the black box at a rate hte; juveniles andadults are harvested with a rate ht randomly, i.e.the harvesting process is independent of thestate of the individuals (age a, reserves e, size l).Furthermore, the ageing process harvests juven-iles and adults at a state-dependent rate hta, whichis beyond experimental control; (see Kooijman,1993 for a specification of the ageing process thatis consistent with the DEB model). Individualsharvested by the ageing process leave the blackbox instantaneously.

The present purpose is to study how foodsupplied to the black box converts to body massand reserves that leave the box in the form ofharvested individuals when the amounts ofoxygen, carbon dioxide, nitrogen waste andfaeces in the black box are kept constant. Thisimplies that these mass fluxes to and from thebox equal the use or production by allindividuals in the box. We do not remove food,which implies that the amount of food in the boxdepends on both the food supply and theharvesting rates of individuals.

In summary, the conversion process has threecontrol parameters: htX, hte and ht and our aim isnow to evaluate all mass fluxes in terms of thesethree control parameters, given the properties ofthe individuals. This result is of direct interest forparticular biotechnological applications, and forthe analysis of ecosystem behaviour provided wewrite the control parameters as appropriatefunctions of other populations and/or environ-mental processes and specify the processes ofdegradation of faeces and dead individuals torecycle the nutrients that are locked into thesecompounds.

We will use the index + to refer to thepopulation, to distinguish fluxes at the popu-lation level from those at the individual level.Embryos are treated separately from juvenilesand adults, not only because we allow differentharvesting rates for both groups, but alsobecause they do not eat, and therefore do notinteract with the environment via food.

Given the initial conditions Fe (0,a,e,l) andF (0,a,e,l), the change of the densities ofembryos and of juveniles plus adults over the

/ 383

state space is given by the McKendrick-vonFoerster hyperbolic partial differential equation:

11t

Fe(t,a,e,l)=−11l0(Fe(t,a,e,l)

ddt

l1−

1

1e 0Fe(t,a,e,l)ddt

e1−

11a

Fe(t,a,e,l)

− hteFe(t,a,e,l) (12)

11t

F(t,a,e,l)=−11l0(F(t,a,e,l)

ddt

l1−

1

1e 0F(t,a,e,l)ddt

e1−

11a

F(t,a,e,l)− ht

+ hta(a,e,l))F(t,a,e,l) (13)

where fa2a1

fl2l1 fe2

e1F(t,a,e,l) de dl da is the number

of individuals (juveniles plus adults) having anage somewhere between a1 and a2, a scaled energydensity somewhere between e1 and e2 and a scaledlength somewhere between l1 and l2. The totalnumber of juveniles plus adults equals N(t)= fa

0

f1lb f1

0 F(t,a,e,l) de dl da. The total number ofembryos likewise equals Ne(t)= fa

0 flb0 fa

0

Fe(t,a,e,l) de dl da, where lb is the scaled lengthat birth (i.e. the transition from embryo tojuvenile).

The boundary condition at a=0 reads, whenRt(e, l) is the reproduction rate:

Fe(t,0,e,l)ddt

a= d(e− e0)d(l− l0) ga

0 g1

0 g1

lp

×R� (e,l)F(t,a,e,l)dl de da for all e,l (14)

where l0 denotes the scaled length at a=0, whichis taken to be infinitesimally small and lp thescaled length at puberty (i.e. the transition fromjuvenile to adult). The quantity e0 is the scaledreserve at a=0, which can be a function of e ofthe mothers. The function d(l− l0) is the Diracdelta function in l (dimension: l−1) and similarly

d(e− e0) is the Dirac delta function in e(dimension: e−1). The boundary condition atl= lb reads:

F(t,a,e,lb)ddt

a=Fe(t,a,e,lb)ddt

a for all a,e

(15)

Hence, the individuals can differ at a=0,because e0 can depend on e, and individuals canmake state transitions at different ages anddifferent scaled reserves.

The dynamics for food amounts to

ddt

MX+ = htXMX + J� X+ (16)

where MX+ denotes the food density in C-molesper black box volume, and J� X+ 0 fa

0 f1lb f1

0

F(t,a,e,l) J� X (e,l) de dl da, where J� X (e,l) denotesthe (negative) ingestion rate of an individual ofscaled energy reserves e and scaled length l, asdiscussed in the previous section. The faecal fluxJ� P+ is simply proportional to the ingestion flux,i.e. J� P+/J� X+ = J� P/J� X.

The molar fluxes of body mass and reserves(*=V, E), are given by

J� *+ = hte ga

0 glb

0 ga

0

Fe(t,a,e,l)M*(e,l) de dl da

+ ga

0 g1

lbg

1

0

(ht+ hta(a))F(t,a,e,l)

× M*(e,l) de dl da. (17)

The mineral fluxes J� M+ and the dissipating heatptT+ follow from (3) and (6)

J� MM+ =−n−1M nOJ� O+ (18)

ptT+ =−mTMJ� M+ − mT

OktO+ = (mTMn−1

M nO − mTOJ� )O+.

(19)

Due to the linear relationships between mass andenergy fluxes, the mass fluxes are simple metricson the densities Fe and F, which are solutions ofthe partial differential equations (12) and (13);the determination of the solution generallyrequires numerical integration.

. . . . .384

3.2.

For dividing organisms, we assume that theageing rate is independent of age and include thishazard rate into the harvesting rate ht; the statevariable age is not used, so the scaled length l andthe scaled reserve density e specify the state ofthe individual. The conversion process of sub-strate into biomass has two control parameters:htX and ht.

Given the initial condition F(0,e,l), thedynamics of density F(t,e,l) is then given by

11t

F(t,e,l)=−11l 0F(t,e,l)

ddt

l1−

11e 0F(t,e,l)

ddt

e1− htF(t,e,l) (20)

with boundary condition

F(t,e,lb)ddt

lb l= lb =2F(t,e,ld)ddt

lb l= ld for all e

(21)

where the scaled length at ‘‘birth’’ relates to thescaled length at division as lb = ld2−1/3, fororganisms that divide into two parts. Thesedynamics imply that both daughters areidentical.

Suppose now that the dynamics of the scaledreserves is independent of the scaled length, andthat the dynamics of the scaled length isproportional to the scaled length. The scaledreserve density then has the property that allindividuals eventually will have the same scaledreserve density, which may still vary with time.(The DEB model for 1D-isomorphs is anexample of such a model.) For simplicity’s sake,we will assume that this also applies at t=0,which removes the need for an individualstructure. The consequence is that a populationthat consists of one giant individual behaves thesame as a population of many small ones.

The partial differential equation (20) collapsesto two ordinary differential equations (Kooi &Kooijman, 1994a), one of these is at the

population level for the total structural bodymass

ddt

ln MV+ =ddt

ln l3 − ht (22)

where the scaled volume kinetics dl3/dt=3l2

dl/dt is given by the model for individuals. Theother differential equation is at the individuallevel for the scaled reserve density kinetics de/dt,which should also be specified by the model forindividuals, see e.g. Table 5. The scaled reservedensity kinetics specifies the (nutritional) state ofa random individual.

The expressions for the dissipating heat (19),and mineral fluxes (18) still apply here, whileJ� O+ = hpt+, with pt+ = fld

lb f10 pt(e,l3) F (t,e,l) de dl,

and pt(e,l3) denotes pt, evaluated at scaled energyreserve e and cubed scaled length l3. (For 1D-isomorphs it is more convenient to use l3 as anargument, rather than l.) This result is directbecause fld

lb f10 F (t,e,l) de dl=MV+/MVm, so that

pt+ = pt(e, gld

lbg

1

0

l3F(t,e,l) de dl)

= pt(e,MV+/MVm)= pt(e,1)MV+/MVm.

The latter equality only holds for models such asthe DEB model for 1S–1D isomorphs, where allpowers are proportional to structural body mass.The dynamics for food amounts to

ddt

MX+ = htXMX + J� X+ = htXMX −ptA(e,1)

mXA

MV+

MVm

,

(23)

where ptA (e,1) does not depend on the scaledreserves e, in the DEB model.

The environment for the population reduces tothe chemostat conditions for the special choice ofthe harvesting rate ht relative to the supply rate:htXMX = ht(MX −MX+).

4. Mass Fluxes in Food Chains of Dividers

Suppose that a food chain of dividers lives ina chemostat, where substrate is supplied at ratehtMX, and population i feeds only at populationi−1. We assume that all individuals have thesame composition of structural body mass andreserves, that all are harvested at the specific rate

/ 385

ht; and that the only cause of removal of anindividual from its population is by being eatenin the food chain or harvested. We have studiedthe rich dynamics of this food chain extensively(Kooi et al., 1997b; Boer et al., 1998), and nowstudy the conversion process from substrate intobiomass with control parameters MX and ht.

Most literature on food chain dynamics allowsgrowth of the zeroth trophic level, which rendersit very difficult, if not impossible, to completemass and energy balances (Kooi et al., 1997a,1998); we do not allow the substrate to grow.

The dynamics of substrate, structural bodymass and scaled reserves in a chemostat is given by

ddt

MX+ = htMX −ptA1(e,1)

mAX1

MV+1

MVm1

− htMX+

= ht(MX −MX+)+ J� X+1 (24a)

ddt

MV+ i =ptGi(ei,1)

mGVi

MV+ i

MVmi−

ptA(s+1)(e,1)mAX(i+1)

MV+(i+1)

MVm(i+1)

− htMV+ i = J� V+ i

+ J� X[mos(i+1) − htMV+ i (24b)

ddt

ei =ptAi(ei,1)− ptDi(ei,1)− ptGi(ei,1)

MEmimEi

−eiptGi(ei,1)MVmimGVi

=0J� E+ i

MEmi− ei

J� V+ i

MVmi1×

MVmi

MV+ifor i=1,2, . . ., N (24c)

where we appended index i to various symbols(e, MV+, MVm, MEm, ptA, ptC, ptG, mE, mGV, mAX) toindicate the species. We take MV+(N+1) =0 in(24b).

The organic, mineral and dissipating heatfluxes for each species are given by (B.8), (18)and (19), respectively. The various mineral fluxesin the food chain are additive, i.e. JM++ =a3

i=1

ktM+ i =−n−1M nO ktO++. The total organic fluxes

in the food chain are given by J� O++ =a3i=1 J� O+ i,

except for the total substrate flux, i.e. the firstelement of J� O++, which should be replaced byJ� X1, because only species 1 eats substrate.

4.1.

We use the experimental data by Dent et al.(1976) as an example of the application of our

theory for mass fluxes in food chains. They grewmyxamoeba (Dictyostelium discoideum), whichfed on bacteria (Escherichia coli), which fed onglucose (CH2O) in a chemostat at 25°C. Figure 3gives their observations on cell volumes duringthe transient states of the chemostat followinginoculation, together with our model fits basedon eqns (24a–24c) and the specifications ofTable 5. We used the equations for 1S–1D-isomorphs, rather than the fully structureddynamics of 1S–3D-isomorphs, since this simpli-fying approximation holds well for organismsthat divide on two parts, (cf. Kooijman & Kooi,1996; Kooijman et al., 1996). The resultingdynamics for a chemostat with throughput rateht and concentration of glucose in the feed MX,are

ddt

MX+ = (MX −MX+)ht−[J� Xm0,1]f0,1MV+1,

(25a)

ddt

MV+1 =MV+1ktE1e1 − ktM1g1

e1 + g1− htMV+1

− [J� Xm1,2]f1,2MV+2, (25b)

ddt

MV+2 =MV+2ktE2e2 − ktM2g2

e2 + g2− htMV+2,

(25c)

ddt

ei = ktEi(fi−1,i − ei) for i=1,2, (25d)

where ei(t), i=1,2 denote the scaled reservedensities, [J� Xmi−1, i] the maximum specific inges-tion rate and fi−1,i =MV+(i−1)/(MK(i−1) +MV+(i−1)) for i=1,2, is the scaled Holling typeII functional response. The parameters are listedin Table 6. Figure 3 gives the model fits to thedata, and the phase diagrams. These dynamicsimply very realistic predictions for the changes inmean cell size (Kooijman & Kooi, 1996).

To arrive at mass fluxes, we assumed that

(nC* nH* nO* nN*)T =(1 2.6 0.9 0.2)T (26)

for * standing for the structural biomass and thereserves of the bacteria and the myxamoeba, andthe faeces of the myxamoeba. The bacteria didnot produce any products or faeces; ammonia(H3N) was the N-waste/source. To couple energyand mass fluxes, we took m−1

PX /m−1AX =0 for the

(a)

0.4

0.2

0.0100500 150

f 0,1,

e 1

f 1,2,

e 2

P1

mm

3

Vc

ml

hr

[Em

] 1

,

.. JC

+1,

.–J

O+

1,–J

N+

1m

m3

ml

hr

[ME

] 1Vc

,

. JC

+2,

.

. .–J

O+

2,–J

N+

2m

m3

ml

hr

[ME

] 2Vc

,

MV

+1

mm

3

ml

[MV

] 1

MX

+

[MX

]

mg

ml

,,

,

MV

+2

mm

3

ml

[MV

] 2

MX

+1

[MX

] 1

,,

(b)

0.6

0.3

0.9

0.0

100500 150

P2

mm

3

Vc

ml

hr

[Em

] 2

(c)1.0

0.5

0.0100500 150

(d)

0.5

1.0

0.0100500 150

(e)1.0

0.5

0.0100500 150

(f)

0.5

0.0

1.0

100500 150(g)

0.2

0.1

0.0100500 150

t (hr)

(h)

0.2

0.1

0.0100500 150

t (hr)

,

.

. . . . .386

F. 3. Transient behaviour of a glucose–bacterium–myxamoeba food chain in a chemostat of volume Vc. (a, b) EnergyFluxes: assimilation (– – –), dissipation (....) and growth (——); (c, d) functional response (——) and scaled energy density(– – –); (e, f) biovolume densities of prey (....) and predator (——), and the data by Dent et al. (1976): glucose (w) andE. coli (W) in (e), E. coli (w) and D. discoideum (W) in (f); (g, h) CO2 (——) production and O2 (– – –) consumption andNH3 (....) consumption/production rates.

/ 387

T 6The parameter estimates for the DEB food chain model, as applied to the dateby Dent et al. (1976). The reactor volume was 200 ml, the temperature 25°C,the concentration in the feed 1 mg ml−1, and the throughput rate h=0.64 hr−1.For glucose we have 1 Cmol=30 mg and for the bacteria and myxamoebaboth 1 Cmol=31.8 mm3, where we assumed that the specific mass equals 1 mgmm−3. For simplicity’s sake, we introduced ktE 0 ktMg/ld (see Appendix A)

Parameter Value Units Interpretation

MX+(0)/[MX] 0.58 mg ml−1 Initial glucose concentrationMV+1(0)/[MV]1 0.46 mm3 ml−1 Initial E. coli densityMV+2(0)/[MV]2 0.070 mm3 ml−1 Initial D. discoideium densitye1(0) 1 (def) — Initial E. coli reserve densitye2(0) 1 (def) — Initial D. discoideum reserve densityMK1/[MX] 0.0004 mg ml−1 Saturation constant E. coliMK2/[MV]1 0.18 mm3 ml−1 Saturation constant D. discoideumg1 0.86 — Investment ratio E. colig2 4.43 — Investment ratio D. discoideumktM1 0.0083 hr−1 Maintenance rate coefficient E. coliktM2 0.16 hr−1 Maintenance rate coefficient D. discoideumktE1 0.67 hr−1 Specific energy conductance E. coliktE2 2.05 hr−1 Specific energy conductance D. discoideum

[J� Xm0,1] 0.65 mgmm3 hr Maximum ingestion rate E. coli

[J� Xm1,2] 0.26 hr−1 Maximium ingestion rate D. discoideum

bacteria and m−1PX /m−1

AX =0.2 for the myxamoeba.Finally, we need the conversion coefficients[MX]1/[ME]1 =1.5, [MX]2/[ME]2 =2.5 and [MV]i/[ME]i =1 for i=1, 2 (see Appendix A). Theresults are given in Fig. 3(g,h). The respirationquotient, i.e. the carbon dioxide flux over theoxygen flux, is almost constant at a value 1.05.

5. Discussion

We have shown the short list of generalassumptions (Table 2) about energy budgets hasfar-reaching consequences for the coupling ofenergy and mass fluxes in biota. The most strictform of the concept homeostasis assumes thatbiomass as a whole does not change in chemicalcomposition. We employ a relaxed version byapplication of homeostasis to structural biomassand reserves separately. Because the amount ofreserves can vary with respect to structural bodymass, the individual can change in relativefrequency of chemical elements in a particularway. More complex models might distinguishmore types of structural body mass and/orreserves to relax the concept of homeostasis evenmore. This is necessary to accommodate themetabolic versatility of plants, for instance. If the

number of types is equal to, or greater than, thenumber of elements, no restriction exists on therelative frequencies of elements in biomass. Anincrease of the number of types of compoundsrapidly becomes counterproductive, however,because the parameter values could be verydifficult to obtain in a reliable way. Some modelsfollow particular compounds. These models areless likely to be useful at the level of the wholeindividual, because of the staggering amount ofcompounds that are present in an organism.

We showed how theory for energy budgetsprovides a theoretical underpinning of themethod of indirect calorimetry. It seems to beessential to delineate one reserve and onestructural component. Both more simple andmore complex models (with more components)are inconsistent with the method of indirectcalorimetry. Although this type of statement ishard to prove formally, it would not surprise uswhen the set of general assumptions in Table 2turns out to be the only one that is consistentwith this method. We also have shown that theoxygen consumption that is associated with thefeeding (i.e. assimilation) process can beunderstood and quantified using mass balanceconsiderations, which is considered to be an

. . . . .388

open problem in the physiological literature(Withers, 1992). The present derivation simplifiesthe one presented in Kooijman (1995). Withsimple supplemental assumptions, energy bud-gets can be used to quantify evaporation of waterin terrestrial animals, which not only affectsdrinking behaviour, but also represents animportant aspect of the thermal balance,especially for endotherms. This opens the routeto advanced models for the control of bodytemperature.

For simplicity’s sake, we excluded complexsituations such as when temporary absence ofoxygen affects energetics, and the method ofindirect calorimetry must be changed. Simul-taneous limitations by energy and mass can beincluded, but require more elaborate models thathave additional state variables to accommodatesuch simultaneous limitations (Kooijman, 1998).

The step from individuals to populationsinvolves a number of assumptions aboutinteractions between individuals, and betweenindividuals and their local environment, that ishere kept as simple as possible. We stress that(Kooijman, 1993) the ‘‘introduction of astructure does not necessarily lead to realisticpopulation models due to the effects of manyenvironmental factors’’. One of the assumptionsimplicitly made here is spatial homogeneity, as isthe case in a well stirred chemostat. Further-more, the large number assumption makes itpossible to eliminate stochastic fluctuations atthe individual level and to model the populationdynamics deterministically using a set of PDEscoupled with ODEs. Sometimes a structuredpopulation dynamic model describing them canbe reduced to an equivalent delay differentialequation (DDE) or even an ODE model. Manyrealistic extensions can be incorporated in thepresented formulation, which frequently docomplicate the analysis of the dynamics con-siderably, but hardly affect the mass–energycoupling that has been discussed here. This isbecause the coupling is effectuated insideorganisms, and mass and energy fluxes at thepopulation level represent simple additions ofthose for individuals. This is why elaborateinteractions between individuals, for instance, donot affect the conclusion that mass fluxes areweighted sums of three energy fluxes. This does

not hold, however, for all possible forms ofcomplicating phenomena.

We conclude that open systems, such aspopulations of living organisms, do not hamperthe application of energy and mass balanceequations. Indeed, the use of such balanceequations can be of great help to modelpopulations realistically. Although we acknowl-edge the fact that the incorporation of dynamicenergy budgets does not automatically lead torealistic population models, we do believe thatuseful population models should be consistentwith the principles of these budgets.

The authors like to thank Cor Zonneveld, PaulHanegraaf and Hugo van den Berg for stimulatingdiscussions.

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APPENDIX A

Volume–Mole Conversions

There does not seem to exist one most usefulnotation for energetics. Volumes are handy inrelation to surface areas, which are needed forthe process of food/substrate uptake in the DEBmodel, while moles are handy for mass fluxes.Coefficients [M

*] convert volume to C-moles

(dimension: mole volume−1; the brackets [] referto volume−1, while the braces 45 refer to surfacearea−1). The energy–mass coupler m*1,*2 couplesenergy flux *1 to mass flux *2 (dimension energyper mole). The chemical potential m

*also has

dimension energy per mass, but cannot beinterpreted as ratio of fluxes. The mass–masscoupler y*1, *2, also known as yield or stoichio-metric coefficient, is a ratio of mass fluxes andtaken to be constant, just like other couplers. Wehave y*1,*2 = y−1

*2,*1, y*1,*2y*2,*3,* = y*1,*3 andh*1*2 = m−1

*2*1 is a mass–energy coupler. Volumesare indicated with V, masses in C-moles with M,structure-specific masses with m

*=M

*/MV.

Mass fluxes in C-moles per time are indicatedwith J�

*(the dot refers to time−1), structure-

specific mass fluxes with jt*= J�

*/MV. Energy

fluxes (i.e. powers) are indicated with pt. Index Xrefers to food, P to product (faeces). Thefollowing conversions between volume-basedand mole-based quantities hold, where the

. . . . .390

dimensions are indicated with l (length), m (mass),e (energy), t (time).

Structure volume

V= MV

[MV]l3

Maximum struc. v.

Vm =0 vtktMg1

3

= MVm

[MV]=0kyVX

ktM4J� XAm5[MV] 1 l3

Reserve energy

E= mEME e

Maximum reserve

Em =[Em]Vm = mEMEm e

Maximum reserve density

[Em]= mE[Me]el3

Energy requirement per structural volume

[EG]= mGV[MV]el3

Scaled length

l=0MV

MVm11/3

=0VVm1

1/3

—

Reserve density

e=mE[MV][Me]

=mEkgyVE —

Maximum spec. assimilation

4ptAm5= mAX4J� XAm5 el2t

Specific maintenance

[ptM]= ktMmGV[MV]el3t

Energy conductance

vt= 4ptAm5[Em] = yEX

4J� XAm5[Me]

lt

Investment ratio

g= [EG]k[Em]=

yEV

k[MV][Me]

—

Maintenance rate

ktM = [ptM][EG]

= jEMyVE1l

Relative reserves

mE = ME

MV= e

mE

[Em][MV]

—

Structural mass

MV =V[MV] m

Maximum structural mass

MVm =Vm[MV]=0kyVX

ktM4J� XAm5[MV] 1

3

[MV] m

Reserve mass

ME = EmE

=MVeyEV

kg m

Maximum reserve

MEm =Em

mE=MVm

[Me][MV]

m

AssimilationFood coupler

mAX = 4ptAm54J� Xm5= mE

yXE

em

AssimilationFood coupler

mAP = 4ptAm54J� Pm5 = mE

yPE

em

Reserve chemical potention

mE = [Em][Me]

em

GrowthStructure coupler

mGV = [EG][MV]

= mE

yVE

em

ProductReserve coupler

yPE = mE

mAP= jPA

jEA

mm

ProductFood coupler

yPX = mAX

mAP= jPA

jEG

mm

StructureReserve coupler

yVE = mE

mVG= jVE

jEG

mm

FoodReserve coupler

yXE = mE

mAX= jXA

jEA

mm

Specific assimilation flux

jEA = jXAyEXmmt

Specific maintenance flux

jEM = ktMyEVmmt

Assimilation flux

J� EA = jEAMV = J� XAyEXmt

Food flux

J� XA = jXAMVmt

/ 391

APPENDIX B

Steady-state Fluxes

This appendix describes the steady-statefluxes. The gist of the argument is that, ifenvironmental conditions change slowly withrespect to the structure of the population interms of the frequency distribution of theindividuals over the state values, the populationcan be treated as a super-individual withrelatively simple rules for mass and energy fluxes.Such a simplification is useful if the populationmodel is conceived as a module in an ecosystemmodel.

At steady state the easiest approach is to relatethe states of the individuals to age. We no longeruse the density F(t, a, e, l), but, instead, therelative density f(t, a)=F(t, a)/N(t). Thisrelative density no longer depends on time atsteady state, so we omit the reference to time. Wewill write J�

*(a) for the flux of compound * with

respect to an individual of age a, where ab is theage at birth and ap the age at puberty. These agesmight be parameters, but the DEB model obtainsthem from MV(ab)=MVb and MV(ap)=MVp.

The characteristic equation applies at steadystate:

ME0 = exp4−hteab5

×ga

ap

exp4−hta−ga

0

hta(a1)da15J� R(a)da (B.1)

We use the characteristic equation to solve forthe food density MX+, so the scaled functionalresponse is f. Given this food density, thetrajectories of the state variables are fixed.

The age distributions of embryos and juvenilesplus adults are given by

fe(a)=hteexp4−htea5

1−exp4−hteab5for a$[0,ab] (B.2)

f(a)=(ht+ hta(a))exp4−hta− fa

0 hta(a1)da15faab

exp4−ht− fa0 hta(a1)da15da

for a$[ab,a] (B.3)

We introduce the expectation operators Oe and O,i.e. OeZ0 fab

0 Z(a)fe(a) da and OZ0 faab

Z(a)f(a)da, for any function Z(a) of age.

The harvesting rates of organic compoundsequal their mass fluxes, i.e.

J� O+ 0GG

G

F

f

J� X+

J� V+

J� E+

J� P+

GG

G

J

j= hNOpt

=GG

G

F

f

−htXMX

NeOehteMV +NO(ht+ hta)MV

NeOehteME +NO(ht+ hta)ME

htXMXmAX/mAP

GG

G

J

j(B.4)

The number of juveniles plus adults in thepopulation, N, and of embryos, Ne, are given by

N=J� X+

OJ� Xand Ne =(1−exp4−hteab5)

NOJ� RhteME0

The population growth rate must be zero atsteady state. We use this to solve the value of thescaled functional response, i.e. f=(ktM + ht)/(ktM/ld − ht/g) in the case of the DEB model. Thismodel has the nice property that e= f at steadystate; it then follows that MX+ =MKf/(1− f),where MK is the saturation constant of theHolling type II functional response in C-mol perreactor volume.

The stable age distribution amounts to

f(a)=2ht exp4−hta5 for a$[0,ht−1 ln 2]

(B.5)

The number of individuals in the population,the total structural body mass, and the organicfluxes are given by

N=J� X+

OJ� X=

MV+

l3dMVm ln 2(B.6)

MV+ 0NOMV =htXMX[MV]

f[J� Xm](B.7)

J� O+ = hpt+ = hpt(f,1)MV+

MVm(B.8)

The mean mass per individual is thus OMV =MV+/N.

. . . . .392

The relative contributions of the three basicpowers depend on the substrate density, andtherefore on throughput rate. Hanegraaf (1997)gives a detailed analysis of mass and energy

transformations in chemostats at steady states,including mixed substrates, fermentations andproduct formation.