The bijection from data to parameter space with the standard DEBmodel quantifies the supply–demand spectrum

Konstadia Lika a, Starrlight Augustine b, Laure Pecquerie c, Sebastiaan A.L.M. Kooijman d,n

a Department of Biology, University of Crete, Voutes University Campus, 70013 Heraklion, Greeceb Center for Ocean Life, National Institute of Aquatic Resources, Technical University of Denmark, Jægersborg Allé 1, 2920 Charlottenlund, Denmarkc Lab. des Sciences de l'Envir. Marin UMR LEMAR - 6539 CNRS/UBO/IRD/Ifremer, Technopôle de la Pointe du Diable B.P.70, 29280 Plouzané, Franced Department of Theoretical Biology, VU University Amsterdam, de Boelelaan 1087, 1081 HV Amsterdam, The Netherlands

H I G H L I G H T S

� We present the bijection between data and parameter space for the standard DEB model.� The boundaries of these spaces involve a new metric: the supply stress.� This metric quantifies the supply–demand spectrum for animal species.� Parameters from 300 species show that invertebrates and ray-finned fish are supply species.� We explain why birds and mammals up-regulate metabolism during reproduction.

a r t i c l e i n f o

Article history:Received 23 October 2013Received in revised form12 March 2014Accepted 13 March 2014Available online 21 March 2014

Keywords:Dynamic energy budget theoryAdd_my_pet collectionMetabolic up-regulationElasticity coefficientsEvolutionary constraints

a b s t r a c t

The standard Dynamic Energy Budget (DEB) model assumes that food is converted to reserve and a fractionκ of mobilised reserve of an individual is allocated to somatic maintenance plus growth, while the rest isallocated to maturity maintenance plus maturation (in embryos and juveniles) or reproduction (in adults).The add_my_pet collection of over 300 animal species from most larger phyla, and all chordate classes,shows that this model fits energy data very well. Nine parameters determine nine data points at abundantfood: dry/wet weight ratio, age at birth, puberty, death, weight at birth, metamorphosis, puberty, ultimateweight and ultimate reproduction rate. We demonstrate that, given a few other parameters, these nine datapoints also determine the nine parameters uniquely that are independent of food availability: maturity atbirth, metamorphosis and puberty, specific assimilation, somatic maintenance and costs for structure,allocation fraction of mobilised reserve to soma, energy conductance, and ageing acceleration. We providean efficient algorithm for mapping between data and parameter space in both directions and foundexpressions for the boundaries of the parameter and data spaces. One of them quantifies the position ofspecies in the supply–demand spectrum, which reflects the internalisation of energetic control. We linkeco-physiological properties of species to their position in this spectrum and discuss it in the context ofhomeostasis. Invertebrates and ray-finned fish turn out to be close to the supply end of the spectrum, whileother vertebrates, including cartilaginous fish, have stronger demand tendencies. We explainwhy birds andmammals up-regulate metabolism during reproduction. We study some properties of the bijection usingelasticity coefficients. The properties have applications in parameter estimation and in the analysis ofevolutionary constraints on parameter values; the relationship between DEB parameters and data hassimilarities to that between genotype and phenotype.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Energetics, i.e. resource acquisition and use of individualorganisms, is basic to behaviour, population and ecosystemdynamics and evolution (Sousa et al., 2010). The actual perfor-mance of individuals very much depends on environmentalfactors, the most important being temperature and resource

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Journal of Theoretical Biology

http://dx.doi.org/10.1016/j.jtbi.2014.03.0250022-5193/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ31 20 5987130.E-mail addresses: lika@biology.uoc.gr (K. Lika),

staug@aqua.dtu.dk (S. Augustine), laure.pecquerie@ird.fr (L. Pecquerie),bas.kooijman@vu.nl (S.A.L.M. Kooijman).

Journal of Theoretical Biology 354 (2014) 35–47

availability. That is why the potential energetic performance ofspecies can best be compared on the basis of parameter values of amodel for this energetics (Kooijman et al., 2008). Dynamic EnergyBudget (DEB) models (Kooijman, 2010) are such models for whichits applicability to data has been demonstrated in many studies(Pecquerie et al., 2010, 2012; Kooijman, 2013; Kooijman and Lika,2014a). The model is based on simple thermodynamic principles(Sousa et al., 2006) and applies to all species (micro-organisms,plants, and animals). Animal energetics is well-captured by thestandard DEB model (see Appendix A for a summary), which hasjust a single reserve and structure. This is demonstrated by theadd_my_pet collection of data and parameters of over 300 animalspecies from most larger phyla, and all chordate classes: thismodel fits energy data very well. The mean FIT mark is 8.4 on therange from minus infinity to 10 (Lika et al., 2011); it stands for 10times one minus the mean relative error of the various data sets.

Data availability is always a problem for application of elabo-rate models, certainly for the purpose of comparing a largenumber of species on the basis of their parameter values.Very little is known about most species. The types of data includeuni-variate data, such as body weight as function of time, length asfunction of body weight, respiration as function of length, clutchsize as function of body weight as well as zero-variate data (datapoints if you like), such as age at birth, weight at puberty, andlength at death. The coupling of traits (feeding, growth, reproduc-tion, and ageing) is key to energetics, implying the necessity toestimate all parameters from all data simultaneously in a single-step procedure (Van der Meer, 2006). While no parameter froma multi-parameter model can be estimated from a single zero-variate data point, parameters might be estimated from a set ofdifferent types of zero-variate data. The add_my_pet collection hasover 100 different types of data in total, but for any individualspecies a very limited selection is available. The completeness ofdata that is available to estimate DEB parameters is scored from 0(maximum body weight only) to 10 (all aspects of energetics arefixed by data) (Lika et al., 2011); the mean completeness level is2.5, the maximum one is 6, which illustrates the problem of lack ofdata even for the best studied species. Kooijman et al. (2008) studythe problem of which measurements, or observations if you like,determine which parameters of the standard DEB model. It turnedout that particular compound parameters, i.e. simple functions ofparameters that have simple dimensions, are much easier toestimate from data than the parameters themselves. Since com-parisons between species are most informative if done on thebasis of all primary parameters, rather than some compound ones,the covariation method (Lika et al., 2011, 2011) has been workedout to estimate all primary parameters from data. This methoduses a selection of parameters for a generalised animal as datapoints. The generalised animal is a hypothetical animal with amaximum structural length of 1 cm that has typical (i.e. frequentlyencountered) body-size corrected parameter values (Kooijman,2010, Table 8.1). The use of (some) parameters of the generalisedanimal as data points still allows for a difference betweenestimated parameter values and those of the generalised animal.This method makes sure that data contain enough information toestimate parameters and reduces the risk of arriving at a good fitthat makes no physical sense. How well parameters are deter-mined by data generally depends on the combination of values ofdata and parameters and model structure, a complex problemindeed. Where model complexity and structure can easily beadapted to the needs of available data in descriptive models, theyare set by internal logic in mechanistic models like the DEB modeland pose conditions for minimal data that is required. Standardstatistical procedures (variance–covariance matrices) can be usedto quantify uncertainty in parameter estimates in regressionsituations (added noise in data). A deep problem is that this

module for stochasticity is very unrealistic from a biologicalperspective. Most noise in energy data originates from variabilityin the environment and in behaviour. DEB parameters are sup-posed to be individual-specific, so ideally, all data should be fromthe same individual. The hope is that if data points are means ofseveral individuals, the corresponding parameter set is represen-tative for the species without claiming that they are actually meanvalues. It is very likely that, in situations where various data setsrelate to various populations of individuals, differences in para-meter values can be expected. This source of stochasticity easilyleads to complex dependencies in data (Bedaux and Kooijman,1994). We come back to this point in the section on the role ofscatter.

This paper aims to study the problem of how well data fixparameters in a new way, working with only a single foodavailability level (abundant food), a single constant temperature,and data only of the most reduced type: zero-variate data. This infact corresponds to what is available for most animal species. Eventhough performance at several food levels is very informativeabout particular parameters, such data are very hard to obtain andare simply not available in most cases. The add_my_pet collectionis presently weak on data on feeding, apart from starving, whichdoubtlessly reflects the experimental problem of measuring it andthe fundamental problem of age and size dependent selectionstrategies and variations in nutritional values of food types inmulti-food type situations. This is why we presently avoid dealingwith this complex problem. In view of the many species that sportmetabolic acceleration in the collection (Kooijman, 2014), we hereinclude the type M acceleration module (see Appendix A). We alsoinclude the ageing module to complete the full life cycle perspec-tive of DEB theory from embryo, juvenile and adult to death byageing. Although parameter estimation aims to step from data toparameter values conceptually, computationally it makes theinverse step from parameter values to model predictions, wherea formalised estimation criterion is used to minimise the differ-ence between predicted and actual data. We here not only mapfrom parameters to data, but also from data to parameter values.Generally this step is not possible, even not unique, but in ourspecial situation of zero-variate data, it is unique and we can studythe bijection from data to parameter space and vice versa. Thisadds to our understanding of the role of scatter, where variationsin environmental conditions, behaviour and measurement errorinduce scatter in data, and adaptation and selection induce scatterin parameter values.

The next section discusses the bijection, its assumptions andthe boundaries of the data and parameter spaces. The focus is onconceptual aspects. The algorithms for the bijection in bothdirections are presented in Appendices B and C and details ofthe boundaries of the parameter and data spaces in Appendices Dand E. These boundaries reveal an important role of a new statistic,called supply stress, which turns out to quantify the position ofspecies in the supply–demand spectrum. This spectrum is workedout in Section 3, followed by a discussion of evolutionary con-straints on parameter values in Section 4. The role of scatter indata and parameter values is discussed in Section 5. Finally ageneral discussion is meant to provide context for our findings,which are summarised in the conclusions section. All symbols arelisted in Tables 1 and A2.

2. Bijection in nine dimensions

DEB theory is imbedded in general systems theory (Bertalanffy,1968), where the state of the system (i.e. individual) is quantifiedby a set of state variables and the changes in state by a set of ODE's.The standard DEB model has reserve, structure and maturity as

K. Lika et al. / Journal of Theoretical Biology 354 (2014) 35–4736

basic state variables, but reproduction in adults needs a reproduc-tion buffer as additional state variable and specific species-specificbuffer handling rules, which also involves extra parameters.As mentioned before, Appendix A presents a quantitative sum-mary. The ageing module involves two extra state variables: theamounts of damage inducing and damage compounds. Bodytemperature can be another state variable for endotherms andterrestrial ectotherms (Kearney et al., 2010); body temperatureapproximates local environmental temperature for most aquaticectotherms. We here only consider situations of constant environ-mental conditions, see assumptions A1–A3 of Section 2.1, with theimplication that all state variables can be written as functions ofage. Since the model assumes weak homeostasis, i.e. the ratio ofthe amounts of reserve and structure remains constant duringgrowth under constant environmental conditions, the ratio doesnot change after birth, and body weight becomes proportional tothe amount of structure.

The parameters of the DEB model can be classified into coreparameters, which affect changes in the state variables andauxiliary parameters, which concern the link between state vari-ables (reserve, structure, maturity, etc.) and measured quantities(length, body weight, respiration, etc.). The standard DEB model(with the acceleration and the ageing modules) has 15 coreparameters: EHb , EHj , EHp , κX , f _Fmg, f _pAmg, _v, κ, f _pT g, ½ _pM �, ½EG�, _kJ , κR,€ha, sG. Tables 1 and A2 (of Appendix A) list all symbols and theirunits. Core parameters differ from compound ones, i.e. simplefunctions of core parameters, in that they relate to a singleunderlying process. So, core parameters have an intimate connec-tion with mechanisms.

The number of required auxiliary (and core) parameters in anyparticular situation depends on the measurements that the modelneeds to predict, where the quest is for the minimum number. Wedo not evaluate respiration here, which DEB theory gets fromconservation of chemical elements. This means that we do notdeal with elemental frequencies of food, faeces, structure andreserve (but see the assumption on molecular weights in assump-tion A5 in Section 2.1), and also not with the yield of faeces onfood. We also do not evaluate dissipating heat, which DEB theorygets from the conservation of energy. This means that we do notdeal with the chemical potentials of food and faeces. DEB theoryhas a module for isotope dynamics, which involves a number ofreshuffling and fractionation parameters (Pecquerie et al., 2010).We do not consider those parameters here, but point to theproblem that the complexity of the DEB model in terms ofnumbers of parameters depends on what the model needs topredict. If the model only needs to predict length as a function oftime since birth at abundant food, the standard DEB model hasonly 3 (compound) parameters, since it reduces to the vonBertalanffy growth model (Pütter, 1920). We mention this to revealthe connection between number of required parameters and typesof data that need to be predicted.

Table 1 lists the nine data points and parameters of thestandard DEB model among which bijection exists under theassumptions of Section 2.1.

2.1. Assumptions

To define the mapping between data and parameter space, weneed to make a number of assumptions.

A1 Temperature is constant and the reference temperature isTref ¼ 293 K. If the actual temperature differs, ab, ap, am and_Rm must first be temperature corrected. A typical Arrheniustemperature is TA ¼ 8 kK, which can be used for this purposein the absence of better information. The correction canbe done by dividing the ages and multiplying the rate bytemperature correction factor expðTA=T�TA=Tref Þ, see(Kooijman, 2010, Eq. (1.2)).

A2 Food is abundantly available. Feeding is not explicitlyincluded here, meaning that we do not include digestionefficiency κX and maximum specific searching rate f _FMg,which are two core parameters of the standard model.

A3 Surface-linked somatic maintenance f _pT g ¼ 0. This primarilyconcerns investment into heating (for endotherms) andosmotic work (for freshwater organisms), which depends onenvironmental conditions. We thus assume that these condi-tions are such that our assumption holds.

A4 The chemical potentials of structure and reserve are μV ¼ 0:5and μE ¼ 0:55 MJ C�mol�1, respectively. The bulk composi-tion of dry biomass is a mixture of carbohydrates, proteinsand fats (Kooijman, 2010, Table 4.2), which is species-specific,but the overall values are assumed to be insensitive to thesevariations. Chemical potentials of reserve and structure canbe estimated from data on energy content of biomass at two(or more) food levels.

A5 Ratios of chemical elements in dry structure as well as dryreserve are C:H:O:N¼1:1.8:0.5:0.15. This fixes the molecularweight of reserve and structure to wE ¼wV ¼ 23:9 g C�mol�1.Notice that similarity of elemental frequencies does not implysimilarity in chemical composition. Avoidance of this assump-tion, including the similarity between structure and reserve,requires measured elemental frequencies of biomass at two(or more) food levels.

A6 Only if the water content of reserve and structure are equal isthe dry/wet weight ratio independent of nutritional condi-tions (as is the standard assumption in the ecological litera-ture). We also make this assumptionwith the implication thatthe dry/wet weight ratio δW has a simple relationship withspecific density of biomass dW and the specific density ofstructure dV, where the specific density of wet structure isdwV ¼ 1 g cm�3 and the specific density of reserve equals thatof structure. The water content of structure and reserve can

Table 1The combination of nine data (left) and parameters (right) that map to each other one-to-one on the basis of the standard DEB model with acceleration and ageing.

Description Symbol Unit Unit Symbol Description

Age at birth ab d J EHb Maturity at birth

Age at puberty ap d J EHj Maturity at metam.

Age at death am d J EHp Maturity at puberty

Dry/wet weight ratio δW – J d�1 cm�2 f _pAmg Specific assimilationWet weight at birth Wb g cm d�1 _v Energy conductanceWet weight at metam. Wj g - κ Allocation fraction to somaWet weight at puberty Wp g J d�1 cm�3 ½ _pM � Spec. somatic maintenanceUltimate wet weight W1 g J cm�3 ½EG� Specific cost for structureMax. reproduction rate _Rm #d�1 d�2 €ha

Ageing acceleration

K. Lika et al. / Journal of Theoretical Biology 354 (2014) 35–47 37

be estimated from data on dry and wet weight trajectoriesduring starvation.

A7 We refrain from the detailed specification of the handlingrules for the reproduction buffer and only consider maximumreproduction rate as a mean over several reproduction cyclesfor a fully grown adult female. To avoid this assumption, weneed reproduction data as a function of time. Buffer handlingrules tend to be species-specific. Maximum weight as datapoint, see Table 1, is assumed to exclude the reproductionbuffer.

A8 Reproduction efficiency κR ¼ 0:95, which stands for the frac-tion of reserve that is allocated to reproduction that ends upin offspring. In the case of reproduction by eggs this repre-sents a conversion from reserve of the mother to that of eggs,so no chemical transformation is involved. This parametercan only be estimated if the full energy balance is availablefrom data.

A9 Growth efficiency κG ¼ 0:8, which stands for the fraction ofreserve that is allocated to growth (of structure) that ends upin structure. This involves a chemical transformation fromreserve to structure. Growth efficiency can be estimated frome.g. growth data at two (or more) food levels. In cases wherethis parameter could be estimated, 0.8 turns out to beconsistent with data.

A10 Maturity maintenance rate coefficient _kJ ¼ 0:002 d�1, whichstands for the maturity specific maintenance costs, if maturityis expressed in cumulative energy investment in maturation.Maturity itself does not have mass or energy. The parametercan only be estimated from data if reproduction is measuredat several food levels. In cases where this parameter could beestimated, 0.002 d�1 turns out to be consistent with data.

A11 Gompertz stress coefficient sG ¼ 10�4, which quantifies howfast ageing accelerates during ontogeny. The parameter canbe estimated from data on relative survival frequency as afunction of age. For ectotherms acceleration is typically verylow, for endotherms it can be in the order of 0.1 (steeperdecline of survival probability as a function of age). The valueonly affects survival by ageing and has no effect on the energybudget, but the energy budget affects ageing.

Assumptions A1–A3 relate to restrictions on environmental con-ditions under which data has been collected. Assumptions A4–A6relate to body composition and assumptions A8–A11 species proper-ties. All these assumptions can be avoided, but this requires morecomplex data, frequently at several food levels, and more advancedparameter identification methods (Lika et al., 2011, 2011). This list ofassumptions leaves nine degrees of freedom for the dynamic energybudgets as specified by the standard DEB model with accelerationand ageing.

2.2. The algorithm of the bijection and its boundaries

The algorithm for the map from the 9-dimensional parameterspace to the data space is presented in Appendix B and that fromdata space to parameter space in Appendix C. Both algorithms arecoded in software package DEBtool (in Matlab) and this softwareshows that the bijection actually exists and its computation is fast.

The algorithm from data to parameters involves the solution of twoimplicit equations, each in one variable, step D8 and D15, where goodinitial values are available. This substantial reduction of computationalcomplexity summarises the significance of this algorithm. Given thatthe mapping between data and parameter space is a bijection,the map from parameters to data can be used to obtain theinverse mapping, by minimising the difference between predictedand observed data (down to zero in this case), which is typicalin parameter estimation. This involves, however, a search in 9

dimensions, and requires high quality initial estimates and substantialcomputation. The present reduction to two times a search in onedimension and good initial estimates reduces the problem substan-tially. On top of that, the algorithm exposes the role of particularcompound parameters that frequently pop-up in DEB theory, e.g. g, k,_kM , tEm, Lm, vHb , and shows why and how they can be obtained fromdata, when primary parameters cannot. Notice for instance that,thanks to DEB's weak homeostasis assumption, scaled structurallengths can be obtained from weights in a simple way (see step D2of Appendix C), but structural lengths themselves take quite a fewmore steps to obtain (step D12).

Not all combinations of parameter values and data values areallowed by the standard DEB model. The restrictions are specifiedin Appendices D and E, respectively. The most remarkable bound-ary is that for κ (BP6 of Appendix D), where κmust be between thepositive roots of κ2ð1�κÞ ¼ ss and the statistic ss � _kJE

pH ½ _pM �2=

f 3s3Mf _pAmg3 ¼ κ2ð1�κÞðkvpH=f 3s3MÞ is called the supply stress. Thename is inspired by the observation that this statistic quantifiesthe distance to the supply-end of the supply–demand spectrum,which will be discussed below. This boundary follows from therequirement that the maximum reproduction must be positive.We could also derive that sso22=33 (BP5 of Appendix D), aconstraint that follows naturally from the structure of the standardDEB model.

Fig. 1 shows that many vertebrates have a (relatively) largevalue for supply stress ss. The metric ss is simpler than it seems,since _kJE

pH ¼ _pJ represents the maturity maintenance rate of an

adult and fsMf _pAmg ¼ f _pAg the specific assimilation of an adult(which is after metamorphosis). Ultimate absolute assimilationis _pA ¼ f _pAgL21 and ultimate absolute somatic maintenance is_pM ¼ ½ _pM�L31. Supply stress can thus also be written as ss ¼_pJ _p

2M= _p

3A evaluated under optimal environmental conditions

(abundant food, thermo-neutral zone). Under constant food con-ditions the energy budget for juveniles and adults amounts to_pA4 _pMþ _pGþ _pJþ _pR, where G stands for growth and R formaturation or reproduction. The inequality sign is because growthis defined as the increase in structure, while reserve also increases,which has to be fuelled from assimilation. A fully grown adult doesnot grow any longer and reserve also does not increase: _pG ¼ 0 and_pA ¼ _pMþ _pJþ _pR. That the supply stress ss must be smaller than4/27 is clear from the observation that for a fully grown adult_pM ¼ κ _pA and _pJr ð1�κÞ _pA as a consequence of the κ-rule. Sub-stitution gives ssr ð1�κÞκ2 for 0oκo1. Supply stress ss onlydepends on κ via maximum structural length, which can be out-divided. Since somatic maintenance _pM is proportional to volumeand assimilation _pA to surface area, ss ¼ _pJ ½ _pM �2=f 3s3Mf _pAmg3 showsthat ss does not depend on the value of κ, but has a direct link withthe possible range of κ, which narrows down to zero at themaximum value of ss ¼ 4=27, where κ ¼ 2=3. If _kJ-0, pubertycan be reached for all food levels that support existence. Theexistence of a lower food level for reproduction demonstrates that_kJ40 in practice (Lika and Kooijman, 2011, Table 2).

The significance of simple explicit borders of the parameterspace is that it allows for the use of genetic algorithms for findingthe best fit. All parameter combinations that are outside theparameter space can be intercepted by a filter and a penaltyreturned to the estimation procedure. The filter should evaluatesM to compute ss, which means that the algorithm for the mapfrom parameters to data needs to be followed till step P9 and thatscaled length at birth needs to be evaluated in step P5. Theboundary at which maturation ceases at birth requires a numericalprocedure that is discussed in Kooijman (2009b) and implementedin software package DEBtool.

Although the notion of the supply–demand spectrum is notnew (Kooijman, 1993, p. 17), the next section further develops theconcept and discusses quantitative aspects.

K. Lika et al. / Journal of Theoretical Biology 354 (2014) 35–4738

3. Supply–demand spectrum

The concept of (energy) budget itself already involves thenotion for matching demand to supply of metabolites (Banavaret al., 2002), which applies to all species. Kooijman (2010) suggeststhat this matching is the main task of hormonal regulation at sub-organismal level. Beyond that, at the organismal-level, species canbe ranked according to a supply–demand spectrum that roughlyreflects where the controls of energetics are: from environmentalto internal. The use of resources is ‘pre-programmed’ in demand-species and the individual tries hard to match this demand byeating enough. Supply-species hardly have such a program, ormodify it in a flexible way, according to the possibilities offered bythe environment. Table 2 compiles stylised eco-physiological

properties of species that link to their position in the spectrum.No species are at the extremes of the spectrum.

Demand species have less metabolic flexibility to handlestarvation in terms of shrinking and rejuvenation, but theycompensate that by a higher talent for finding the last food item(¼ low half saturation constant), for which they need complexbehaviour and good memory and sensors. The half saturationconstant is in fact the ratio of the (specific) ingestion and searchingrates, meaning that demand species have a large specific foodsearching rates. High peak metabolic activity, relative to thestandard one, is part of the skills they need to capture (fast) preyand is connected to searching rate. Capillaries (in a closedcirculatory system) make that an increase in heart beat is felt inall corners of the body (where muscles contract), which explainsthe link with a high peak metabolic rate. Only annelids, cephalo-pods and vertebrates have capillaries, all other animal specieswork with open circulatory systems, where an increase in heartbeat has less consequences for tissues that are further away fromthe heart. Annelids probably have them to build up pressure whenpushing their body through soil; without capillaries muscle con-traction would transport fluid inside the body too easily and doesnot have the effect that the body is pushed forward. We see this asan adaptation to life in soils that has little to do with the evolutionfrom supply to demand systems. Cephalopods and vertebrateshave telolecithal eggs, which possibly relates to their closedcirculatory system with which they mobilise yolk. The embryobeing on the outside of yolk facilitates access to environmentaldioxygen and allows for high metabolic rates, compared to iso- orcentro-lecithal eggs. Mammals have alecithal eggs, which probablyrelates to their foetal development.

Demand systems have (food) acquisition homeostasis, with ther-mal homeostasis as pinnacle. Many species (insects, reptiles) devel-oped in the direction of thermal homeostasis via behaviour (sitting insun or shade), some species (insects, tunas, sharks) sport metabolicheating (endothermy), but mammals and birds have fully masteredthis art (after birth). Endothermy induces timing problems of ageingrelative to maturation; food availability has seasonal controls and lifecycles must fit seasonal cycles. If an endothermic mouse and anectothermic lizard of the same body size and energy budget para-meters would also have the same ageing parameters, the (warmer)mouse would live too short. Endotherms accelerate ageing (Gompertzstress coefficient sG40), starting with an extra-low ageing rate. Thisgives age-dependent survival probabilities that are high for a longtime, and then suddenly drop. Survival curves of ectotherms dropmuch more gradually, as far as ageing is concerned (Kooijman, 2010,Chapter 6). Notice that many factors affect survival and ageing is rarelythe most important one in field conditions. Birds and mammals alsosport upregulation of metabolism before egg laying or during preg-nancy and lactation: maximum feeding rate is temporarily increased(Kooijman, 2010). The rationale of this pattern is discussed below. Zookeepers (and farmers) know that most birds can be stimulated to laymore eggs by removing freshly-laid eggs, which shows that egg-production is not energy limited. Offspring production in birds istypically limited by parental care just before fledging, when foodrequirement is at maximum (Kooijman and Lika, 2014a).

Because supply stress ss relates assimilation (supply) to main-tenance (demand) we suggest that it quantifies the distance to thesupply-end in the classification of species in the supply–demandspectrum and consequently sd ¼ 4=27�ss is the distance to thedemand-end of the spectrum. The width of the supply–demandspectrum is thus ssþsd ¼ 4=27. Fig. 1 also shows that κ isfrequently very close to the upper boundary for zero reproduction,but not to the lower boundary. This is further discussed inKooijman and Lika (2014a). Although maximum reproduction isaround κ ¼ 0:45 for supply-species (Kooijman and Lika, 2014a),κ approaches 2/3 for demand-species.

supply demand

Fig. 1. Allocation fraction κ as a function of supply stress ss for over 300 species ofthe add_my_pet collection. It can only take values between the two black curvesand the maximum supply stress is 4/27. Colours indicate high-level taxa: blue forradiata, brown for bilateria, turquoise for platyzoa, dark turquoise for lophotrocho-zoa, green for ecdysozoa, black for invertebrate deuterostomes, magenta forectothermic vertebrata and red for endothermic vertebrata. Open symbols indicateacceleration. (For interpretation of the references to colour in this figure caption,the reader is referred to the web version of this paper.)

Table 2Stylised eco-physiological properties that relate to the position of a species in thesupply–demand spectrum.

Supply Demand

Eat what is available Eat what is neededLarge half saturation coefficient Small half saturation coefficientRather passive, simple behaviour Rather active, complex behaviourSensors less developed Sensors well developedCan handle large range of intake Can handle small range of intakeLow peak metabolic rate High peak metabolic rateOpen circulatory system Closed circulatory systemIso- & centro-lecithal eggs A- & telo-lecithal eggsTypically ectothermic Typically endothermicReserve density varies strongly Reserve density varies littleLarge range of ultimate sizes Small range of ultimate sizesSurvives some shrinking well Survives shrinking badlySurvives rejuvenation well Survives rejuvenation poorlyEnergetic birth control Behavioural birth controlNo upregulation for reproduction Upregulation for reproductionNo acceleration of ageing Acceleration of ageingEvolutionary original Evolved from supply systemsHas demand components Has supply components(maintenance) (some food must be available)

K. Lika et al. / Journal of Theoretical Biology 354 (2014) 35–47 39

This interpretation of ss is confirmed in Fig. 2, which presents itas a function of the minimum functional response, i.e. the foodingestion rate as a fraction of the maximum one of an individual ofthat size, that is required to reach puberty. It amounts tofmin ¼ ð _kJE

pH ½ _pM �2=κ2ð1�κÞs3Mf _pAmg3Þ1=3 ¼ ðf 3ss=κ2ð1�κÞÞ1=3 and

has a minimum for κ ¼ 2=3 for non-accelerating species. Thefigure also shows that smax

s ðfminÞ ¼ f 3min4=27, which directly followsfrom the previous expression for f¼1 and κ2ð1�κÞosso4=27.The figure clearly shows that supply species can reach puberty fora much broader range of food intake levels, compared to demandspecies. Some mammals and birds have a minimum scaled func-tional response for reaching puberty close one. This explains whythese taxa have upregulation of metabolism linked to reproduc-tion. This upregulation is an extra module in DEB models, that isnot part of the standard DEB model.

Fig. 3 shows that endotherms with low values for fmin and sshave a high value for κ. The coupling between κ and ss followsfrom the increase of the possible range of κ with decreasing ss. Yetit is remarkable that none of the endotherms in the add_my_petcollection have a low supply stress ss in combination with a low κ.These couplings require further investigation.

Another strong confirmation for the interpretation of ss comesfrom taxa that have a large value of ss: these are exactly the taxa that

can considered to be demand species on the basis of the criteria ofTable 2: all invertebrates have a small supply stress ss, but vertebrateshave higher values. Fig. 4 is the same as Fig. 1 but now highlights the5 classes of fish. While hagfish (Myxini) and ray-finned fish (Actinop-terygii) are close to the supply end of the spectrum, cartilaginous fish(Chondrichthyes) tend to be closer to the demand end. One species oflobe-finned fish (Sarcopterygii), the coelanth, turns out to be a supplyspecies, while the Australian lungfish has demand tendencies. Thisstrategy is probably open to lungfish, because they can switch offmaintenance (torpor, although the non-Australian species can do thisbetter). Lampreys (Cephalaspidomorphi) seem to have some demandtendencies, which possibly relates to their life style of ‘milking’ fish.Only the European brook lamprey is presently in the collection; morespecies are required for confirmation. Ray-finned fish (Actinopterygii)tolerate a very wide range of food levels (Kooijman, 2009a), whichconfirms their classification as supply species.

Birds and mammals are close to the demand end, where foodintake is primarily controlled by metabolic needs, while mostinvertebrates are close to the supply end, where food intake isprimarily controlled by food availability. Cnidarians are possiblythe most extreme supply species with extreme capacity of shrink-ing and rejuvenation in response to starvation: some medusea caneven rejuvenate till polyps (Piraino et al., 1996). Because cephalo-pods have a closed blood circulation system, telolecithal eggs, highpeak metabolic rate, complex behaviour and superb vision, weexpected to see tendencies for demand species in this taxon. Yettheir ss values are small, which probably relates to their life style ofsuicide reproduction. They do not die by ageing and their size atdeath is considerably smaller than their asymptotic size, whilemost species approximate that size (insects being an exception,Kooijman, 2014). This means that their value for sMf _pAmg isrelatively very large for species with that size at death, so ss issmall; their range in body sizes at death is quite large.

Fig. 5 shows supply stress ss as a function of ultimate structurallength of species. DEB theory has rules for the co-variation ofparameter values (Kooijman, 1986), which are based on simplephysico-chemical arguments. These rules imply that ss is expectedto be independent of maximum body size, since EH

p increases withcubed maximum structural length, f _pAmg with structural length,while _kJ , ½ _pM � and sM are independent of maximum structural

demand

supply

Fig. 2. Distance to the supply end in the supply–demand spectrum as a function ofthe minimum scaled functional response that is required to reach puberty. Thecurve is smax

s ðfminÞ ¼ f 3min4=27. Colour coding as in Fig. 1. All supply species hide inthe lower left corner of the figure.

demand

supply

Fig. 3. As Fig. 2, but only endotherms are highlighted. The red-level for mammals(bigger dots) and the blue-level for birds (smaller dots) are proportional to κ. Thehighest values of κ (bright red and blue) occur at low values for fmin and ss. (Forinterpretation of the references to colour in this figure caption, the reader isreferred to the web version of this paper.)

supply demandFig. 4. As Fig. 1, but colours now indicate the 5 fish classes: blue for Myxini,turquoise for Cephalaspidomorphi, green for Chondrichthyes, red for Actinopterygiiand black for Sarcopterygii. Open symbols indicate acceleration. (For interpretationof the references to colour in this figure caption, the reader is referred to the webversion of this paper.)

K. Lika et al. / Journal of Theoretical Biology 354 (2014) 35–4740

length. The figure supports this idea by illustrating that theevolution from supply to demand systems is possible for speciesof all maximum body sizes.

4. Evolutionary constraints on parameter values

DEB parameter values are individual-specific and are partlyunder genetic control, in the sense of quantitative genetics, whiledata, as presented in Table 1, has an intimate link with eco-physiological performance. Natural selection is on eco-physiological performance, so the relationship between parametersand data parallels that between genotype and phenotype. Let usconsider a particular case study. Different species of ctenophores ofsimilar ultimate body size lay eggs of different size that all hatch in asingle day. This invites for the question of how this constraint ondata translates into constraints on parameter values. Not all para-meters contribute to age at birth ab ¼ _kMτb, where τb given in stepP6 of the map from parameters to data (see Appendix B). It turns outto be a function of _kM ¼ ½ _pM �=½EG�, k¼ _kJ= _kM , g¼ _v½EG�=κf _pAmg andvbH ¼ ðκ=ð1�κÞÞEbH=L3m. Ultimate structural length is L1 ¼ sMLm withacceleration factor sM ¼ lj=lb and maximum structural length isgiven by Lm ¼ κf _pAmg=½ _pM �. Scaled length at birth lb in sM is also afunction of k, g and vH

b and scaled length at metamorphosis ljinvolves vjH ¼ ðκ=ð1�κÞÞEjH=½EG�L3m. Suppose that evolution does notaffect ½EG� and _kJ between related species, the constraints thatabo1 d and L1 is fixed must now be translated into constraintson ½ _pM �, _v, f _pAmg, κ, EHb and EH

j . If we focus on the upper boundaryab¼1 d, we can translate this into the problem of how theseparameters can vary while sM _v=τbg remains fixed at value L1=ab.We still need a numerical analysis to go to actual numbers, but thecase beautifully illustrates the variety of evolutionary pathways thatare open to achieve aparticular eco-physiological performance. Onepossible route to increase size at birth, without affecting age at birthand ultimate length, is to increase EH

b and _v (in a special way).Another one is to increase κ via g and compensate by changing sMand _v. Yet another one is to increase f _pAmg and compensate by sM;the route is via the maternal effect that reserve density at birthequals that of the mother at egg formation.

5. Role of scatter

Scatter is an unavoidable aspect of biological data. The relativecontribution of measurement error is typically small, however.

Most scatter relates to ‘biological variability’ which has genotypicand phenotypic aspects, especially if different types of dataoriginate from different biological subjects and/or differentresearch workers. The question about role of scatter is highlyrelevant, but also difficult to address. We here study the role ofscatter in the form of sensitivity of the mapping using elasticitycoefficients and present a numerical example. We explore the linkbetween the elasticities on both directions.

The bijection P from data d¼ ðd1;…; d9ÞT to parameters p¼ðp1;…; p9ÞT has a differentiable inverse D, so it classifies as aC1-diffeomorphism. In other words PðDðpÞÞ ¼ p and DðPðdÞÞ ¼ d.A 9�9 matrix of elasticity coefficients ep is associated to eachpoint in the 9 dimensional parameter space and ed to each point inthe 9 dimensional data space.

Table 3 gives (9 dimensional) parameters and data that areconnected by the bijection, including the relative error between dand DðPðdÞÞ and between p and PðDðpÞÞ, respectively. The absoluterelative errors vary from 0 till 1:8� 10�4. These errors reflect theaccuracy of the numerical procedures that are used in the algo-rithm of the bijection, where numerical integration and rootfinding occurs. We randomly sampled the data and parameterspace for mapping and noticed that the relative error couldincrease above 0.1 if abo0:5 d or am4104 d or Wm41 Mg. Afterfiltering the random trials for these boundaries, the mapping inboth directions had a typical relative error of 0.0005, but couldoccasionally increase till 0.05, while the errors in both directionscorrelated. These errors reflect accuracy settings in the numericalprocedures in the mapping.

The product eped ¼ edep ¼ I must hold. This can be proved asfollows. Let D¼ diagðdÞ and P ¼ diagðpÞ. The elasticity matricescan now be written as ed ¼D�1ð∂=∂pT ÞPðdÞP and ep ¼ P�1ð∂=∂dT ÞDðpÞD. The inverse function theorem (Apostol, 1974, p. 372) learnsthat ðð∂=∂pT ÞPðdÞÞ�1 ¼ ð∂=∂dT ÞDðpÞ. As a result we have

e�1d ¼ D�1 ∂

∂pTPðdÞP� ��1

¼ P�1 ∂∂pTPðdÞ� ��1

D¼ P�1 ∂∂dTDðpÞD¼ ep

Likewise we have e�1p ¼ ed and eped ¼ edep. The elasticities for the

parameters, ep, could not be obtained reliably by numericaldifferentiation; many values sensitively depend on the perturba-tion factor that was used, specially for small factors. The values inTable 3 were obtained from those for ed. The values for ed couldonly be obtained by plotting the numerical derivatives as afunction of the perturbation factor and graphically back-extrapolate to perturbation zero. Many values were approximatelylinear in the perturbation factor; not a good sign, but still work-able. The determinant of the matrix for the data elasticities wasfound to be detðedÞ ¼ 0:447 in this numerical example.

The position of the zeros indicates the absence of information,so _Rm has information for €ha, but €ha has no information for _Rm. Thereason is that, in the present simple implementation of the ageingmodule in the standard DEB model, energetics affects ageing, butageing does not affect energetics. Since ½EG� is proportional to dV,the elasticity ðdV=½EG�Þ∂½EG�=∂dV ¼ 1, while none of the otherparameters affect dV.

The most extreme elasticity is ð∂½ _pM �=∂apÞap=½ _pM � ¼ �7:4, that of½ _pM � for ap (lower panel of Table 3). The map from parameters to datais most sensitive to the parameter κ, with the weights and maximumreproduction to be affected the most. The map from data to para-meters is sensitive mainly to the ages at birth and puberty, ab and ap,which affect all parameters except κ and ½EG� (small elasticities).

6. Discussion

Although application and testing of a theory or model arefrequently combined in practice, their aims can be very different.

demand

supply

Fig. 5. Supply stress as a function of ultimate structural length using the colourcoding as in Fig. 1. Open symbols indicate metabolic acceleration. The demand endof the supply–demand spectrum is at the top of the ordinate. (For interpretation ofthe references to colour in this figure caption, the reader is referred to the webversion of this paper.)

K. Lika et al. / Journal of Theoretical Biology 354 (2014) 35–47 41

We here study applicability of the standard DEB model and inparticular the problem of parameter identification. The problemthat we addressed is that some statistical aspects of parameteridentification can be successfully studied with elasticity coeffi-cients in the situation that a bijection exists between data andparameter values. In typical situations, where the number of datapoints (greatly) exceeds the number of parameters, identificationdepends on a large number of properties. Feeding and survival areintrinsically stochastic and this source of stochasticity can easilyhave rather complex implications for the statistical aspects ofparameter estimation (Kooijman, 2009a). Many other sources ofstochasticity also exist, such as variation of parameter valuesamong individuals. With the bijection we can separate effects ofstochasticity on parameter estimation from effects of the modelstructure and focus on the latter.

The structure of the standard DEB model is a mix of componentswith supply and demand organisation. Somatic maintenance isproportional to the amount of structure and maturity maintenanceto the level of maturity, irrespective of the nutritional condition. Sothey have a demand-organisation. Allocation to growth andmaturation or reproduction has a supply-organisation and fullydepends on what is available. Shrinking and rejuvenation arerequired in the absence of resources, where the demand can nolonger be satisfied (Augustine et al., 2011).

In our search for the boundaries of the parameter and dataspaces, we identified a simple metric, called supply stressss ¼ _pJ _p

2M= _p

3A for fully grown adults, which seems to quantify the

distance to the supply-end in the supply–demand spectrum onwhich species can be ranked. We came to this particular inter-pretation by the direct link between ss and the range of functionalresponses that allow puberty to be reached and by the taxa thathave a small demand stress. It is in itself already very remarkablethat ss, as estimated from parameters, does segregate taxa. Theaccuracy of parameter values is difficult to address but theexistence of clear taxon-related patterns supports the idea thattheir values do reflect some eco-physiological properties. As far aswe know, this is the first time that the supply–demand spectrumis formally recognised as a spectrum and quantified.

Table 2 lists a number of diagnostic properties of supply anddemand species. Application of these ideas should reveal if theseproperties can be extended and/or refined. Parental care, forinstance, is best developed in demand systems, but also welldeveloped in e.g. particular groups of (social) insects. We presentlywork on the further extension of the add_my_pet collection with(holometabolic) insects, which will allow us to test if parental carematches patterns in supply stress. Parental care was previouslyidentified as an alternative or supplementary strategy for meta-bolic acceleration (Kooijman, 2014) to ease the start up of feedingand digestion by the neonate, when experience is low andrequirement of high quality food is high. Fig. 4 weakly suggests alink between acceleration and supply stress in ray-finned fish.Territorial defence might be connected to parental care sinceterritorial defence is typically only shown during the breedingseason. Storing food in deposits in preparation for bleak seasons,frequently combined with hibernation or torpor, might also be atrait connected to demand systems. A more systematic study onvarious taxa is required to link these traits to the position in thesupply–demand spectrum.

Most animal research concerns species near the demand end ofthe spectrum, where food intake is not very sensitive to foodavailability. The result is that weight becomes age-dependent, whichexplains why many growth models in the literature are age-based.Size-based growth models do exist for demand species, e.g. Stratheet al. (2009), but are less frequent. Species at the supply-end of thespectrum show, however, that a huge weight variation is possibleamong individuals of the same age but experienced differences inTa

ble

3Elasticity

coefficien

tse d

ofdataforparam

eters,e.g.

ð∂d V

=∂f

_ pAmgÞf

_ pAmg=d V

¼0an

de p

ofparam

etersfordata,

e.g.

ð∂f_ p

Amg=∂d

VÞd

V=f

_ pAmg¼

1:00

4,in

case

ofparam

eter

anddatava

lues

that

map

onto

each

other

asindicated

inthelast

columns.Th

eseco

ndlast

columngive

stherelative

errorof

map

pingforw

ard,followed

byba

ckward.T

hedata-elasticities

wereob

tained

byex

trap

olatingthenumerical

derivatives

toze

roperturbation;theparam

eter-elasticities

wereco

mputedfrom

thedata-elasticities.

e df_ p

Amg

_ vκ

½_ pM�

½EG�

E HbE Hj

E Hp€ ha

Error

Valued

d V0

00

01

00

00

00.1

a b�0.06

0�0.91

21.56

0�0.04

8�0.21

6�0.32

40

00

3.1e

�6

115.8d

a p�0.25

40.75

91.43

5�0.03

1�0.05

30.23

0�0.16

70.27

50

�3.8e

�5

464.1d

a m0.33

3�0.33

30.33

3�0.33

30

00

0�0.33

3�1.7e

�4

1288

dW

b0.84

9�0.72

64.82

0�0.17

9�1.63

00.95

90

00

9.1e

�6

2.15

6g

Wj

0.87

7�0.70

94.77

0�0.25

1�1.57

2�0.05

00.99

50

0�1.7e

�4

20.39g

Wp

0.92

5�0.64

84.63

0�0.42

9�1.40

4�0.09

10.08

90.91

10

�2.4e

�4

316g

W1

3.87

0�0.81

42.94

5�3.07

0�0.77

2�1.01

00.99

50

0�1.8e

�4

55.7

kg_ Rm

2.28

60.69

1�6.680

�1.94

50.65

0�1.99

01.000

�0.002

0�1.9e

�4

7.23

d�1

e pd V

a ba p

a mW

bW

jW

pW

1_ Rm

Error

Valuep

f_ pAmg

1.004

4.13

1�4.95

70

�0.01

3�0.88

21.49

7�0.26

50.18

12.8e

�7

225Jd

�1cm

�2

_ v0

�3.52

92.46

40

0.52

64.43

8�0.74

4�0.111

�0.06

9�1.1e

�7

0.02

cmd�1

κ�0.003

0.06

7�0.31

60

�0.28

60.02

80.09

50.16

2�0.25

1�3.8e

�7

0.8

½_ pM�

1.006

6.33

8�7.35

20

�0.78

3�0.89

02.22

0�0.54

1�0.000

1.9e

�16

18Jd

�1cm

�3

½EG�

10

00

00

00

00

2615

Jcm

�3

E Hb1.01

6�5.47

66.46

60

2.74

40.80

6�1.95

0�0.59

91.05

01.1e

�5

275J

E Hj1.01

7�5.15

66.117

01.70

11.77

8�1.84

4�0.63

31.04

9�1.7e

�4

2750

JE Hp

1.01

6�4.10

74.98

60

1.58

40.55

5�0.40

4�0.73

11.04

7�2.5e

�4

50kJ

€ ha

�0.003

1.39

3�0.39

2�3.003

�0.04

3�0.40

30.118

0.32

7�0.001

5.0e

�4

10�6d�2

K. Lika et al. / Journal of Theoretical Biology 354 (2014) 35–4742

food availability. Age-based growth models do not work for them, ifdifferences in environmental conditions are involved. Research onageing suffers from the same problem. Most research is again doneon endotherms with a limited range of intake rates among indivi-duals of the same body size. This restricts the effect of energetics onageing, which, according to DEB theory, should be much bigger insupply species.

The evolution from supply to demand reflects an increase inhomeostatic control: the capacity of individuals to run theirmetabolism independent from environmental conditions (tem-perature, food). It is obviously never perfect. DEB theory delineatesfive different types of homeostasis: (1) strong (metabolic poolshave a constant chemical composition), (2) weak (ratios of poolsizes become constant during growth in constant environments),(3) structural (body shape does not change during growth),(4) acquisition (food intake less dependent on food availability),(5) thermal (body temperature independent from environment).The role of shape (structural homeostasis) concerns, in the contextof DEB theory, the scaling of surface areas (transport) to volumes(mass conservation). It is remarkable that this role of shape comesback in the scaling of assimilation versus somatic maintenance inthe metric ss.

We studied all 21 organisational alternatives of models that havereserve and structure as state variables and comply to weak home-ostasis and concluded that the κ-rule as implemented in the standardDEB model is the only possibility that is consistent with a number ofstylised empirical facts on energetics (Lika and Kooijman, 2011). Themost convincing empirical support for the κ-rule possibly comesfrom studies where κ changes with coherent consequences forgrowth, development and respiration (Mueller et al., 2012).

The κ-rule has profound effects on metabolic organisation andthe distribution of its value over the species has surprises(Kooijman and Lika, 2014a, 2014b). The rule has the counter-intuitive implication that species can boost their growth andreproduction by wasting assimilates: the waste-to-hurry phenom-enon (Kooijman, 2013). It now seems that the κ-rule also plays adeep role in supply–demand spectra and really shaped the evolu-tion of metabolism.

7. Conclusions

� Given a number of assumptions on auxiliary parameters, anine-dimensional bijection exists between core-parameter anddata space. We present an efficient algorithm for this bijection,which can be used in the specification of initial values inparameter estimation methods for large data sets. The relation-ship between DEB parameters and data has similarities to thatbetween genotype and phenotype.

� We found expressions for the boundaries of the data and theparameter space. These expressions can be used to filterparameter combinations in parameter estimation methods forlarge data sets.

� One of the boundaries involves a new metric, the supply stress,which turns out to quantify the location of species in thesupply–demand spectrum. This spectrum reflects the interna-lisation of energetic control.

� We identified diagnostic properties of species for their positionin the supply–demand spectrum and discuss the links betweenthese properties.

� Invertebrates and ray-finned fish turn out to be close to thesupply end of the spectrum, while other vertebrates havedemand tendencies. We explain why demand species showup-regulation of metabolism in association with reproductiveevents and discuss evolutionary aspects.

� We discuss some properties of the bijection using elasticitycoefficients. The most extreme elasticity coefficient is that ofspecific somatic maintenance for age at puberty; age at pubertystrongly decreases for increasing specific somatic maintenanceamong species.

� We consider evolutionary constraints on parameter valuesand present alternative metabolic routes for changes in eggsize under constraints on incubation time, as observed inctenophores.

� The κ-rule for allocation of mobilised reserve to soma wasfound to be responsible for the supply–demand spectrum, incombination with surface area–volume relationships. Theκ-rule was also found to be responsible for the waste-to-hurry phenomenon.

Acknowledgements

We like to thank all who contributed to the add_my_petcollection and Mike Kearney, Bob Kooi and Gonçalo Marques forhelpful discussions. Jean-Christophe Poggiale contributed with theproof that the elasticity matrices for parameters and data areinverse to each other.

Appendix A. The standard DEB model with acceleration

Table A2 lists all symbols that are used in this paper, apart fromthe ones listed in Table 1. The notation follows the DEB rules; theirrationale is explained in the DEB-notation document (see underreferences).

The standard DEB model has three state variables, energy inreserve E, structural length L and maturity EH. Maturity has nomass or energy and is quantified as cumulative (dissipating)energy investment. Development starts with age, structural lengthand maturity all zero, and an amount of reserve such that reservedensity E=L3 at birth equals that of the mother at egg formation(maternal effect). Foetal (and bud) development (mammals, sev-eral fish, salps, cnidarians) represents a variation on egg-develop-ment, where _pA=f _pAmgL2m ¼ el2, where e is the scaled reservedensity of the mother. The maternal effect rule determines theenergy cost of an egg or foetus, apart from the reproductionefficiency κR, which quantifies the overhead costs for reproduc-tion. Life stage switches are linked to maturity EH exceedingthreshold values, which are fixed parameter values: birth b (startof feeding and metabolic acceleration), metabolic metamorphosis j(ceasing of metabolic acceleration), puberty p (ceasing of matura-tion, start of allocation to reproduction). The lengths at whichthese switches occur (lb, lj and lp) are not parameters, but dependon food history, i.e. food density X(t) (so on scaled functionalresponse f(t)). Temperature affects all rate parameters. Thechanges in the state variables are simple functions of the powers_pn, which are given in Table A1: ðd=dtÞE¼ _pA� _pC , ðd=dtÞL3 ¼_pG=½EG�, with _pG ¼ κ _pC� _pS and ðd=dtÞEH ¼ _pR, with _pR ¼ ð1�κÞ_pC� _pJ . Maturation is ceased, and allocation to reproduction isstarted if EH ¼ EpH (or lZ lp); this allocation involves the samepower _pR. During acceleration in the early juvenile stage, max-imum specific assimilation f _pAmg and energy conductance _vincrease with length; the scaling in Table A1 uses the values off _pAmg and _v at birth. No metabolic acceleration occurs if EjH ¼ EbH(i.e. lj¼ lb); metabolic metamorphosis might (e.g. in bivalves) ormight not (e.g. in cephalopods) coincide with a morphological one.The ratio lj=lb is the acceleration factor with which the values off _pAmg and _v at birth are multiplied to arrive at those aftermetamorphosis. The juvenile stage is absent if EpH ¼ EbH (i.e.

K. Lika et al. / Journal of Theoretical Biology 354 (2014) 35–47 43

lp ¼ lj ¼ lb); this occurs in e.g. Oikopleura and insects. The expres-sion for the mobilisation power _pC follows from the weak home-ostasis requirement: the chemical composition of the individualdoes not change during growth in constant environments (possi-bly after a short adaptation period). Feeding is proportional tostructural surface area, so to squared length in isomorphs, i.e.individuals that do not change in shape during growth. Environ-mental conditions (temperature and osmotic value) are assumedto be such that surface-coupled maintenance costs (heating inendotherms, osmotic work in freshwater organisms) are negligi-ble. The powers for ingestion and defecation occur in the environ-ment, not in the individual, so they are excluded from Table A1.Food searching is not discussed here.

Survival as affected by ageing is modelled by assuming that thehazard rate of the individual is proportional to the density ofdamage compounds (modified proteins), which are continuouslyproduced by damage-inducing compounds (modified DNA). Theproduction of the latter is taken to be proportional to the rate atwhich reserve mobilised, since this reflects dioxygen consumptionthat is not associated to assimilation. Ageing is accelerated in sometaxa (endotherms) by also increasing the density of damage-inducing compounds proportional to their concentration. Thequantitative details and motivation are presented in Kooijman(2010, Chapter 6). The simplified expression for age at death that isused here assumes that life span is large relative to the growthperiod. In that case, the survival probability as affected by ageingreduces to

Prfam4tg ¼ exp6 _h

3W

_h3G

1�expð _hGtÞþ _hGtþ_h2Gt

2

2

!0@

1A

with _h3W ¼ €ha _v=6Lm and _hG ¼ sG _vf

3=Lm, see Kooijman (2010,Eq. (6.5)). The mean age at death, am, equals the integral of thisfunction over all positive times t.

The nine parameters (with units) of the standard DEB modelwith acceleration are: specific maximum assimilation rate f _pAmg(J d�1 cm�2), energy conductance _v (cm d�1), specific somaticmaintenance ½ _pM � (J d�1 cm�3), maturity maintenance rate coeffi-cient _kJ (d�1), specific cost for structure ½EG� (d cm�3), reproduc-tion efficiency κR (–), maturity at birth EH

b (J), maturity atmetamorphosis EH

j (J), and maturity at puberty EHp (J).

The remaining four primary DEB parameters that are notdiscussed here are: specific searching rate f _Fmg (dm3 d�1 cm�2),digestion efficiency κX (–), ageing acceleration €ha (d�2), andGompertz stress coefficient sG (–).

A number of auxiliary parameters are required for various typesof conversion: dry–wet weight, length–weight, mass–energy. Theyare shape coefficient δM (–), specific density of wet structure dV

w

(g cm�3), chemical potential of reserve μE (J C-mol�1), and che-mical indices for hydrogen, oxygen and nitrogen of reserve and

structure nHV, nOV, nNV, nHE, nOE, nNE (–). These parameters dependon the type of measurement, not on the structure of the DEBmodel. The DEB model obtains respiration from the conservationlaw for chemical elements, which involves the chemical indices.

Appendix B. The algorithm for map from parameters to data

The algorithm has 18 steps and requires the computation of anumber of compound parameters that play a role in DEB theory:

P1 maintenance ratio k¼ _kJ= _kM , with somatic maintenance ratecoefficient _kM ¼ ½ _pM�=½EG�;

P2 dry/wet weight ratio δW ¼ dV=dwV , with specific density of

structure dV ¼ κG½EG�wV=μV ;P3 energy investment ratio g ¼ ½EG� _v=κf _pAmg and maximum

structural length Lm ¼ κf _pAmg=½ _pM �;P4 scaled maturity Un

H ¼ En

H=f _pAmg, Vn

H ¼ Un

H=ð1�κÞ and vnH ¼Vn

Hg2 _k

3M= _v

2 with n¼ b; j; p;P5 scaled length at birth lb is solved from xbgvbH=vðxbÞl3b ¼R xb

0 ðgþ lðxÞÞ=vðxÞ dx with xb ¼ g=ð1þgÞ and vðxÞ ¼expð� R x0 ððk�x1Þ=ð1�x1ÞÞðlðx1Þ=gx1Þ dx1Þ and lðxÞ ¼ ðð1=lbÞðxb=xÞ1=3�ðBxb ð43;0Þ�Bxð43;0ÞÞ=3gx1=3Þ�1, where Bxða; bÞ is theincomplete beta function;

P6 scaled age at birth τb ¼ 3R xb0 dx=ð1�xÞx2=3ð3gx1=3b =

lb�Bxb ð43;0ÞþBxð43;0ÞÞ;P7 scaled exponential growth rate ρj ¼ gð1=lb�1Þ=ð1þgÞ

(between birth and metamorphosis);

P8 scaled length at metamorphosis lj ¼ lbþR vjHvbHðd=dvHÞl dvH with

ðd=dvHÞl¼ ρjl=3=ðl3ð1=lb�ρj=gÞ�kvHÞ and lðvbHÞ ¼ lb;P9 scaled ultimate length l1 ¼ sM with acceleration factor

sM ¼ lj=lb;

P10 scaled length at puberty lp ¼ ljþR vpHvjHðd=dvHÞl dvH with ðd=dvHÞl

¼ ððg=3Þðl1� lÞ=ðgþ1ÞÞ=ðl2ðgl1þ lÞ=ðgþ1Þ�kvHÞ and lðvjHÞ ¼ lj;P11 scaled age at metamorphosis τj ¼ τbþ3ρ�1

j log sM;P12 scaled von Bertalanffy growth rate ρB ¼ ð3þ3=gÞ�1 (after

metamorphosis);P13 scaled age at puberty τp ¼ τjþρ�1

B log ðl1� ljÞ=ðl1� lpÞ;P14 ages an ¼ τn= _kM with n¼ b; p;P15 structural lengths Ln ¼ lnLm with n¼ b; j; p;1;P16 wet weights Wn ¼ dwV L

3nð1þ½Em�wE=dVμEÞ with n¼ b; j; p;1

and ½Em� ¼ f _pAmg= _v;P17 maximum reproduction rate _Rm ¼ ðκR=E0Þððð1�κÞ=κÞ ½ _pM �s3M

L3m� _kJEpHÞ with initial reserve E0 ¼ u0

EL3m½EG�=κ and initial

scaled reserve u0E ¼ ð3g=ð3gx1=3b =lb�Bxb ð43;0ÞÞÞ3 and xb ¼

g=ð1þgÞ;P18 age at death am ¼Γð43Þð6=g _kM

€haÞ1=3, where ΓðxÞ is the gammafunction.

Table A1

The scaled powers _pn=f _pAmgL2m , for n¼ A;C; S; J, as specified by the standard DEB model with acceleration and without heating costs for an

isomorph of scaled length l¼ L=Lm , scaled reserve density e¼ ðE=L3Þ _v=f _pAmg and scaled maturity density uH ¼ EH=½EG�L3m=κ at scaled functionalresponse f � X=ðKþXÞ, where X denotes the food density and K the half saturation constant. Maximum length Lm ¼ κf _pAmg=½ _pM � has only thisinterpretation in the absence of acceleration, where lj¼ lb. Parameters: allocation fraction κ, investment ratio g¼ ½EG� _v=κf _pAmg, maintenance

ratio k¼ _kJ ½EG�=½ _pM �.

power

f _pAmgL2mEmbryo 0o lr lb Early juvenile lbo lr lj Late juvenile ljo lr lp Adult lpo lo1

Assimilation, _pA 0fl2

llb

fl2ljlb

fl2ljlb

Mobilisation, _pC el2gþ lgþe

el2gl=lbþ lgþe el2

glj=lbþ lgþe

el2glj=lbþ lgþe

Som. maint., _pS κl3 κl3 κl3 κl3

Mat. maint., _pJ kuH kuH kuH kupH

K. Lika et al. / Journal of Theoretical Biology 354 (2014) 35–4744

The computation of scaled length at birth lb in step P5 is by farthe most demanding, but efficient routines based on Kooijman(2009b) are available in software package DEBtool. Numericalintegrations are required in lj, lp and τb as well, which reducesaccuracy. This algorithm has been coded in function iget_pars_9 ofDEBtool.

Appendix C. The algorithm for map from data to parameters

This algorithm has 16 steps:

D1 acceleration factor sM ¼ ðWj=WbÞ1=3. For non-acceleratingspecies: Wj¼Wb and sM ¼ 1 and aj¼ab;

D2 scaled length at birth lb ¼ Lb=Lm ¼ sMðWb=W1Þ1=3, scaledlength at metamorphosis lj ¼ Lj=Lm ¼ sMðWj=W1Þ1=3 and

scaled length at puberty lp ¼ Lp=Lm ¼ sMðWp=W1Þ1=3 can beobtained from wet weights. Although structural lengthsthemselves cannot be accessed yet, their ratios can. Ultimatestructural length L1 ¼ LmsM and maximum structural lengthLm will be given below when using ultimate wet weight W1.Maximum structural length Lm will be treated as a compoundparameter and the interpretation only applies to non-accelerating species; ultimate structural length L1 exceedsLm for accelerating species at abundant food;

D3 age at metamorphosis aj ¼ ðap log sMþab log sÞ=ðlog sMþlog sÞ with s¼ ððsM� ljÞ=ðsM� lpÞÞ1=lb �1. This is based on therelationship between the exponential growth rate_r j ¼ 3 log sM=ðaj�abÞ during acceleration between birth and

metamorphosis and the von Bertalanffy growth rate_rB ¼ ð1=ðap�ajÞ log ðsM� ljÞ=ðsM� lpÞ after metamorphosis.Their links with DEB parameters at abundant food are_r j ¼ _kMð1=lb�1Þ=ð1þ1=gÞ and _rB ¼ ð _kM=3Þ=ð1þ1=gÞ, seeKooijman (2010, Eq. (2.24)). So _rB ¼ rj=ð3=lb�3Þ and substitu-tion gives the result;

D4 cost for structure ½EG� ¼ μVdV=wVκG ¼ 26 151 δW J cm�3,given the assumptions. This directly follows from the defini-tion of growth efficiency κG ¼ μVdV=wV ½EG�;

D5 von Bertalanffy growth rate _rB ¼ ð1=ðap�ajÞÞ log ð1� ljÞ=ð1� lpÞ, which directly follows from the definition LðtÞ ¼L1�ðL1�L0Þ expð� _rBtÞ;

D6 maximum reserve residence time tEm ¼ Lm= _v ¼ 1=g _kM ¼ab=3:7lb, where scaled length at birth lb is given above. Thisis based on ðd=dtÞLð0Þ ¼ _v=3 and the approximation that thisholds during the full incubation time, leading to Lb ¼ _vab=3,see Kooijman (2010, Eq. (2.47)). However, reserve becomesdepleted during incubation of eggs, increasing incubationtime by a mean factor of 1.226 among species that are presentin the collection. This value for tEm is an approximation thatwill be replaced in step D9;

D7 specific somatic maintenance cost ½ _pM � ¼ 3_rB½EG�=ð1�3_rBtEmÞ.This is based on the von Bertalanffy growth rate _rB ¼ ½ _pM�=3=ð½EG�þ½ _pM �tEmÞ at abundant food (Kooijman, 2010, Eq.(2.24)) after metamorphosis. Since tEm is an approximation,this value for ½ _pM � is also approximative (see next step);

D8 somatic maintenance rate coefficient _kM ¼ ½ _pM�=½EG�, from itsdefinition, see Kooijman (2010, Section 2.5.1). This value for _kM

can be used as an initial value for a numerical procedure to solve_kM from the exact value for ab ¼ τb= _kM , with τb given

Table A2Symbols, units and descriptions that are used in this paper, in addition to the ones listed in Table 1.

Bxða; bÞ – Incomplete beta function d Vector of nine datadW g cm�3 Spec. density of dry biomass dV

w g cm�3 Spec. density of wet structureD Map from parameters to data e – Scaled reserve densityed – Matrix of elasticities for data ep – Matrix of elasticities for parametersE J Reserve E0 J Reserve at age zeroEH J Maturity f – Scaled functional response

f _Fmg L d�1 cm�2 Specific searching rate g – Energy investment ratio€ha

d�2 Ageing acceleration _hG d�1 Gompertz ageing rate_hW d�1 Weibull ageing rate k – Maintenance ratio_kJ d�1 Maturity maintenance rate coeff. _kM d�1 Somatic maintenance rate coeff.

K M Half saturation coefficient l – Scaled structural lengthlb – Scaled structural length at birth lj – Scaled structural length at metam.lp – Scaled structural length at puberty L cm Structural lengthL1 cm Ultimate structural length Lm cm Maximum structural lengthnij Chemical index for i in j p Vector of 9 parameters_pA J d�1 Assimilation rate _pC J d�1 Mobilisation rate_pG J d�1 Growth investment rate _pJ J d�1 Maturity maintenance rate_pM J d�1 Somatic maintenance rate _pR J d�1 Reproduction investment rate_pS J d�1 Total somatic maint. rate _pMþ _pT f _pAg J d�1 cm�2 Specific assimilation ratef _pT g J d�1 cm�2 Specific heating rate P Map from data to parameters_rB d�1 von Bertalanffy growth rate _r j d�1 Exponential growth ratesd – Demand stress sG – Gompertz stress coefficientsM – Acceleration factor ss – Supply stresstEm d Maximum reserve residence time T K TemperatureTA K Arrhenius temperature T ref K Reference temperatureuE0 – Scaled energy invested per egg uH – Scaled maturity

uH – Scaled maturity at puberty vHb – Scaled maturity at birth

vHj – Scaled maturity at metamorphosis vH

p – Scaled maturity at pubertywE g C-mol�1 Molecular weight of reserve wV g C-mol�1 Molecular weight of structurex – Transformed scaled reserve density xb – Transf. scaled reserve dens. at birthX M Food density ΓðxÞ – Gamma functionδM – Shape coefficient δW – Dry/wet weight ratioκG – Growth efficiency κR – Reproduction efficiencyκX – Digestion efficiency μE J C-mol�1 Chemical potential for reserveμV J C-mol�1 Chemical potential for structure ρB – Scaled von Bertalanffy growth rateρj – Scaled exponential growth rate τ – Scaled ageτb – Scaled age at birth τp – Scaled age at puberty

K. Lika et al. / Journal of Theoretical Biology 354 (2014) 35–47 45

by Kooijman (2010, Eq. (2.38)). So _kM must be solvedfrom _kMab ¼ 3

R xb0 dx=ð1�xÞx2=3ðαb�Bxb ð43;0ÞþBxð43;0ÞÞ, where

Bxða; bÞ is the incomplete beta function, αb ¼ 3gx1=3b =lb, xb ¼ g=ðebþgÞ, g¼ _rB=ð _kM=3� _rBÞ (see D9), while _rB is given in D5, aj inD3, lb, lj, lp in D2 and sM in D1. With this correct value for _kM , weobtain ½ _pM � ¼ _kM½EG� to replace the value obtained in step D7;

D9 energy investment ratio g¼ _rB=ð _kM=3� _rBÞ. This is based onthe von Bertalanffy growth rate _rB ¼ _kM=3=ð1þ1=gÞ, seeKooijman (2010, Eq. (2.24)) and its definition g ¼ ½EG�=κ½Em�,see Kooijman (2010, Eq. (2.21)). The value for tEm ¼ ðg _kMÞ�1

that was obtained in step D6 can now be replaced;

D10 allocation fraction κ ¼ 1� _RmtEms�3M l3bð1:75þgÞ=κR, with

κR ¼ 0:95 as default. This is based on the maximum repro-

duction rate _Rm ¼ ðκR=E0Þððð1�κÞ=κÞ _kM ½EG�s3ML3m� _kJEpHÞ

(Kooijman, 2010, Eq. (2.58)), the costs per (foetal) offspring

E0 ¼ L3bð½Em�7=4þ½EG�=κÞ, see Kooijman (2010, Eq. (2.51)) and

neglecting maturity maintenance ( _kJ ¼ 0). The cost of anegg is slightly larger, due to retardation of development bythe reserve becoming limiting, which increases cumulativemaintenance costs. Placental costs are ignored as well,assuming that most of it is recovered by eating the placenta

after birth. Substitution gives _Rm ¼ ðð1�κÞκR _kM=ð1:75=gþ1ÞÞs3ML3m=L

3b . This value for κ will be replaced in step D15;

D11 reserve capacity ½Em� ¼ ½EG�=κg, based on the definition of theenergy investment ratio g ¼ ½EG�=κ½Em�, see Kooijman (2010,Eq. (2.21));

D12 maximum structural length Lm ¼ s�1M ðW1=dwV ð1þð½Em�

wE=dVμEÞÞÞ1=3 where the coefficients dVw, dV, wE and μE are

specified by the assumptions. This is based on W1 ¼dwV L

31ð1þ½Em�wE=dVμEÞ see Kooijman (2010, Eq. (3.2)). We

now have access to Lb ¼ lbLm, Lj ¼ ljLm and Lp ¼ lpLm;D13 energy conductance _v ¼ Lm=tEm, which directly follows from

tEm ¼ Lm= _v;D14 specific assimilation rate f _pAmg ¼ _v½Em� ¼ ½ _pM�Lm=κ. This

follows from the definition of reserve capacity ½Em� ¼f _pAmg= _v, see Section 2.3.1, and from maximum structurallength Lm ¼ κf _pAmg=½ _pM�, see Kooijman (2010, Section 2.6);

D15 maturity level at birth EbH ¼ ðð1�κÞ=κÞ½EG�L3b , metamorphosis

EjH ¼ ðð1�κÞ=κÞ½EG�L3j and puberty EpH ¼ ðð1�κÞ=κÞ½EG�L3p . This isbased on k¼ _kJ= _kM ¼ 1, where maturity density does notchange, see Kooijman (2010, Eq. (2.32)). The exact values canbe obtained by integrating the ode's for reserve E and maturityEH over structural length L. For that purpose the initial reserveEð0Þ ¼ E0 is obtained by step D9 of the previous subsection,while EHð0Þ ¼ 0. The value for κ can be used as an initial value

for a numerical procedure to solve κ from the exact value for_Rm, repeating steps D11 till D15 till conversion;

D16 ageing acceleration €ha ¼ 4:27=a3m_kMg, for small Gompertz

stress coefficient sG ¼ 10�4. This is based on am ¼Γð4=3Þ=_hW , with _h

3W ¼ €ha _v=6Lm and Γð4=3Þ ¼ 0:893, see Kooijman

(2010, Eq. (6.6)).

This algorithm has been coded in function get_pars_9 of DEBtool.

Appendix D. The boundaries of the parameter space

The following formal constraints on parameter values apply atconstant food at scaled functional response f, with 0o f r1.

BP1 All parameters, i.e. f _pAmg, _v, κ, ½ _pM�, ½EG�, EHb , EHj , EHp , €ha, must bepositive. The specific costs for structure ½EG�4dVμV=wV fol-low from κGo1. Constraint BP6 is more restrictive thanconstraint BP1 for κ, however.

BP2 Allocation fraction κ must be smaller than 1.BP3 The maturity levels must increase: 0oEbHrEjHrEpH .BP4 Scaled length at birth lb cannot exceed 1. For given energy

investment ratio g and maintenance ratio k, scaled length atbirth lb increases with scaled maturity at birth vH

b . Themaximum value for lb equals 1 for eb¼1, so a maximum valuefor vH

b exists. To find this value, we rewrite (Kooijman, 2010,Eq. (2.28)) as

ddτ

vH ¼ uEl2 gþ l

uEþ l3�kvH

and remove scaled time be considering

dduE

l¼ �1

3uEl2

guE� l4

gþ l;

dduE

vH ¼ kvHuEl

2

uEþ l3

gþ l�1

For eb¼1 and lb¼1, we have xb ¼ g=ð1þgÞ and αb ¼ 3gx1=3b .Moreover u0

E ¼ ð3g=ðαb�Bxb ð4=3;0ÞÞÞ3 and ubE ¼ 1=g. This set of

2 ODE's should now be integrated for uE from uE0 till uEb, where

lðu0EÞ ¼ ϵ and vHðu0

EÞ ¼ ϵ3 for very small ϵ, e.g. some 10�4. Weshould test that lðub

EÞ ¼ lb ¼ 1, since the numerical integrationis not trivial in this case.

BP5 Supply stress ss ¼ _kJEpH ½ _pM�2=f 3s3Mf _pAmg3r22

33, else allocationfraction κ cannot be between 0 and 1 (see BP1 and BP2).

BP6 Allocation fraction κ must be between the two positive rootsof κ2ð1�κÞ ¼ ss. If κ is at one of the boundaries, maturity atpuberty is only reached asymptotically, maximum reproduc-tion _Rm ¼ 0. If ss ¼ 22

33, the two positive roots coincide and we

have κ ¼ 23.

BP7 The constraint apoam (see BD1) translates to €haoΓð43Þð6=g _kMa3pÞ. The detailed argument is a bit more complexbecause death by ageing is stochastic and not all individualsneed to reach puberty.

These boundaries have been coded in function filter_pars_9 ofDEBtool.

Appendix E. The boundaries of the data space

The following formal constraints on data values apply atconstant food at scaled functional response f, with 0o f r1.

BD1 All data, i.e. dV, ab, ap, am, Wb, Wj, Wp, Wm, _Rm, must bepositive.

BD2 Ages must increase during the life-cycle: 0oabrapoam.BD3 Weights must increase during the life-cycle: 0oWbr

WjrWprW1.BD4 A solution for _kM from D8 must exist, which translates into

ab _rBo lim_kM↓3_rB

Z xb

0

dxð1�xÞx2=3 αb�Bxb

43;0� �þBx

43;0� �� �

BD5 Puberty can be reached if maximum reproduction _Rm40.Allocation fraction κ only has a solution if for _pm

M ¼ s3ML3m½ _pM�and _pm

J ¼ _kJEpH and _pm

R ¼ E0 _Rm=κR

_pmR o _pm

M1�κκ

� _pmJ for κ-0

The quantities sM, _Rm, ½ _pM �, _kJ , Lm, EHp and uE0 are treated hereas functions of data (see algorithm of the bijection) andE0 ¼ u0

EL3m½EG�=κ. So for small κ the condition reduces to

u0E ½EG� _Rm=κRos3M½ _pM�� _pm

J =L3m

Step D10 of the map from data to parameters shows that_RmtEml

3bð1:75þgÞoκRs3M follows naturally from the

K. Lika et al. / Journal of Theoretical Biology 354 (2014) 35–4746

approximative estimate for κ, but is an approximative con-straint only.

These boundaries have been coded in function filter_data_9 ofDEBtool.

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Web references

Information about the DEB research program and its results can be found at⟨http://www.bio.vu.nl/thb/deb/⟩; the software package DEBtool that has beenused at ⟨http://www.bio.vu.nl/thb/deb/deblab/debtool/⟩; the add_my_pet col-lection is presented at ⟨http://http://www.bio.vu.nl/thb/deb/deblab/add_my_pet/Species.html⟩; and its manual at ⟨http://www.bio.vu.nl/thb/deb/deblab/add_my_pet/add_my_pet.pdf⟩. The DEB notation and its rules are presentedat ⟨http://www.bio.vu.nl/thb/deb/deblab/Kooy2010_n.pdf⟩.

K. Lika et al. / Journal of Theoretical Biology 354 (2014) 35–47 47