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Opinion

Have We Outgrown the Existing Models ofGrowth?

Dustin J. Marshall1,* and Craig R. White1

HighlightsMost models of growth assume thatreproductive output scales proportio-nately with body size. However, arecent meta-analysis of fish contra-dicts this assumption, and shows thatreproduction scales hyperallometri-cally across a wide variety of taxa.

Hyperallometric reproduction repre-sents a profound challenge to mostmechanistic theories of growth.

We argue that growth slows notbecause of mechanistic constraintbut because of increasing allocation

Theories of growth have a long history in biology. Two major branches of theory(mechanistic and phenomenological) describe the dynamics of growth andexplain variation in the size of organisms. Both theory branches usually assumethat reproductive output scales proportionately with body size, in other wordsthat reproductive output is isometric. A meta-analysis of hundreds of marinefishes contradicts this assumption, larger mothers reproduce disproportion-ately more in 95% of species studied, and patterns in other taxa suggest thatreproductive hyperallometry is widespread. We argue here that reproductivehyperallometry represents a profound challenge to mechanistic theories ofgrowth in particular, and that they should be revised accordingly. We suspectthat hyperallometric reproduction drives growth trajectories in ways that arelargely unanticipated by current theories.

to reproduction, and that many modelsof growth have been seeking to under-stand the wrong phenomenon.

Resolving these issues has majorimplications for how we manage andharvest species, and predict theimpacts of global change on the pro-ductivity of biological systems.

1Centre for Geometric Biology/Schoolof Biological Sciences, MonashUniversity, Melbourne, Australia

*Correspondence:dustin.marshall@monash.edu(D.J. Marshall).

Do We Understand Body Size As Well As We Think?Body mass is a universal feature of all organisms, varies across 21 orders of magnitude, andhas fascinated biologists for centuries [1]. Body mass covaries with most of the key physiologi-cal, ecological, and evolutionary parameters. For example, larger organisms have lower mass-specific metabolic rates and tend to live for longer than smaller organisms [1]. Larger organismshave lower intrinsic rates of increase and population sizes, smaller effective population sizes,and lower rates of evolution than tiny organisms [1]. Body mass shows conspicuous macro-ecological patterns, covarying with latitude, temperature, and mode of life [1]. Despite itsimportance and its commonality to all animals, we argue that we still have a remarkably poorunderstanding of the drivers of body size. In this opinion article, we briefly describe the twomajor branches of theory concerning growth, their similarities and differences, and discuss howa recent empirical discovery challenges the core assumptions of both.

Understanding Growth and ReproductionBottom-Up (Mechanistic) ApproachesTheoretical attempts to understand ontogenetic changes in body size (growth) have a longhistory [2–14], and debate continues to rage about the mechanistic validity and utility of theoriesof metazoan growth [15–22]. The most venerable of these theories was proposed by Pütter [2],but is now most commonly associated with von Bertalanffy [3,4,23,24]. The von Bertalanffygrowth function (VBGF, see Glossary) estimates the rate of increase in mass (growth) as thedifference between rates of ‘anabolism’ and ‘catabolism’, where the rate of catabolism isproportional to mass m, and the rate of anabolism is proportional to mb, where b is positive andless than 1. Although von Bertalanffy considered a range of potential values for b [4], a value of2/3 is the most common based on the assumption that synthesis is proportional to the uptakeof resources over a resorbing surface [3]. The mechanistic interpretation of the VBGF hasshifted through time, and it has since been variously restated as representing the difference

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GlossaryDynamic energy budget (DEB)theory: a mechanistic model ofgrowth to quantitatively describemass and energy budgets at thelevel of organisms. It has numerousassumptions about energy uptake,storage, and utilisation. Similarly toVBGF, growth slows down in thisframework because of energydynamics rather than because ofincreasing allocation to reproduction.Hyperallometric reproductivescaling: an empirical relationshipbetween reproductive energy outputand body size (mass) for matureindividuals that has a scalingexponent greater than 1.0. Ifreproduction is hyperallometric, thenlarger individuals reproducedisproportionately more than smallerindividuals.Integral projection models: a classof demographic population modelsthat allow for continuous stagestructure and heterogeneity amongindividuals. Life-history variables arelinked to vital rate functions toproduce demographic parameters forpopulations or individual phenotypes.Isometric reproductive scaling: anempirical relationship betweenreproductive energy output and bodysize (mass) for mature individuals thathas a scaling exponent that is 1.0. Ifreproduction is isometric, then largerindividuals have the same relativereproductive energy output to smallerindividuals. Note that modelsassuming isometric allocation to

between rates of energy assimilation and expenditure (e.g., [25,26]), or the difference betweenthe rates of energy use for metabolism (including growth) and maintenance (e.g., [27]). Theseconsiderations generate the impression of a sound theoretical foundation for the VBGF. Oddly,however, the VBGF does not consider reproduction at all. The exclusion or minimisation of therole of reproduction in mechanistic models of growth continued well beyond von Bertalanffyand into extant models today.

The mechanistic framework used in VBGF to understand the slowing of growth to a final orasymptotic size started a school of thought that focused on ‘bottom-up’ drivers of growth andbody size. Under this framework, body size is the product of proximal, mechanistic constraintson resource supply and demand. Variations of this approach have proliferated ever since, andinclude well-known frameworks such as dynamic energy budget (DEB) theory [6], theontogenetic growth model (OGM) [7,9,12], and Pauly’s limiting gill surface model(LGSM) [11]. Although each has different foci and emphases, each of these models essentiallyassumes that the size-dependence of resource (energy or oxygen) acquisition is shallower thanthe size-dependence of resource use, such that these two relationships eventually convergeand growth ceases (Box 1). In other words, growth slows or ceases at some asymptotic sizebecause the organism is no longer able to acquire [3,4,6,11,23], distribute [7,9,12], or use[3,23] resources faster than it expends them on self-maintenance. VBGF, DEB, OGM, LGSMand other mechanistic models often do an excellent job of describing the slowing of growth ofmost organisms as they approach a final or asymptotic size (Box 1).

As was pointed out over 20 years ago [28], despite their fit to the data, the mechanisticinferences derived from bottom-up models are problematic because such models fail toadequately consider reproduction. Reproduction is costly, both in terms of mortality riskand energy – reproductive tissues can represent up to 75% of body mass in highly fecundinvertebrates [29]. The onset of reproduction therefore involves very different mechanisticdynamics than the preceding juvenile phase as resources are shunted from growth to repro-duction [28]. Most mechanistic models make a simple but absolutely crucial assumption: thatreproduction is proportionate to body size – in other words, reproduction energy output scalesisometrically with size. These models assume that allocation to reproduction occurs from birthand remains a constant fraction of total body size throughout ontogeny (Table S1 in the

reproduction generally apply acrossthe entire life history, and not onlypost-maturity.Mechanistic models of growth:there is no absolute definition ofwhat constitutes a mechanisticversus phenomenological model ofgrowth. For our purposes, we definemechanistic models of growth asthose that parameterise modelsbased on explicit physiologicalmechanisms and focus on statevariables that alter physiology (e.g.,temperature) rather than affectingselection (e.g., mortality risk) on bodysize.Ontogenetic growth model(OGM): a mechanistic model ofgrowth based on assumptions aboutthe drivers of metabolic scaling andconstraints on resource distribution.The model assumes that allocation

Box 1. Models Fit Growth Trajectories Well, Even If Their Assumptions Are Not Met

The von Bertalanffy growth function (VBGF; Figure I, i) [4] and the ontogenetic growth model (OGM; Figure I, ii) [12] bothdescribe energy allocation to growth (the shaded area in Figure I) as the difference between resource use and eitherresource uptake in the case of VBGF or resource distribution in the case of OGM. Both assume that resource use scalesisometrically and that uptake or distribution scales hypoallometrically (2/3 in the case of VBGF and 3/4 in the case of OGM).Neither VBGF or OGM consider reproduction, which has recently been shown to scale hyperallometrically with b > 1.0for fish [38]. We propose here an alternative and very simple growth model (iii) that incorporates hypoallometric scaling ofboth energy intake and expenditure, as well as hyperallometric scaling of reproductive output (all of which are wellsupported by data). We assume that total production rate is proportional to rates of energy acquisition and use, and thatthe rate of energy allocation to growth (the shaded area in Figure I) is the difference between total production rate and therate of energy allocation to reproduction, which scales hyperallometrically. To illustrate that this model provides adescription of growth patterns that is at least as good as the VBGF and OGM, we illustrate this ‘hyperallometricreproduction’ model using data for cod Gadus morhua (iv; see online supplemental materials and Figure S2 for a moredetailed description of the modeling approach and underlying data). Non-linear least-squares fits for the integrals of theVBGF (orange) and OGM (blue) to weight-for-age data for cod are shown in (iv) and are colored to match (i) and (ii); thesefits overlap with the nearly identical (r2 > 0.995 in all cases) fit for the hyperallometric reproduction model, shown in red in(iv). Importantly, only the hyperallometric model allows for hyperallometric reproduction. (iv) Grand mean � SEM ofannual means of weight-for-age of wild cod in the northeastern Scotian Shelf from commercial landings from 1971–2001 [60]. Illustration credit: Diane Rome Peebles.

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to reproduction is a constantproportion of size, begins at birth,and remains unchanged byontogeny.Pauly’s limiting gill surface model(LGSM): a mechanistic model ofgrowth for aquatic organisms,particularly fish, in which theacquisition of oxygen is limited by gillsurfaces. The model assumes that,because the ability to acquire oxygenby gills scales with size less steeplythan oxygen demands scale withsize, growth slows as largerindividuals approach theirphysiological limits. Reproductionplays no role in the slowing ofgrowth.Phenomenological models ofgrowth: it is impossible to definitivelyargue that one model is mechanisticand another is phenomenological.We define phenomenological modelsas those that focus on how externalstate variables alter selection on lifehistories, and how life histories areoptimised given assumed constraintsand trade-offs.Reproductive energy output: thetotal reproductive output(number � size � energy content ofoffspring) by individual per unit time.von Bertalanffy growth function(VBGF): a model that describes thegrowth of an organism with theunderlying mechanistic assumptionthat growth slows down because therate at which resources are acquiredcannot keep pace with the rate atwhich resources are required.Reproduction is assumed to play norole in growth dynamics.

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supplemental information online) that is unchanged before and after reproductive maturity. Aswe shall show later, such assumptions are unrealistic and contradicted by data. Note that wehave not included a discussion of information-based models of growth, but these models alsotend to disregard the costs of reproduction.

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Top-Down (Phenomenological) ApproachesLife-history theory assumes that organisms evolve to maximise their reproductive output, andthat life-history strategies are the product of optimising various trade-offs among manycompeting traits for the allocation of key limiting resources [28,30]. In contrast to mechanisticapproaches, life-history theory tends to make simplifying assumptions about how the limitingresources are acquired or can be deployed. For example, such theories might assume that theproduction of new tissue must scale with body mass [28,31], or that resource acquisition scaleswith body size in particular ways [32], while remaining agnostic as to why such scaling occurs.Phenomenological, life-history models assume that the external context of the organism drivesthe optimisation of growth and reproduction (and hence size) within the constraints of assumedtrade-offs [33]. Thus, size at maturity is predicted to be inversely related to mortality rate under asimple life-history model [30]. From this perspective, life-history models are ‘top-down’ – theyare driven by selection on body size, and the underlying physiology evolves in response to theseselection pressures. Such phenomenological models of growth are good at predictingchanges in size, based on variation in external context across populations or between closelyrelated species (e.g., [34,35]).

Phenomenological and mechanistic models are the two major branches of theory that considergrowth and body size, but there have been several attempts to combine them. One group,broadly classed as biphasic models (reviewed in [36]), considers the mechanistic drivers ofgrowth before and after maturity separately. Similarly, links between DEB theory and integralprojection models consider the demographic consequences of different physiologies [37].These hybrid approaches seek to maximise their relevance and explanatory potential bydrawing the different strengths of both the mechanistic and phenomenological schools ofthought about how and why organisms grow. Thus, models of growth occur along a contin-uum, from exclusively mechanistic to exclusively phenomenological, but most tend to makeassumptions about the costliness of reproduction.

Most Mechanistic and Life-History Models of Growth Assume ReproductiveIsometryAs the above makes clear, there is tremendous diversity in the theories we use to understandbody size. Despite this diversity, most models assume that reproductive output scales directlyand proportionally to size (Table S1).

Of the major branches of theory, the mechanistic models in particular tend to have a coreassumption that the fraction of energy allocated to reproduction remains unchanged frombirth to death. Under these models, growth slows because resources become more limited,not because more resources are directed to reproduction. Were these models to haveincreasing allocation to reproduction, net resource flux would become negative and theorganism should shrink. Phenomenological and biphasic models tend to have differentdynamics for growth before and after maturity. Before maturity, growth in these models isunaffected by reproduction because it is either excluded or the costs are insignificant. Aftermaturity, increasing allocation to reproduction reduces growth (Table S1). Phenomenologicalmodels typically allow reproductive scaling to emerge from the model rather than assuming itsform a priori and a wide array of reproductive scaling exponents have been predicted –

spanning hypoallometry, isometry, and hyperallometry. Two questions therefore seem to beworth asking. (i) How reasonable is an assumption of isometric reproductive output (iso-metric reproductive scaling)? (ii) If reproduction is not isometric, how does this alter ourunderstanding of growth?

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Reproductive Hyperallometry Is Probably CommonWe define reproductive hyperallometry as the intraspecific relationship between reproductiveenergy output per unit time and body size for mature individuals experiencing the sameconditions. Such a definition best matches theories of growth. A recent study showed that, for342 species of fish across 15 orders, fecundity scales hyperallometrically for most [38]. For95% of fish species in the study, larger mothers have disproportionately higher reproductiveoutput via fecundity effects, offspring size effects, or both. For marine fish it therefore seemsthat hyperallometric reproduction is the rule, but does it occur more generally?

Initial indications are that hyperallometric reproduction is not restricted to marine fishes(Figure 1). A compilation of weight–fecundity relationships for 23 species of freshwater fishfound a mean exponent of 1.08 [39]. Mass–fecundity scaling exponents are often greater than1.0 for invertebrates from a range of phyla (mean � S.E. for mussels, b = 1.42 � 0.17; seaurchin, 1.224 � 0.048 [40]). For 20 species of crab, a range of scaling exponents have beenreported, ranging from �1.0 to 1.4, with most sitting above 1.0 [41]. Some taxa show relativelyextreme hyperallometry; for example, four species of congeneric bugs show scaling exponentsof mass and fecundity between 5.1 and 7.8 [42]. Importantly, the studies described here, otherthan Barneche et al. [38], only examine the scaling of fecundity. The true costs of reproductionare the number and the energy content of offspring, and estimates of reproductive scalingshould ideally include both. Estimates of offspring energy content within species are rare, but,for marine fish at least, both fecundity and per offspring energy content scale hyperallometri-cally with body size [38]. We are not aware of any other compilations of how offspring energycontent scales with body size within species, but Lim et al. [43] show that larger mothers tend toproduce larger offspring (with presumably more energy) across a wide variety of taxa. It seemslikely therefore that estimates of fecundity should estimate the lower bound of reproductivescaling.

We suspect that reproductive hyperallometry is the rule for most taxa, but it has simply beenoverlooked. Extensive and formal meta-analyses will be necessary to confirm or contradict thissuspicion. We anticipate that there will some species with isometric or even hypoallometricreproduction, and determining patterns and drivers of deviations from hyperallometry will be aninteresting challenge. At this point it seems reasonable to suggest that hyperallometric repro-duction is the rule in a major taxonomic group (marine fish) and, at the very least, also occurs in awide range of other taxa. Based on these patterns of reproduction, we would argue thattheories of growth should be modified.

What Does Reproductive Hyperallometry Mean for Theory?Mechanistic models of growth are particularly vulnerable to the violation of the assumption ofreproductive isometry. As Roff [44] shows, deviations from isometric reproduction profoundlyalter the predictions of growth models: increasing allocation to reproduction slows growth morerapidly than many models predict, leading to negative growth or shrinkage (sensu [44]). The firstversion of the OGM [7] explicitly assumes that reproductive allocation is isometric and that theenergy density of eggs stays constant across ontogeny (an additional assumption contradictedby [38]). Under these assumptions, OGM predicts that reproduction should not qualitativelyaffect the trajectory of growth, instead it should simply lower asymptotic size. Incorporatingboth an increasing allocation of body mass to reproduction and changes in energy density ofeggs means that such equations (equation 5 in their paper) no longer apply. Instead, growthwould be predicted to be negative at larger body sizes. Subsequent updates to the OGM model[9] make no mention of reproduction, and we therefore cannot assess the impacts of incorpo-rating reproductive hyperallometry into these. Charnov et al. [45] retain the assumptions of the

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Figure 1. Relationship between Body Mass (Relative to Largest Individual Measured for That Population)and Relative Batch Fecundity (Relative to the Most Fecund Individual for That Population). (A) For three marineinvertebrates (Mytilus edulis shown in green and Strongylocentrotus droebachiensis in light blue, both from [40], andPocillopora damicornis in dark blue, from [57]). (B) for three terrestrial animals (Gopherus agassizii in yellow, from [58], Ranatemporaria in purple, from [59], and Nephotettix virescens in orange, from [42]). In all cases, solid lines depict hyperallo-metric fit based on data; dotted lines show the often-assumed isometric relationship.

original OGM with regards to isometric reproduction and, again, the conclusions of this modelmust change when reproductive hyperallometry is included. DEB models allow for reproductivehyperallometry, and some explicitly predict it [46], but it has been argued strongly that a fixedallocation to reproduction (reproductive isometry) is the best assumption [47].

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It seems that many of the influential ‘bottom-up’ models of growth need revising in light of thefinding that reproduction is often hyperallometric. These models had a decreasing relativeresource supply function ‘hardwired’ into them such that reproduction costs must stayconstant or organisms would shrink once they start to reproduce. We would argue that thesemodels are trying to explain dynamics that are driven by increasing allocation to reproduction(Box 1), but they do not allow for it. Whether these models can incorporate reproductivehyperallometry while maintaining other realistic assumptions is unclear.

Some life-history models are more amenable to reproductive hyperallometry, indeed someeven require or assume hyperallometry (Table S1). For example, Mangel assumes hyper-allometry, and ‘reverse engineers’ a prediction of natural mortality in the field, providing newinsights on what has long been thought of as a black box of aquatic ecology (Table S1). Othermodels predict reproductive hyperallometry. For example, a model by Audzijonyte andRichards predicts that reproductive hyperallometry emerges from reproduction becomingrelatively less costly at larger sizes (Table S1). Models by Kozlowski and coworkers, somebiphasic models, and the pioneering work of Gadgil and Bossert (who presciently predictedfrom their model that reproduction should be hyperallometric in fish specifically) all have someparameter space in which hyperallometric reproductive scaling should be favoured(Table S1). We believe that the models of Kozlowski in particular have not received therecognition they deserve – these models anticipated both hyperallometric reproduction andthe evolutionary impacts of intense harvesting in fish [35].

Even life-history models often assume that relative scope for reproduction decreases withlarger body sizes. The hyperallometry predicted by these models emerges from increasedallocation of an ever-decreasing amount (relative to size) of resource to reproduction. Moregenerally, the assumption that relative scope for production decreases with size is perhaps theone assumption common to both life-history and mechanistic models of growth (Table S1).Eighteen years ago, Kozlowski [48] and other since [49] have noted the importance of thisassumption for driving the outcomes of a range of models of growth. Given how crucial andubiquitous this assumption is, it is remarkable how little empirical scrutiny it has received.

The Scaling of Resource Acquisition – An Underexplored ParameterRelatively few studies estimate the scaling of resource acquisition with body size. Peters [1], inhis excellent book, concluded that energy acquisition scales at the roughly same exponent asmetabolic rate based on among-species comparisons. If the exponents for acquisition andexpenditure are the same, the net scope for production should increase with mass in anabsolute sense (contradicting most mechanistic theories of growth) but decrease with mass ina relative sense (if both energy acquisition and expenditure use scale at 0.75, then net scope forproduction also scales with an exponent of 0.75). Thus, Kozlowski and colleagues [31,50]reasonably followed the conclusion of Peters in assuming scope for production scales hypo-allometrically. Interestingly however, Peters’ actual data show that energy acquisition scalesslightly higher than energy use (exponents of 0.82 and 0.76, respectively) such that the netscope for production should actually increase in both an absolute and a relative sense (netscope for production should scale a M1.16). However using interspecific scaling exponents toinfer intraspecific scaling exponents is problematic, and intraspecific comparisons are moreinformative and direct.

Unfortunately, studies of the intraspecific scaling of ingestion with size remain surprisingly rare.Reiss [51] reports scaling exponents of energy ingestion of between 0.63 and 0.89 for 10ectotherms, and for some of these species the exponent for ingestion exceeds the exponent for

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energy use. Based on this albeit limited number of studies, we would argue that the long-heldand highly influential assumptions that available resource flux decreases either absolutely(in mechanistic models) or relatively (in life-history models) with body size require formalre-examination. Such empirical studies exist (e.g., [52,53]) but are rare. Estimating this param-eter may also help to resolve unrelated controversies among theoreticians regarding the scalingof ingestion more generally [54,55]. In the meantime, it seems more reasonable to assume thatresource intake has a scaling exponent that is at least as high as the metabolic scalingexponent.

Making the simple assumption that both resource acquisition and use have the same exponent,and combining these parameters with hyperallometric scaling of reproductive output, it isremarkable how well growth trajectories can be predicted. Box 1 shows that combining theseparameters fits the data for cod Gadus morhua at least as well as the VBGF [3,4,23,24] andOGM [7,9,12]. Table S1 shows that the same is true for an additional 12 species of fish,employing metabolic rate and reproductive scaling exponents taken from published compi-lations [38,56]. It seems, to us at least, that the growth dynamics that biologists have longsought to understand emerge simply from hyperallometric reproductive scaling.

Concluding Remarks: Where to from Here?Three important issues for further exploration emerge from this discussion. First, the ubiquity ofhyperallometric fecundity scaling should be explored for a broader range of taxa (as well as thescaling of other components of reproductive energy output; Box 2). Second, models across themechanistic and phenomenological spectrum should be revised to better accommodatehyperallometric reproductive scaling. Third, estimates of the size scaling of energy acquisitionand energy use for the same individuals under the same conditions will be necessary tounderstand how scope for production scales with size.

The relationship between body size and reproductive output and the drivers of this relationshipare important to resolve, both from fundamental and applied perspectives. Fisheries modelsmake strong and explicit assumptions about reproductive isometry and the scaling of energyintake. For example, ‘balanced harvesting’ practices are predicated on net production beingrelatively higher in smaller fish than larger fish [38]. Global climate change is driving down thebody sizes across a wide variety of taxa – the impacts of these declines on populationproductivity depend in part on how reproduction scales with body size [21,38]. We have

Box 2. Why Does the Energy Density of Offspring Decrease with Maternal Size?

In addition to showing that fecundity scales hyperallometrically with size, Barneche et al. [38] revealed systematicpatterns in egg energy content across mothers of different sizes: larger mothers produced larger, but less energy-denseeggs than did smaller mothers. Overall, the relationship between maternal size and offspring size is steeper than therelationship between maternal size and offspring energy density. Larger mothers thus produce offspring that each havemore energy in them in total. We have recognised that larger mothers produce larger offspring across a wide variety oftaxa, with diverse theories to explain such patterns [61]. However, the finding that larger mothers produce larger andless energy-dense offspring is novel and, as far as we are aware, is unanticipated by theory. This finding directlycontradicts the explicit assumptions of a suite of biphasic models and mechanistic theories of growth which assumeconstant energy density of reproductive components across the life cycle (Table S1). Furthermore, the majority of otherlife-history and mechanistic models of growth are silent with regards to ontogenetic changes in energy density ofreproductive components, and thus implicitly assume that energy density is constant. It is difficult to explain thissystematic and widespread change in energy composition of eggs across females of different sizes. A recent studyshowed larger eggs are more efficient at developing to the feeding stage [62] – perhaps because this increasedefficiency allows larger eggs to be less energy dense. Future studies should determine whether other taxa showcovariance between size and the energy density of offspring, and new theory should be developed to account for thiscovariance.

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Outstanding QuestionsHow ubiquitous is hyperallometricreproduction across organisms?

Does egg energy density scale withfemale size in taxa other than fish?

How should models of growth changewhen reproductive hyperallometry isincluded?

How does net production scale withbody size?

argued here that the longstanding and pervasive assumption of reproductive isometry is nolonger supported by the data in a major group of animals and for a wide variety of other taxa. Wesuggest that this should cause us to re-evaluate how we model organismal growth(see Outstanding Questions).

AcknowledgmentsThis work was supported in part by the Monash University Centre for Geometric Biology. We thank Lesley Alton and

Candice Bywater for compiling the weight-for-age data for fish. An anonymous reviewer and Asta Audzijonyte provided

extremely thoughtful comments that greatly improved the manuscript. Belinda Comerford assisted with document

preparation.

Supplemental InformationSupplemental information associated with this article can be found online at https://doi.org/10.1016/j.tree.2018.10.005.

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