Allometric exponents do not support a universal metabolic allometry

The debate about the value of the allometric scaling exponent (b) relating metabolic rate to body mass (metabolic rate = a x mass(b)) is ongoing, with published evidence both for and against a 3/4-power scaling law continuing to accumulate. However, this debate often revolves around a dichotomous distinction between the 3/4-power exponent predicted by recent models of nutrient distribution networks and a 2/3 exponent predicted by Euclidean surface-area-to-volume considerations. Such an approach does not allow for the possibility that there is no single "true" exponent. In the present study, we conduct a meta-analysis of 127 interspecific allometric exponents to determine whether there is a universal metabolic allometry or if there are systematic differences between taxa or between metabolic states. This analysis shows that the effect size of mass on metabolic rate is significantly heterogeneous and that, on average, the effect of mass on metabolic rate is stronger for endotherms than for ectotherms. Significant differences between scaling exponents were also identified between ectotherms and endotherms, as well as between metabolic states (e.g., rest, field, and exercise), a result that applies to b values estimated by ordinary least squares, reduced major axis, and phylogenetically correct regression models. The lack of support for a single exponent model suggests that there is no universal metabolic allometry and represents a significant challenge to any model that predicts only a single value of b.     

Testing the metabolic theory of ecology: allometric scaling exponents in mammals

Many fundamental traits of species measured at different levels of biological organization appear to scale as a power law to body mass (M) with exponents that are multiples of 1/4. Recent work has united these relationships in a "metabolic theory of ecology" (MTE) that explains the pervasiveness of quarter-power scaling by its dependence on basal metabolic rate (B), which scales as M(0.75). Central to the MTE is theory linking the observed -0.25 scaling of maximum population growth rate (rm) and body mass to the 0.75 scaling of metabolic rate and body mass via relationships with age at first reproduction (alpha) derived from a general growth model and demographic theory. We used this theory to derive two further predictions: that age at first reproduction should scale inversely to mass-corrected basal metabolic rate alpha infinity (B/M)(-l) such that rm infinity (B/M)1. We then used phylogenetic generalized least squares and model selection methods to test the predicted scaling relationships using data from 1197 mammalian species. There was a strong phylogenetic signal in these data, highlighting the need to account for phylogeny in allometric studies. The 95% confidence intervals included, or almost included, the scaling exponent predicted by MTE for B infinity M(0.75), rm infinity M(-0.25), and rm infinity alpha(-1), but not for alpha infinity M(0.25) or the two predictions that we generated. Our results highlight a mismatch between theory and observation and imply that the observed -0.25 scaling of maximum population growth rate and body mass does not arise via the mechanism proposed in the MTE.     

A universal approach to estimate biomass and carbon stock in tropical forests using generic allometric models

Allometric equations allow aboveground tree biomass and carbon stock to be estimated from tree size. The allometric scaling theory suggests the existence of a universal power-law relationship between tree biomass and tree diameter with a fixed scaling exponent close to 8/3. In addition, generic empirical models, like Chave's or Brown's models, have been proposed for tropical forests in America and Asia. These generic models have been used to estimate forest biomass and carbon worldwide. However, tree allometry depends on environmental and genetic factors that vary from region to region. Consequently, theoretical models that include too few ecological explicative variables or empirical generic models that have been calibrated at particular sites are unlikely to yield accurate tree biomass estimates at other sites. In this study, we based our analysis on a destructive sample of 481 trees in Madagascar spiny dry and moist forests characterized by a high rate of endemism (> 95%). We show that, among the available generic allometric models, Chave's model including diameter, height, and wood specific gravity as explicative variables for a particular forest type (dry, moist, or wet tropical forest) was the only one that gave accurate tree biomass estimates for Madagascar (R2 > 83%, bias < 6%), with estimates comparable to those obtained with regional allometric models. When biomass allometric models are not available for a given forest site, this result shows that a simple height-diameter allometry is needed to accurately estimate biomass and carbon stock from plot inventories.     

One size fits all: stability of metabolic scaling under warming and ocean acidification in echinoderms

Responses by marine species to ocean acidification (OA) have recently been shown to be modulated by external factors including temperature, food supply and salinity. However the role of a fundamental biological parameter relevant to all organisms, that of body size, in governing responses to multiple stressors has been almost entirely overlooked. Recent consensus suggests allometric scaling of metabolism with body size differs between species, the commonly cited ‘universal’ mass scaling exponent (b) of ¾ representing an average of exponents that naturally vary. One model, the Metabolic-Level Boundaries hypothesis, provides a testable prediction: that b will decrease within species under increasing temperature. However, no previous studies have examined how metabolic scaling may be directly affected by OA. We acclimated a wide body-mass range of three common NE Atlantic echinoderms (the sea star Asterias rubens, the brittlestars Ophiothrix fragilis and Amphiura filiformis) to two levels of pCO₂ and three temperatures, and metabolic rates were determined using closed-chamber respirometry. The results show that contrary to some models these echinoderm species possess a notable degree of stability in metabolic scaling under different abiotic conditions; the mass scaling exponent (b) varied in value between species, but not within species under different conditions. Additionally, we found no effect of OA on metabolic rates in any species. These data suggest responses to abiotic stressors are not modulated by body size in these species, as reflected in the stability of the metabolic scaling relationship. Such equivalence in response across ontogenetic size ranges has important implications for the stability of ecological food webs.     

The top-down mechanism for body-mass-abundance scaling

Scaling relationships between mean body masses and abundances of species in multitrophic communities continue to be a subject of intense research and debate. The top-down mechanism explored in this paper explains the frequently observed inverse linear relationship between body mass and abundance (i.e., constant biomass) in terms of a balancing of resource biomasses by behaviorally and evolutionarily adapting foragers, and the evolutionary response of resources to this foraging pressure. The mechanism is tested using an allometric, multitrophic community model with a complex food web structure. It is a statistical model describing the evolutionary and population dynamics of tens to hundreds of species in a uniform way. Particularities of the model are the detailed representation of the evolution and interaction of trophic traits to reproduce topological food web patterns, prey switching behavior modeled after experimental observations, and the evolutionary adaptation of attack rates. Model structure and design are discussed. For model states comparable to natural communities, we find that (1) the body-mass abundance scaling does not depend on the allometric scaling exponent of physiological rates in the form expected from the energetic equivalence rule or other bottom-up theories; (2) the scaling exponent of abundance as a function of body mass is approximately -1, independent of the allometric exponent for physiological rates assumed; (3) removal of top-down control destroys this pattern, and energetic equivalence is recovered. We conclude that the top-down mechanism is active in the model, and that it is a viable alternative to bottom-up mechanisms for controlling body-mass-abundance relations in natural communities.     

Scaling toxicity data across species

The response of various species to doses of chemicals can often give the impression that some (such as cattle in the case of molybdenum) are much more susceptible than others to these chemicals. These impressions usually rely on an underlying assumption that equivalent doses are based on mg of the chemical per kg body weight of the animal. That is, that doses scale as the first power of body weight. This assumption is more often wrong than right. When viewed in a more general way, where the scaling is proportional to a power of the body weight and the exponent determined empirically, it is often found that equivalent doses scale with an exponent in the range of 0.6 to 0.8. As a result, larger animals are indeed more susceptible to toxicity on a mg kg^–1 body weight basis, but this is not because of unique differences in the species, but only because of different body sizes. This method of scaling is called allometry or allometric scaling. An early version of this approach was based on body surface area where the exponent is 2/3. More recently, pharmacokinetics has revealed that the reason for the different response of larger animals is related to the slower metabolic and clearance rates for larger animals which give rise to larger biological half-lives for chemicals in the body and to higher tissue concentrations per given dose.     

Consequences of paternal care on pectoral fin allometry in a desert-dwelling fish

Positive static allometry is a scaling relationship where the relative size of traits covaries with adult body size. Traditionally, positive allometry is thought to result from either altered physiological requirements at larger body size or from strongly condition-dependent allocation under sexual selection. Yet, there are no theoretical reasons why positive allometry cannot evolve in fitness-related traits that are solely under the influence of natural selection. We investigated scaling and sexual dimorphism of a naturally selected trait, pectoral fin size, in comparison to a trait important in male–male combat, head width in natural populations of a fish, the desert goby Chlamydogobius eremius. Male desert gobies provide uniparental care and use their pectoral fins to fan the brood (often under hypoxic conditions); hence, larger fins are expected to be more efficient. Male pectoral fins do not appear to fulfil a signalling function in this species. We found that, for both pectoral fin size and head width, males exhibited positive allometric slopes and greater relative trait size (allometric elevation) than females. However, for head width, females also showed positive allometry, albeit to a lesser degree than males. Because fin locomotory function typically does not result in positive allometry, our findings indicate that other naturally selected uses, such as paternal care, can exaggerate trait scaling relationships.     

On estimating the exponent of power-law frequency distributions

Power-law frequency distributions characterize a wide array of natural phenomena. In ecology, biology, and many physical and social sciences, the exponents of these power laws are estimated to draw inference about the processes underlying the phenomenon, to test theoretical models, and to scale up from local observations to global patterns. Therefore, it is essential that these exponents be estimated accurately. Unfortunately, the binning-based methods traditionally used in ecology and other disciplines perform quite poorly. Here we discuss more sophisticated methods for fitting these exponents based on cumulative distribution functions and maximum likelihood estimation. We illustrate their superior performance at estimating known exponents and provide details on how and when ecologists should use them. Our results confirm that maximum likelihood estimation outperforms other methods in both accuracy and precision. Because of the use of biased statistical methods for estimating the exponent, the conclusions of several recently published papers should be revisited.     

Scaling mass and morphology in leaves: an extension of the WBE model

Recent advances in metabolic scaling theory have highlighted the importance of exchange surfaces and vascular network geometry in understanding the integration and scaling of whole-plant form and function. Additional work on leaf form and function has also highlighted general scaling relationships for many leaf traits. However, it is unclear if a common theoretical framework can reveal the general rules underlying much of the variation observed in scaling relationships at the whole-plant and leaf level. Here we present an extension of the general model introduced by G. B. West, J. H. Brown, and B. J. Enquist that has previously been applied to scaling phenomena for whole plants to predict scaling relationships in leaves. Specifically, the model shows how the exponents that describe the scaling of leaf surface area, length, and petiole diameter should change with increasing leaf mass (or with one another) and with variation in leaf dimensionality. The predictions of the model are tested and found to be in general agreement with a large data set of leaves collected from both temperate and arid sites. Our results demonstrate that a general model based on the scaling properties of biological distribution networks can also be successfully applied to understand the diversity of leaf form and function.     

Crown ratio influences allometric scaling in trees

Allometric theories suggest that the size and shape of organisms follow universal rules, with a tendency toward quarter-power scaling. In woody plants, however, structure is influenced by branch death and shedding, which leads to decreasing crown ratios, accumulation of heartwood, and stem and branch tapering. This paper examines the impacts on allometric scaling of these aspects, which so far have been largely ignored in the scaling theory. Tree structure is described in terms of active and disused pipes arranged as an infinite branching network in the crown, and as a tapering bundle of pipes below the crown. Importantly, crown ratio is allowed to vary independently of crown size, the size of the trunk relative to the crown deriving from empirical results that relate crown base diameter to breast height diameter through crown ratio. The model implies a scaling relationship in the crown which reduces to quarter-power scaling under restrictive assumptions but would generally yield a scaling exponent somewhat less than three-quarters. For the whole tree, the model predicts that scaling between woody mass and foliage depends on crown ratio. Measurements on three boreal tree species are consistent with the model predictions.     

Diversity–stability relationships revisited: scaling rules for biological communities near equilibrium

The debate on diversity–stability relationships has a long history of theoretical interest and plays a central role in development of modern ecology. But such debate has recently re-opened under critical scrutiny both in theoretical and empirical studies. In this paper we use allometric (or energetic) scaling and statistical physics approaches to this problem. On the basis of local Damuth symmetry, a Markov model of transfer of energy between different species, and the fluctuation–dissipation theorem, scaling rules of species number and population variability of biological communities near equilibrium are derived. These scaling rules indicate that the diversity–stability relationship may be an energetic and thermodynamic consequence of ecological systems near equilibrium, not a simple statistical consequence as derived by other recent theoretical work.     

Problems of the gonad index and what can be done: analysis of the purple sea urchin <Emphasis Type="Italic">Strongylocentrotus purpuratus</Emphasis>

The gonad index, GI, is widely used as a measure of changes in reproductive state. There are, however, problems with its use because it is based on the implicit assumption of an isometric relationship between gonad size and some measure of total size. If, for example, gonad weight and total weight are used, the exponent for an allometric relationship usually is ignored and hence assumed to be 1.0. It is further assumed that this exponent is fixed for all states of the reproductive cycle and that gonads begin to develop at size = 0. Data for the purple sea urchin Strongylocentrotus purpuratus at Gregory Point, Oregon, USA, gathered over a period of 31 months showed that these assumptions cannot be supported. The relationship is better modeled with a function that (1) takes into account size of initial gonad production and (2) allows allometric exponents that vary with site or season. Thus, a better approach is to use a wide range of sizes to estimate size when gonads begin to develop and then, with this correction, ANCOVA to test for differences of gonad size among samples. Gonad changes at Gregory Point were estimated using fixed sizes of 5 cm diameter and 60 g total weight. Publishing means for X and Y, the standard error of the estimate, R ², and slope for each regression are shown to be sufficient to compare our results with results across studies.     

Life history traits predict relative abundance in an assemblage of forest caterpillars

Species in a given trophic level occur in vastly unequal abundance, a pattern commonly documented but poorly explained for most taxa. Theoretical predictions of species density such as those arising from the metabolic theory of ecology hold well at large spatial and temporal scales but are not supported in many communities sampled at a relatively small scale. At these scales ecological factors may be more important than the inherent limits to energy use set by allometric scaling of mass. These factors include the amount of resources available, and the ability of individuals to convert these resources successfully into population growth. While previous studies have demonstrated the limits of macroecological theory in explaining local abundance, few studies have tested alternative generalized mechanisms determining abundance at the community scale. Using an assemblage of forest moth species found co-occurring as caterpillars on a single host plant species, we tested whether species abundance on that plant could be explained by mass allometry, intrinsic population growth, diet breadth, or some combination of these traits. We parameterized life history traits of the caterpillars in association with the host plant in both field and laboratory settings, so that the population growth estimate was specific to the plant on which abundance was measured. Using a generalized least-squares regression method incorporating phylogenetic relatedness, we found no relationship between abundance and mass but found that abundance was best explained by both intrinsic population growth rate and diet breadth. Species population growth potential was most affected by survivorship and larval development time on the host plant. Metabolic constraints may determine upper limits to local abundance levels for species, but local community abundance is strongly predicted by the potential for population increase and the resources available to that species in the environment.     

Theoretical analysis of the weight-length relationship in fish juveniles

The weight-length relationship in fish juveniles was investigated theoretically, to assess the significance of the allometric factor and the validity of the condition factor; these biological factors often remain undetermined, because most fishery studies have been conducted for commercial-sized and/or adult populations. The exponent b (allometric factor) seemed to be the main parameter, performing a key role in the equation W=aL ^b , where W=weight, a is a constant and L=length. The parameters a (condition factor) and K (ponderal index; K=10³ W/L ³) were judged to be less important in comparative studies, since these parameters were closely correlated with b. It is recommended that the assumed theoretical value of b=3 not be used in applied ichthyological surveys, since this value was rarely obtained in the studies, and since a much wider range is usually seen. These analyses led to a new working hypothesis — not yet verified — which opens a new approach to understanding the biological significance of the allometric factor. This approach involves the fractal theory (where b may be considered as a fractal dimension equivalent) linked to the theory of saltatory ontogeny [where b is a threshold characteristic in the (early) life history of fishes].     

Effects of temperature and body mass on metabolic rates of sprat, Sprattus sprattus L.

Sprat, Sprattus sprattus L., is a small schooling clupeid forming large stocks in several ecosystems. Despite its high trophodynamic impact, little is known about its energy consumption rates. As a central component of a bioenergetic budget, metabolic rates of sprat from 3.11 to 9.71 g wet weight (WW) were measured at nine different temperatures (T) ranging from 9 to 21°C using a computer-controlled intermittent-flow respirometer. Routine metabolism (R _R) was related to T (°C) and WW (g) by R _R = 0.074 WW^1.077 e^{0.080 T}. Standard metabolic rates (R _S) as calculated from the 10% percentiles of the repeated measurements were on average 12% lower and still influenced by continuous swimming activity: R _S = 0.069 WW^1.073 e^{0.078 T}. We interpret the deviation of the scaling exponent b from typically found exponents of b ~ 0.8 as a consequence of permanently elevated activity level. The high permanent swimming activities also indicated that the concept of standard metabolism may not be meaningful in schooling planktivorous fish. These results suggest that generally in bioenergetic models for clupeid schooling fish the activity multipliers should be chosen very conservatively.     

Oxygen consumption of the semi-terrestrial crab <Emphasis Type="Italic">Pachygrapsus marmoratus</Emphasis> in relation to body mass and temperature: an information theory approach

Pachygrapsus marmoratus is a semi-terrestrial crab and the most common grapsid crab in the intertidal belt of rocky shores throughout the Mediterranean Sea, Black Sea and northeastern Atlantic. In this study, the combined effects of temperature (T), body mass (M), and sex (S) on the routine oxygen consumption rate (R) in P. marmoratus were quantified. The blotted wet body mass of the specimens ranged between 43 mg and 18.0 g, and five test temperatures were used between 13.5 and 28.0°C. Six candidate models that reflected different assumptions regarding the dependence of R on S and T were compared. Model selection was based on Kullback–Leibler’s information theory and Akaike’s information criterion (AIC). The model had the highest support by the data (E is the activation energy, B = 8.618 × 10⁻⁵ eV K⁻¹ is Boltzmann’s constant, T _a is the absolute temperature in Kelvin, and b the allometric scaling exponent); for P. marmoratus it was found that No sex dependence of R was supported by the data. Following a multi-model inference (MMI) approach, the mean (± SE) allometric exponent was 0.750 (± 0.013) having a 95% (bootstrap) confidence interval of 0.726–0.774. Thus, it was established that P. marmoratus follows Kleiber’s 3/4 law, as seems to be generally true for intertidal crabs. The allometric exponent was independent of temperature as has also been reported for many other marine invertebrates (at normal temperatures). Q ₁₀ values were relatively low, indicating wide thermal tolerance of the species. Model selection based on information theory is recommended for respiration studies, as an effective method in finding a parsimonious approximating model. MMI by model averaging, based on Akaike weights, is an effective way to make robust parameter estimations and deal with model selection uncertainty.     

The metabolic costs of building ant colonies from variably sized subunits

Ant colonies are superorganisms with emergent traits that, for some species, reflect the combined activity of physically distinct worker castes. Although larger castes have high production costs, they are thought to save their colonies energy by efficiently performing specialized tasks. However, because workers are generally idle until sensing specific stimuli, their maintenance costs may be an important component of colony-level investment. I used metabolic scaling to examine the maintenance costs of dimorphic major and minor Pheidole castes across levels of colony organization (e.g., individual, group, and colony). Majors from three species had lower mass-specific metabolic rates than minors because of allometries at both individual and group levels and subsequently lived longer when starved. Thus, large major castes may offset their production costs in both their idle and active states. The slope scaling metabolic rate from incipient to reproductive colonies of Pheidole dentata (∼colony mass^0.89) fell between the slopes for minor groups (∼group mass^1.04) and major groups (∼group mass^0.79) and appears to reflect developmental shifts in subunit mass and number and their offsetting effects on per capita energy demands. These results highlight how metabolic scaling may help visualize the energetic correlates of emergent behavior and unravel the mechanisms governing colony organization.     

Physioecology of zooplankton. III. Effects of phytoplankton concentration,temperature, and body size on the metabolic rate ofCalanus pacificus

Weight-specific rates of oxygen consumption of actively feeding copepodite stages ofCalanus pacificus Brodsky were measured under various combination of phytoplankton concentration and temperature. The rate decreased logarithmically with a logarithmic increase in dry body weight of copepods, and the relationship between these variables was described using a log-transformed allometric equation. The body-size dependence of the metabolic rate was independent of changes in food concentration and temperature, but the metabolic level increased linearly with a logarithmic increase in temperature and was not significantly affected by changes in food concentration. Respiration rates measured in this study forC. pacificus were about twice as high as rates reported for unfed closely related species of the same genus. An analysis of the metabolic cost of feeding processes suggests that metabolic models derived from feeding models may be of little ecological value at present.Contribution No. 1129 from the Department of Oceanography, University of Washington, Seattle, Washington 98195, USA