Testing the energetic equivalence rule with helminth endoparasites of vertebrates

As a general test of the energetic equivalence rule, we examined macroecological relationships among abundance, density and host body mass in a comparative analysis of the assemblages of trophically transmitted endoparasitic helminths of 131 species of vertebrate hosts. Both the numbers and total volume of parasites per gram of host decreased allometrically with host body mass, with slopes roughly consistent with those expected from the allometric relationship between host basal metabolic rate and body mass. From an evolutionary perspective, large body size may therefore allow hosts to escape from the deleterious effects of parasitism.     

Discontinuities and alternative scalings in the density–mass relationship of anuran larvae

Size–density relationships (SDRs) frequently follow a power-law relationship, with exponents that compensate for the increase in population-level metabolic demand—the energetic equivalence rule. However, these exponents present a range of values, and elucidating its methodological and biological determinants has become a main issue. So far, a restricted set of potential relationships, mechanisms, and taxa have been considered. Here, we analyzed SDR in a population of tadpoles inhabiting a network of 16 temporal ponds. Alternative scaling regimes were detected using pacewise regressions and estimating exponents with maximum likelihood (ML). If discontinuities in the SDR are ignored, a scaling close to values reported elsewhere is observed. However, estimated slopes between discontinuities are steeper (threefold to fivefold) than those often reported, but congruent with the performance predicted for ML and the biases reported for other methods. Our estimations largely deviate from an energetic equivalence, suggesting that large individuals use less energy per unit area. The detection of different SDRs in the same database, with a strong decay in abundance with body size, points to a pattern poorly considered in previous studies, widening the range of patterns, mechanisms, and ecological, or evolutionary consequences of the SDRs.     

Body mass, population density, and offspring number in mammals

The negative relationship between population density and body mass with the body mass exponent of -0.75 implies that the energy flow through populations of small- and large-bodied species is the same, for individual metabolism scales to body mass raised to the power of +0.75. This relationship called the energetic equivalence rule, has often been observed for mammal species assemblages studied at regional scales. Here we suggest a demography-based mechanism that may generate it. Having analyzed about 130 literature sources, mostly in Russian, we collected demography and body-mass data for 88 mammalian species from the territory and coastal waters of the former Soviet Union. The data were used to construct a number of interspecific relationships. It is shown that (1) the number of offspring per lifetime is approximately inversely proportional to the relative mass at birth (the exponent is not significantly different from -1), (2) the average lifespan is proportional to body mass to the 0.25 power, (3) body mass at birth is proportional to the adult body mass. We develop a simple theory to demonstrate that relations (1) to (3) entail the energetic equivalence rule. The theory also allows us to explain violation of this rule (in non-flying birds, for example), namely, to predict the exponent of relation (1) for any given exponent of the relation between population density and body mass. This is possible because relations (2) and (3) are likely to more universally hold than relation (1). Finally, since natural selection acts on individual traits rather than on population-level ones such as population density, the theory opens up the way to an evolutionary explanation for the energetic equivalence rule.     

WBE 模型及其在生态学中的应用：研究概述

介绍了WBE模型,综述了该模型在生态学中的应用进展。WBE模型,以及以该模型为基础的MTE模型,假设生物体为自相似分形网络结构,提出代谢速率和个体大小之间存在3/4指数关系,分别预测了从个体到生物圈多个尺度上的生物属性之间的异速生长关系,而且部分得到了验证。WBE模型的应用涵盖了个体组织生物量、年生长率,种群密度和生态系统单位面积产量、能量流动率等多个方面;即使在生物圈大尺度上,WBE模型也可用来预测试验中无法直接测量的特征变量的属性,如全球碳储量的估算等。至今,关于WBE和MTE模型仍然存在各种褒贬争论,讨论焦点主要集中于模型建立的前提假设以及权度指数的预测。今后的研究工作应规范试验技术和方法,考虑物种多样性和环境等因素的影响,提出符合各类生物的模型结构体系,使其具有更广泛的应用性和预测性。     

Allometric scaling of intraspecific space use

Allometric scaling relationships enable exploration of animal space-use patterns, yet interspecific studies cannot address many of the underlying mechanisms. We present the first intraspecific study of home range (HR) allometry relative to energetic requirements over several orders of magnitude of body mass, using as a model the predatory fish, pike Esox lucius. Analogous with interspecific studies, we show that space use increases more rapidly with mass (exponent = 1.08) than metabolic scaling theories predict. Our results support a theory that suggests increasing HR overlap with body mass explains many of these differences in allometric scaling of HR size. We conclude that, on a population scale, HR size and energetic requirement scale allometrically, but with different exponents.     

Energetic inequivalence in eusocial insect colonies

The energetic equivalence rule states that population-level metabolic rate is independent of average body size. This rule has been both supported and refuted by allometric studies of abundance and individual metabolic rate, but no study, to my knowledge, has tested the rule with direct measurements of whole-population metabolic rate. Here, I find a positive scaling of whole-colony metabolic rate with body size for eusocial insects. Individual metabolic rates in these colonies scaled with body size more steeply than expected from laboratory studies on insects, while population size was independent of body size. Using consumer-resource models, I suggest that the colony-level metabolic rate scaling observed here may arise from a change in the scaling of individual metabolic rate resulting from a change in the body size dependence of mortality rates.     

Allometric Convergence in Savanna Trees and Implications for the Use of Plant Scaling Models in Variable Ecosystems

Theoretical models of allometric scaling provide frameworks for understanding and predicting how and why the morphology and function of organisms vary with scale. It remains unclear, however, if the predictions of ‘universal’ scaling models for vascular plants hold across diverse species in variable environments. Phenomena such as competition and disturbance may drive allometric scaling relationships away from theoretical predictions based on an optimized tree. Here, we use a hierarchical Bayesian approach to calculate tree-specific, species-specific, and ‘global’ (i.e. interspecific) scaling exponents for several allometric relationships using tree- and branch-level data harvested from three savanna sites across a rainfall gradient in Mali, West Africa. We use these exponents to provide a rigorous test of three plant scaling models (Metabolic Scaling Theory (MST), Geometric Similarity, and Stress Similarity) in savanna systems. For the allometric relationships we evaluated (diameter vs. length, aboveground mass, stem mass, and leaf mass) the empirically calculated exponents broadly overlapped among species from diverse environments, except for the scaling exponents for length, which increased with tree cover and density. When we compare empirical scaling exponents to the theoretical predictions from the three models we find MST predictions are most consistent with our observed allometries. In those situations where observations are inconsistent with MST we find that departure from theory corresponds with expected tradeoffs related to disturbance and competitive interactions. We hypothesize savanna trees have greater length-scaling exponents than predicted by MST due to an evolutionary tradeoff between fire escape and optimization of mechanical stability and internal resource transport. Future research on the drivers of systematic allometric variation could reconcile the differences between observed scaling relationships in variable ecosystems and those predicted by ideal models such as MST.     

植物种群自疏过程中构件生物量与密度的关系

不论是在对植物种群自疏规律还是在对能量守衡法则的研究中,个体大小(M)大多针对植物地上部分生物量,地下部分和构件生物量及其动态十分重要又多被忽视。以1年生植物荞麦为材料研究了自疏种群地下部分生物量、包括地下部分的个体总生物量以及各构件生物量与密度的关系。结果表明:平均地上生物量和个体总生物量与密度的异速关系指数(γabove-ground和γindividual)分别为-1.293和-1.253,与-4/3无显著性差异(P>0.05),为-4/3自疏法则提供了有力证据;平均根生物量-密度异速指数γroot(-1.128)与-1无显著性差异(P>0.05),与最终产量恒定法则一致;平均茎生物量-密度异速指数γstem(-1.263)接近-4/3(P>0.05),平均叶生物量-密度异速指数γleaf(-1.524)接近-3/2(P>0.05),分别符合-4/3自疏法则与-3/2自疏法则;而繁殖生物量与密度的异速关系指数γreproductive(-2.005)显著小于-3/2、-4/3或-1(P<0.001)。因此,不存在一个对植物不同构件普适的生物量-密度之间的关系。光合产物在地上和地下构件的生物量分配格局以及构件生物量与地上生物量之间特异的异速生长关系导致不同构件具有不同的自疏指数。无论对于地上生物量还是个体总生物量,荞麦种群能量均守衡,而对于地下生物量,荞麦种群能量不守衡。     

7种木本植物的分支指数与代谢指数

West、Brown和Enquist提出的植物分形网络模型(简称WBE模型)认为: 植物的分支指数(1/a, 1/b)决定植物的代谢指数, 当分支指数1/a、1/b分别为理论值2.0、3.0时, 代谢速率与个体大小的3/4次幂成正比, 但是恒定的3/4代谢指数并不能全面地反映植物的代谢情况。基于分支指数的协同变化, Price、Enquist和Savage对WBE模型进行扩展, 提出植物分支参数协同变化模型(简称PES模型)。该文借助于PES模型分析了7种木本植物的分支指数和代谢指数。结果表明: 物种间叶面积与叶生物量呈异速生长关系, 基于叶面积得到的分支指数1/a和代谢指数θ在物种间无显著差异, 基于叶生物量得到的分支指数1/a、1/b和代谢指数θ在物种间均存在显著差异, 但基于叶面积和叶生物量分别拟合出的整体分支指数1/a、1/b和代谢指数θ与理论值均无显著差异, 且用叶面积作为代谢速率的替代指标比用叶生物量分析得出的代谢指数与理论值更接近。今后研究应当关注植物叶面积与叶生物量的异速生长关系对植物代谢速率及相关功能特性的影响。     

Branching and metabolic exponents in seven woody plants

West、Brown和Enquist提出的植物分形网络模型(简称WBE模型)认为: 植物的分支指数(1/a, 1/b)决定植物的代谢指数, 当分支指数1/a、1/b分别为理论值2.0、3.0时, 代谢速率与个体大小的3/4次幂成正比, 但是恒定的3/4代谢指数并不能全面地反映植物的代谢情况。基于分支指数的协同变化, Price、Enquist和Savage对WBE模型进行扩展, 提出植物分支参数协同变化模型(简称PES模型)。该文借助于PES模型分析了7种木本植物的分支指数和代谢指数。结果表明: 物种间叶面积与叶生物量呈异速生长关系, 基于叶面积得到的分支指数1/a和代谢指数θ在物种间无显著差异, 基于叶生物量得到的分支指数1/a、1/b和代谢指数θ在物种间均存在显著差异, 但基于叶面积和叶生物量分别拟合出的整体分支指数1/a、1/b和代谢指数θ与理论值均无显著差异, 且用叶面积作为代谢速率的替代指标比用叶生物量分析得出的代谢指数与理论值更接近。今后研究应当关注植物叶面积与叶生物量的异速生长关系对植物代谢速率及相关功能特性的影响。     

Size–abundance relationships in Florida ant communities reveal how ants break the energetic equivalence rule

1. Ants are among the most abundant terrestrial organisms, yet little is known of how ant communities divide resources because it is difficult to measure the number of individuals in colonies and the density of colonies. 2. The body size–abundance relationships of the ants of five upland ecosystems in Florida were examined. The study tested whether abundance, energy use, and total biomass were distributed among species and body sizes as predicted by Damuth's energetic equivalence rule. Estimates of average worker body size, colony size, colony mass, and field metabolic rates were used to examine the relationships among body sizes, energy use, and total biomass. 3. Analyses revealed significant variation in energy use and did not support the energetic equivalence hypothesis. Specifically, the energy use and total standing biomass of species with large workers and colonies was much greater than smaller species. 4. These results suggest that larger species with larger colonies account for a disproportionate fraction of the total abundance and biomass of ants. A general model of resource allocation in colonies provides a possible explanation for why ants do not conform to the predictions of the energetic equivalence rule and for why ants are so abundant.     

On the relationship between mass and diameter distributions in tree communities

It has been suggested that frequency distributions of individual tree masses in natural stands are characterized by power-law distributions with exponents near -3/4, and that therefore tree communities exhibit energetic equivalence among size classes. Because the mass of trees is not measured directly, but estimated from diameter, this supposition is based on the fact that the observed distribution of tree diameters is approximately characterized by a power-law with an exponent approximately -2. Here we show that diameter distributions of this form are not equivalent to mass distributions with exponents of -3/4, but actually to mass distributions with exponents of -11/8. We discuss the implications of this result for the metabolic theory of ecology and for understanding energetic equivalence and the processes structuring tree communities.     

Comparative energetics of the giant hummingbird (Patagona gigas)

Hummingbirds (family Trochilidae) represent an extreme outcome in vertebrate physiological design and are the only birds capable of sustained hovering. The giant hummingbird (Patagona gigas) is the largest trochilid, with a mass of ~20 g, and is found over an altitudinal range from 0 to 4,500 m above sea level. We report here measurements of daily, basal, and hovering rates of oxygen consumption in the giant hummingbird; compare these values with data from smaller hummingbirds; and assess overall metabolic and allometric limits to trochilid body size. The sustained metabolic scope (i.e., the ratio of daily energy expenditure to basal metabolic rate) in the giant hummingbird is higher than that in smaller hummingbirds but lies below a proposed theoretical maximum value for endotherms. Scaling exponents in the allometric relationships for different modes of energetic expenditure were comparable, suggesting that the giant hummingbird, although a clear outlier in terms of body size, does not obviously deviate from metabolic relationships derived from other trochilid taxa.     

Size-abundance relationships in an Amazonian bird community: implications for the energetic equivalence rule

We studied size-abundance relationships in a species-rich Amazonian bird community and found that the slope of the logarithmic relationship between population density and bodymass (b = -0.22) is significantly shallower than expected under Damuth's energetic equivalence rule (EER), which states that population energy use (PEU) is independent of species body mass. We used estimates of avian field metabolic rates to examine the logarithmic relationship between PEU and body mass and its variation among ecological guilds. The relationship for all species had a significantly positive slope (b = 0.46), indicating that PEU of larger species was greater than that of smaller species. Analyses of guilds revealed significant variation. The slopes of the frugivore-omnivore, insectivore, and granivore guilds were all significantly positive, with that of the frugivore-omnivore guild being the steepest. In contrast, PEU did not vary significantly with species body mass among raptors. These results were confirmed, in analyses using both species values and phylogenetically independent contrasts, and the results do not support the EER in this community. The spatial distribution of resources and mechanisms of interference competition within guilds may explain why most patterns differed from the predictions of the EER. Other sources of variation, including the effects of scale, are also discussed.     

Energy In-Equivalence in Australian Marsupials: Evidence for Disruption of the Continent’s Mammal Assemblage,or Are Rules Meant to Be Broken?

The energy equivalence rule (EER) is a macroecological hypothesis that posits that total population energy use (PEU) should be independent of species body mass, because population densities and energy metabolisms scale with body mass in a directly inverse manner. However, evidence supporting the EER is equivocal, and the use of basal metabolic rate (BMR) in such studies has been questioned; ecologically-relevant indices like field metabolic rate (FMR) are probably more appropriate. In this regard, Australian marsupials present a novel test for the EER because, unlike eutherians, marsupial BMRs and FMRs scale differently with body mass. Based on either FMR or BMR, Australian marsupial PEU did not obey an EER, and scaled positively with body mass based on ordinary least squares (OLS) regressions. Importantly, the scaling of marsupial population density with body mass had a slope of −0.37, significantly shallower than the expected slope of −0.75, and not directly inverse of body-mass scaling exponents for BMR (0.72) or FMR (0.62). The findings suggest that the EER may not be a causal, universal rule, or that for reasons not yet clear, it is not operating for Australia’s unique native fauna.     

Dynamic relationships between body size,species richness,abundance, and energy use in a shallow marine epibenthic faunal community

We study the temporal variation in the empirical relationships among body size (S), species richness (R), and abundance (A) in a shallow marine epibenthic faunal community in Coliumo Bay, Chile. We also extend previous analyses by calculating individual energy use (E) and test whether its bivariate and trivariate relationships with S and R are in agreement with expectations derived from the energetic equivalence rule. Carnivorous and scavenger species representing over 95% of sample abundance and biomass were studied. For each individual, body size (g) was measured and E was estimated following published allometric relationships. Data for each sample were tabulated into exponential body size bins, comparing species‐averaged values with individual‐based estimates which allow species to potentially occupy multiple size classes. For individual‐based data, both the number of individuals and species across body size classes are fit by a Weibull function rather than by a power law scaling. Species richness is also a power law of the number of individuals. Energy use shows a piecewise scaling relationship with body size, with energetic equivalence holding true only for size classes above the modal abundance class. Species‐based data showed either weak linear or no significant patterns, likely due to the decrease in the number of data points across body size classes. Hence, for individual‐based size spectra, the SRA relationship seems to be general despite seasonal forcing and strong disturbances in Coliumo Bay. The unimodal abundance distribution results in a piecewise energy scaling relationship, with small individuals showing a positive scaling and large individuals showing energetic equivalence. Hence, strict energetic equivalence should not be expected for unimodal abundance distributions. On the other hand, while species‐based data do not show unimodal SRA relationships, energy use across body size classes did not show significant trends, supporting energetic equivalence.     

Allometry of territory size and metabolic rate as predictors of self-thinning in young-of-the-year Atlantic salmon

1. Self-thinning is a progressive decline in population density caused by competitively induced losses in a cohort of growing individuals and can be depicted as: log₁₀ (density) = c − β log₁₀ (body mass).
2. In mobile animals, two mechanisms for self-thinning have been proposed: (i) the space hypothesis predicts that maximum population density for a given body size is the inverse of territory size, and hence, the self-thinning slope is the negative of the slope of the allometric territory-size relationship; (ii) the energetic equivalence hypothesis predicts that the self-thinning slope is the negative of the slope of the allometric metabolic rate relationship, assuming a constant supply of energy for the cohort.
3. Both hypotheses were tested by monitoring body size, population density, food availability and habitat for young-of-the-year Atlantic salmon ( Salmo salar ) in Catamaran Brook, New Brunswick. The results were consistent with the predictions of the space hypothesis. Observed densities did not exceed the maximum densities predicted and the observed self-thinning slope of −1·16 was not significantly different from the slope of −1·12, predicted by the allometry of territory size for the population under study.
4. The observed self-thinning slope was significantly steeper than −0·87, predicted by the allometry of metabolic rate, perhaps because of a gradual decline in food abundance over the study period. The decline in density was more rapid in very shallow sites and may have been partly caused by a seasonal change in water depth and an ontogenetic habitat shift rather than solely by competition for food or space.
5. The allometry of territory size may be a useful predictor of self-thinning in populations of mobile animals competing for food and space.     

Mass invariance of population nitrogen flux by terrestrial mammalian herbivores: an extension of the energetic equivalence rule

According to the energetic equivalence rule, energy use by a population is independent of average adult body mass. Energy use can be equated with carbon flux, and it has been suggested that population fluxes of other materials, such as nitrogen and phosphorus, might also be independent of body mass. We compiled data on individual nitrogen deposition rates (via faeces and urine) and average population densities of 26 species of mammalian herbivores to test the hypothesis of elemental equivalence for nitrogen. We found that the mass scaling of individual nitrogen flux was opposite to that of population density for the species in our dataset. By computing the product of individual nitrogen flux and average population density for each species in our dataset, we found that population-level nitrogen flux was independent of species mass, averaging c. 3.22 g N ha⁻¹ day⁻¹. Results from this analysis can be used to understand the influence of mammalian herbivore communities on nitrogen cycling in terrestrial ecosystems.     

Evidence for "diminishing returns" from the scaling of stem diameter and specific leaf area

Research indicates that increases in total leaf area (A(T)) may fail to keep pace with increases in total leaf mass (M(L)) across plants differing in size (e.g., as measured by stem diameter, D). This "diminishing returns" hypothesis predicts that the scaling exponent for A(T) vs. M(L) will be less than one and that the exponent for specific leaf mass (i.e., A(T) / M(L)) vs. D will be negative. These predictions were examined using data from 46 plants ranging between 0.125 cm ≤ D ≤ 0.485 m across 25 woody dicot species. Standardized major axis slopes were used to quantify scaling exponents and random effects models were used to quantify species and size effects on the numerical values of exponents. The exponents for A(T) vs. M(L) and A(T) / M(L) vs. D differed among species and different species groupings. In general, the exponent for A(T) vs. M(L) was less than one and the exponent for A(T) / M(L) vs. D was negative, as predicted. However, random effects models indicated that species effects overshadowed size effects, although size effects were statistically significant. The diminishing returns hypothesis therefore receives statistical support, i.e., although the numerical values of exponents are "species-dependent," they are less than unity, as predicted by theory.     

Evolution of body size in food webs: does the energetic equivalence rule hold?

The energetic equivalence rule (EER), which is derived from empirical observations linking population density and body size and from the allometric law linking metabolism and body size, predicts that the amount of energy used by the various species should be independent of body size. Here, we examine this hypothesis using a model that allows entire food webs to emerge from coevolution of interacting species. Body size influences both individual metabolism and interactions among species in the model. Overall, population density does decrease with body size roughly following a power law whose exponent is variable. We discuss this variability in the light of empirical data sets. The emerging relationship between the flux of resources exploited by the various species and their body size follows a decreasing power law, thus contradicting the EER. Our model emphasizes the importance of considering the influence of body size on species interactions in attempting to explain large-scale patterns related to body size.