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。     

植物代谢速率与个体生物量关系研究进展

植物的各项生理生态功能（例如,呼吸、生长和繁殖）都与个体生物量成异速生长关系。West, Brown及Enquist基于分形网络结构理论所提出的WBE模型认为：植物的代谢（呼吸）速率正比于个体生物量的3/4次幂。然而,恒定的“3/4异速生长指数”与实测数据、植物生理生态学等研究之间存在矛盾,引发激烈的争论。论文分析了不同回归方法对代谢指数的影响,重点对植物代谢速率与个体生物量异速生长关系研究进展进行了综述,分析并得出了植物代谢指数在小个体时接近1.0,并随着生物量的增加而系统减小,且其密切依赖于氮含量的调控的结论。据此,提出了进一步深入研究植物代谢速率个体生物量关系需要解决的一些科学问题。     

Metabolic scaling in insects supports the predictions of the WBE model

The functional association between body size and metabolic rate (BS-MR) is one of the most intriguing issues in ecological physiology. An average scaling exponent of 3/4 is broadly observed across animal and plant taxa. The numerical value of 3/4 is theoretically predicted under the optimized version of West, Brown, and Enquist's vascular resource supply network model. Insects, however, have recently been proposed to express a numerically different scaling exponent and thus application of the WBE network model to insects has been rejected. Here, we re-analyze whether such variation is indeed supported by a global deviation across all insect taxa at the order and family levels to assess if specific taxa influence insect metabolic scaling. We show that a previous reported deviation is largely due to the effect of a single insect family (Termitidae). We conclude that the BS-MR relationship in insects broadly supports the core predictions of the WBE model. We suggest that the deviation observed within the termites warrants further investigation and may be due to either difficulty in accurately measuring termite metabolism and/or particularities of their life history. Future work on allometric scaling should assess the nature of variation around the central tendencies in scaling exponents in order to test if this variation is consistent with core assumptions and predictions of the WBE model that stem by relaxing its secondary optimizing assumptions that lead to the 3/4 exponent.     

Plant physiology in theory and practice: An analysis of the WBE model for vascular plants

The theoretical model of West, Brown and Enquist (hereafter WBE) proposed the fractal geometry of the transport system as the origin of the allometric scaling laws observed in nature. The WBE model has either been criticized for some restrictive and biologically unrealistic constraints or its reliability debated on the evidence of empirical tests. In this work, we revised the structure of the WBE model for vascular plants, highlighting some critical assumptions and simplifications and discuss them with regard to empirical evidence from plant anatomy and physiology. We conclude that the WBE model had the distinct merit of shedding light on some important features such as conduit tapering. Nonetheless, it is over-simplistic and a revised model would be desirable with an ontogenetic perspective that takes some important phenomena into account, such as the transformation of the inner sapwood into heartwood and the effect of hydraulic constraints in limiting the growth in height.     

最大密度法则研究进展

 该文从理论推导和研究方法等方面对近几十年来关于最大密度法则的研究进展进行了综述,得出结论：1)关于最大密度法则理论主要有几何关系的3/2法则和空间填充分行支状网运输结构的WBE模型。进一步研究发现它们都是建立在一种静态的统计分析基础之上的,因而近几年研究者们开始尝试用动态的个体植物之间的竞争来建立模型。尽管如此,关于最大密度法则的模型仍然没有逃出固有的模式,如用平均植物大小代替整个植物种群。因此,关于最大密度法则理论需要进一步的研究。2)最大密度法则理论在假设条件、数学推导、 用于估计参数的原始数据选择等方面存在争议。任何模型的建立都是基于一些特定的条件和假设建立的,因而得到的关系并不是一个万能的定律。所以在分析数据时,这些模型可结合使用。3)在研究方法上,由于大家对最大密度法则的理解不同,标准不同,造成研究方法多种多样。因而建议在以后的研究中建立一个客观统一的方法。     

Sizing up allometric scaling theory

Metabolic rate, heart rate, lifespan, and many other physiological properties vary with body mass in systematic and interrelated ways. Present empirical data suggest that these scaling relationships take the form of power laws with exponents that are simple multiples of one quarter. A compelling explanation of this observation was put forward a decade ago by West, Brown, and Enquist (WBE). Their framework elucidates the link between metabolic rate and body mass by focusing on the dynamics and structure of resource distribution networks-the cardiovascular system in the case of mammals. Within this framework the WBE model is based on eight assumptions from which it derives the well-known observed scaling exponent of 3/4. In this paper we clarify that this result only holds in the limit of infinite network size (body mass) and that the actual exponent predicted by the model depends on the sizes of the organisms being studied. Failure to clarify and to explore the nature of this approximation has led to debates about the WBE model that were at cross purposes. We compute analytical expressions for the finite-size corrections to the 3/4 exponent, resulting in a spectrum of scaling exponents as a function of absolute network size. When accounting for these corrections over a size range spanning the eight orders of magnitude observed in mammals, the WBE model predicts a scaling exponent of 0.81, seemingly at odds with data. We then proceed to study the sensitivity of the scaling exponent with respect to variations in several assumptions that underlie the WBE model, always in the context of finite-size corrections. Here too, the trends we derive from the model seem at odds with trends detectable in empirical data. Our work illustrates the utility of the WBE framework in reasoning about allometric scaling, while at the same time suggesting that the current canonical model may need amendments to bring its predictions fully in line with available datasets.     

Lack of Evidence for 3/4 Scaling of Metabolism in Terrestrial Plants

Scaling, as the translation of information across spatial, temporal, and organizational scales, is essential to predictions and understanding in all sciences and has become a central issue in ecology. A large body of theoretical and empirical evidence concerning allometric scaling in terrestrial individual plants and plant communities has been constructed around the fractal volume-filling theory of West, Brown, and Enquist （the WBE model）. One of the most thought-provoking findings has been that the metabolic rates of plants, like those of animals, scale with their size as a 3/4 power law. The earliest, single most-important study cited in support of the application of the WBE model to terrestrial plants claims that whole-plant resource use in terrestrial plants scales as the 3/4 power of total mass, as predicted by the WBE model. However, in the present study we show that empirical data actually do not support such a claim. More recent studies cited as evidence for 3/4 scaling also suffer from several statistical and data-related problems. Using a forest biomass dataset including 1 266 plots of 17 main forest types across China, we explored the scaling exponents between tree productivity and tree mass and found no universal value across forest stands. We conclude that there is not sufficient evidence to support the existence of a single constant scaling exponent for the metabolism-biomass relationship for terrestrial plants.     

Lack of Evidence for 3/4 Scaling of Metabolism in Terrestrial Plants

Scaling, as the translation of information across spatial, temporal, and organizational scales, is essential to predictions and understanding in all sciences and has become a central issue in ecology. A large body of theoretical and empirical evidence concerning allometric scaling in terrestrial individual plants and plant communities has been constructed around the fractal volume-filling theory of West, Brown, and Enquist (the WBE model). One of the most thought-provoking findings has been that the metabolic rates of plants, like those of animals, scale with their size as a 3/4 power law. The earliest, single most-important study cited in support of the application of the WBE model to terrestrial plants claims that whole-plant resource use in terrestrial plants scales as the 3/4 power of total mass, as predicted by the WBE model.However, in the present study we show that empirical data actually do not support such a claim. More recent studies cited as evidence for 3/4 scaling also suffer from several statistical and data-related problems. Using a forest biomass dataset including 1 266 plots of 17 main forest types across China, we explored the scaling exponents between tree productivity and tree mass and found no universal value across forest stands. We conclude that there is not sufficient evidence to support the existence of a single constant scaling exponent for the metabolism-biomass relationship for terrestrial plants.     

Allometric scaling laws for water uptake by plant roots

This paper develops scaling laws for plant roots of any arbitrary volume and branching configuration that maximize water uptake. Water uptake can occur along any part of the root network, and thus there is no branch-to-branch fluid conservation. Maximizing water uptake, therefore, involves balancing two flows that are inversely related: axial and radial conductivity. The scaling laws are tested against the root data of 1759 plants from 77 herbaceous species, and compared with those from the WBE model. I further discuss whether the scaling laws are invariant to soil water distribution. A summary of some of the results follows. (1) The optimal radius for a single root (no branches) scales with volume as . (2) The basic allometric scaling for root radius branches (r_i+1=β*r_i) is of the form , where f(N)=A(N)/(n_b*(1+A(N))), n_b is the number of branches, and A(N) and ε(N) are functions of the number of root diameter classes (not constants as in the WBE model). (3) For large N, β converges to the β from the WBE model. For small N, the β's for the two models diverge, but are highly correlated. (4) The fractal assumption of volume filling of the WBE model are also met in the root model even though they are not explicitly incorporated into it. (5) The WBE model for rigid tubes is an asymptotic solution for large root systems (large N and biomass). (6) The optimal scaling solutions for the root network appears to be independent of soil water distribution or water demand. The data set used for testing is included in the electronic supplementary archive of the journal.     

Allometric scaling among tree components in Pinus massoniana stands with different sites

The WBE model was used to predict intraspecific scaling relationships among mean branch, needle, stem, root, and above-ground masses across eight stands of Pinus massoniana to test whether the scaling exponent was (1) dependent on site and (2) in accordance with WBE theory. The results showed that mean stem and root masses as well as mean above-ground and root masses scaled in a near-isometric manner across sites, except at two sites, which exhibited an exponent slightly less than unity. Mean needle mass scaled as 3/4 power of mean stem mass, except at one site, which exhibited an exponent slightly higher than 3/4. Mean branch mass scaled isometrically with mean stem mass at each site. These results supported the WBE theory. However, mean branch mass across sites scaled neither as 3/4 nor 1 power of mean stem mass, indicating that the scaling relationship predicted by WBE theory for these two components did not hold in P. massoniana stands.     

Scaling relationship between tree respiration rates and biomass

The WBE theory proposed by West, Brown and Enquist predicts that larger plant respiration rate, R, scales to the three-quarters power of body size, M. However, studies on the R versus M relationship for larger plants (i.e. trees larger than saplings) have not been reported. Published respiration rates of field-grown trees (saplings and larger trees) were examined to test this relationship. Our results showed that for larger trees, aboveground respiration rates R_A scaled as the 0.82-power of aboveground biomass M_A, and that total respiration rates R_T scaled as the 0.85-power of total biomass M_T, both of which significantly deviated from the three-quarters scaling law predicted by the WBE theory, and which agreed with 0.81–0.84-power scaling of biomass to respiration across the full range of measured tree sizes for an independent dataset reported by Reich et al. (Reich et al. 2006 Nature 439, 457–461). By contrast, R scaled nearly isometrically with M in saplings. We contend that the scaling exponent of plant metabolism is close to unity for saplings and decreases (but is significantly larger than three-quarters) as trees grow, implying that there is no universal metabolic scaling in plants.     

Evaluating scaling models in biology using hierarchical Bayesian approaches

Theoretical models for allometric relationships between organismal form and function are typically tested by comparing a single predicted relationship with empirical data. Several prominent models, however, predict more than one allometric relationship, and comparisons among alternative models have not taken this into account. Here we evaluate several different scaling models of plant morphology within a hierarchical Bayesian framework that simultaneously fits multiple scaling relationships to three large allometric datasets. The scaling models include: inflexible universal models derived from biophysical assumptions (e.g. elastic similarity or fractal networks), a flexible variation of a fractal network model, and a highly flexible model constrained only by basic algebraic relationships. We demonstrate that variation in intraspecific allometric scaling exponents is inconsistent with the universal models, and that more flexible approaches that allow for biological variability at the species level outperform universal models, even when accounting for relative increases in model complexity.     

Testing metabolic ecology theory for allometric scaling of tree size, growth and mortality in tropical forests

The theory of metabolic ecology predicts specific relationships among tree stem diameter, biomass, height, growth and mortality. As demographic rates are important to estimates of carbon fluxes in forests, this theory might offer important insights into the global carbon budget, and deserves careful assessment. We assembled data from 10 old-growth tropical forests encompassing censuses of 367 ha and > 1.7 million trees to test the theory's predictions. We also developed a set of alternative predictions that retained some assumptions of metabolic ecology while also considering how availability of a key limiting resource, light, changes with tree size. Our results show that there are no universal scaling relationships of growth or mortality with size among trees in tropical forests. Observed patterns were consistent with our alternative model in the one site where we had the data necessary to evaluate it, and were inconsistent with the predictions of metabolic ecology in all forests.     

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。     

Scaling Relationships between Leaf Mass and Total Plant Mass across Chinese Forests

Biomass partitioning is important for illustrating terrestrial ecosystem carbon flux. West, Brown and Enquist (WBE) model predicts that an optimal 3/4 allometric scaling of leaf mass and total biomass of individual plants will be applied in diverse communities. However, amount of scientific evidence suggests an involvement of some biological and environmental factors in interpreting the variation of scaling exponent observed in empirical studies. In this paper, biomass information of 1175 forested communities in China was collected and categorized into groups in terms of leaf form and function, as well as their locations to test whether the allocation pattern was conserved or variable with internal and/or environmental variations. Model Type II regression protocol was adopted to perform all the regressions. The results empirically showed that the slopes varied significantly across diverse forested biomes, between conifer and broadleaved forests, and between evergreen and deciduous forests. Based on the results, leaf form and function and their relations to environments play a significant role in the modification of the WBE model to explore more accurate laws in nature.     

Modelling below- and above-ground biomass for non-woody and woody plants

BACKGROUND AND AIMS: Intraspecific relationships between below- and above-ground biomass (MB and MA, respectively) have been studied extensively to evaluate environmental effects on growth and development at the level of the individual plant. However, no current theoretical model for this relationship exists for broad interspecific trends. The aims of this paper are to provide a model and to test its predictions using a recently assembled, large database (1406 data entries for 257 species). METHODS: An allometric model was derived to predict the relationship between MB and MA for non-woody and woody plants based on previously developed scaling relationships for leaf, stem and root standing biomass and annual growth rates. The predictions of this model were tested by comparing the numerical values of predicted scaling exponents (the slopes of log-log regression curves) with those observed for the database. KEY RESULTS AND CONCLUSIONS: For non-woody plants and the juveniles of woody species, the model predicts an isometric scaling relationship (i.e. MB proportional, variant MA). For woody plants, a complex scaling function is predicted. But, for a particular set of biologically reasonable conditions, the model predicts MB proportional, variant MA across woody plants. These predictions accord reasonably well with observed statistical trends when non-woody and woody plants are studied separately (n=1061 and 345 data entries, respectively). Although the reliability of regression formulas to estimate MB based on MA measurements increased with increasing plant size, estimates of MB can be as much as two orders of magnitude off, even when using regression formulas with r2 >0.90 and F >53,000.     

Intra-plant versus Inter-plant Root Competition in Beans: avoidance,resource matching or tragedy of the commons

Root competition inhibits root proliferation. All else equal, a plant should invest roots in a nutrient patch devoid of roots rather than one already occupied by roots. Less clear is how a plant should respond to intra-plant versus inter-plant root competition. We consider three responses for how a plant may select habitats based on intra-versus inter-plant root competition: inter-plant avoidance, resource matching, or intra-plant avoidance. The first assumes that plants prefer to have their own space and preferentially proliferate roots away from neighboring plants. The second response, based on the ideal free distribution, assumes that plants invest so as to equalize average returns from roots, regardless of the identity of the neighboring roots. The third, based on game theory, assumes that the plant proliferates roots so as to maximize whole-plant fitness, in which case it is better to proliferate plants among a neighbor's roots than to continue proliferating amongst one's own roots. To test among these models we grew beans (Phaseolus varigaris, var. Kenya) in a greenhouse under two planting scenarios. Both scenario were tested under 0.5 and 0.1 strength of nutrient solution. Under scenario A (fence-sitters), two split-root plants each shared two patches by virtue of having roots in each. Under scenario B (owners) two plants each had their own patch. The results supported the game theory model of intra-plant avoidance (whole plant habitat selection). Fence-sitters produced 150% more root mass per individual than owners. Owners produced 90% more yield (dry mass of pods) than fence-sitters. Furthermore, owners had significantly higher shoot-root ratios than fence-sitters. These effects did not vary with high or low nutrient levels. The over-proliferation of roots under inter-plant competition (fence-sitters) was manifest by the tenth day after planting. In short, the fence-sitters engaged in a tragedy of the commons in which they competed with each other through root proliferation. At the ESS, the fitness maximizing strategy of the individual is to sacrifice collective yield in a quest to `steal' nutrients from its neighbor. The research has three implications. First, plants may be able to assess and respond to local opportunities in a manner that maximizes the good of the whole plant. Second, nutrient foraging as a game may provide a fresh perceptive for viewing root competition either intra-specifically or inter-specifically. Third, it may be possible to increase the yield of certain crop species by breeding more `docile' cultivars that do not overproduce roots in response to inter-plant competition.     

Evolutionary senescence in plants

Senescence is a decline in age-specific survival and reproduction with advancing age. Studies of evolutionary plant senescence are designed to explain this decline in life history components within the context of natural selection. A review of studies of plant demography reveals senescent declines in both annual and perennial plants, but also suggests that there are some plant species which may not be expected to show senescence. Thus, future comparative studies of closely related species, with and without senescence, should be possible. The assumptions of the major evolutionary theories of senescence are evaluated for their validity with respect to plants. Different plant species violate one or more of the assumptions of the theories, yet the consequences of violating these assumptions have never been investigated. Whereas, to date, evolutionary senescence has been studied only indirectly in plants, it is concluded that plants provide good experimental systems for clarifying our understanding of senescence in natural populations.