{"title":"论植物结垢","authors":"Cory Matthew","doi":"10.1002/glr2.70020","DOIUrl":null,"url":null,"abstract":"<p>This editorial revisits the topic of plant allometry. This topic is the subject of a large volume of literature, so coverage here is necessarily selective, focusing on points of interest for grassland research. In my final year of undergraduate study (1983), three different courses I took included a module based on Yoda's 1963 study, “Self-thinning in overcrowded pure stands” (Yoda et al., <span>1963</span>). Principles elucidated in that paper were seen as fundamental to the theoretical understanding of crop-specific husbandry recommendations for yield optimization. Meanwhile, Hutchings (<span>1983</span>) published an article “Ecology's law in search of a theory,” indicating a lack of consensus among researchers of that era as to what ecological drivers were operating to produce the plant behaviour patterns Yoda and colleagues had described.</p><p>Briefly, the self-thinning rule (Yoda et al., <span>1963</span>) states that when values for single plant mean dry weight (<i>w</i>) for plants in a crowded stand are plotted against stand density on a log–log scale, the points for plants of different species or plants of the same species at different ages will fall along a line of slope −3/2, which became known as the “−3/2 boundary line.” As a stand approaches the boundary line, for example through an increase in plant size over time or through increased planting density, some plants will be lost from the population so that size/density (i.e., <i>w:d</i>) trajectories over time or across planting densities follow the boundary line. The intensity of competition increases and plant allocation between body parts changes as the boundary line is approached. This also is important in crop husbandry. For example, height or leaf accumulation may be favoured at the expense of reproductive yield or bulb development.</p><p>Data from such studies suggest that an effective tactical approach for fodder beet production involves planting at 8 plants per m<sup>2</sup>, allowing approximately 60 days for leaf area development, followed by 90 days for carbohydrate translocation to support bulb fill. At this plant density, bulbs are comparatively large (which is desirable), and during the bulb-fill growth stage, the crop accumulates bulb dry weight at rates that can exceed 350 kg DM ha<sup>−1</sup> day<sup>−1</sup>. During the leaf area development phase, there is opportunity for weeds to colonize bare soil, and weed control—often requiring a costly herbicide combination—is critical (Matthew et al., <span>2011</span>). For maize, experimental data from Wisconsin showed that the optimal plant density for grain production was approximately 6000 plants m<sup>−2</sup> lower than that for silage production. This occurred because for silage the forage biomass gains from a higher planting density of around 80 000 plants ha<sup>−1</sup> outweighed the competition-induced loss in grain yield above 75 000 plants ha<sup>−1</sup> (Cusicanqui & Lauer, <span>1999</span>). In oil palm plantations, the optimal tree spacing in a triangular planting pattern represents a compromise between higher tree density for increased fruit yield and wider spacing to slow height growth and extend the period during which manual harvesting will remain feasible. In one study, a decrease in planting distance from 9.5 to 7.5 m increased tree height by 50 cm from 3.5 to 4.0 m after 14 years (Bonneau & Impens, <span>2022</span>). In forestry science, the self-thinning line has been parameterized on a species-specific basis (Pretzsch & Biber, <span>2005</span>). This is because it is an important reference line for density control in the context of thinning. If the species-specific lines are known, suboptimal densities and the resulting production losses can be avoided.</p><p>Davies (<span>1988</span>) in her Figure 3.6 observed that data for tiller weight and density from perennial ryegrass swards fit the −3/2 boundary rule, citing other research in support. However, on closer scrutiny, this conclusion is an oversimplification as data for larger tillers sit above the trend line, while data for smaller tillers sit below it. The actual slope for Davies' data set is more like −5/2, rather than −3/2. This discrepancy between presumed and actual slope calls to mind a comment of Mrad et al. (<span>2020</span>) who reviewed eight potential mechanisms to explain the −3/2 power rule and in their closing comments cited Russian physicist Lev Landau: “Money is in the exponent, and the exponent needs to be calculated precisely.” On reflection, it is evident that the operation of the self-thinning rule will be a far more complex process for a grass sward than for a forest tree stand. Forest trees are perennial and must “average” their population density across seasons. In most tree species and forests, the population density is fixed across time after establishment, or recruitment is slow. A few tree species can produce new shoots from roots, meaning the population is dynamic. Grass tillers by contrast often have a life of less than 1 year and may have a dormant period during winter cold or summer drought, when leaves have senesced and the shoot apical meristem positioned near the ground level survives as a quiescent bud cocooned in undeveloped leaf primordia or sheaths of mature dead leaves. They can also readily generate new shoots from axillary buds or crown buds in the case of alfalfa. Hence, shoot population in grass swards and in forage crops such as alfalfa can fluctuate dynamically through a season or through a regrowth cycle following defoliation.</p><p>Matthew et al. (<span>1995</span>), in their analysis of tiller and shoot density data for ryegrass swards and alfalfa stands, conceptualized the −3/2 boundary line as a constant leaf area line for different shoot size–density combinations, representing the maximum leaf area index (LAI) that a given environment can support. For swards with coordinates plotting above the boundary line, the rate of leaf senescence would exceed that of leaf formation, whereas for those below the line, leaf formation rate would exceed senescence—thus defining the boundary line, or positions along it, as points of equilibrium. These authors proposed four phases of size–density dynamics during the defoliation and regrowth cycle of a grass sward: (i) initiation of new shoots in early regrowth to accelerate LAI recovery, with population density and LAI increasing; (ii) near −5/2 self-thinning, where LAI continues to rise while smaller or younger shoots die due to basal shading by larger, expanding shoots; (iii) size–density compensation along the −3/2 self-thinning line at constant sward LAI, with herbage mass increasing from pseudostem accumulation to support leaves as surviving shoots increase in height; and (iv) a phase of constant herbage mass, representing a “ceiling herbage mass” characteristic of a specific vegetation type. Phase (iv), by mathematical necessity, requires 1:1 self-thinning if there is any increase over time in mean shoot size. Using this model, the authors developed a slope correction to account for the “steeper than −3/2” self-thinning linked to LAI increase in phase (ii) and a plant shape correction to account for the impact of change in plant shape during regrowth on the population required to provide the environmentally sustainable LAI. The plant shape correction was based on a simple dimensionless parameter: m<sup>2</sup> leaf per (m<sup>3</sup> volume)<sup>2/3</sup>, which was designated “<i>R</i>” and seemed effective in understanding the impact of treatments such as shading on plant behaviour. For example, in alfalfa, shading reduced <i>R</i> by suppressing branching (unpublished data). We also proposed that the distance of a point, defined by tiller size–density coordinates, from an arbitrarily positioned self-thinning line, could be used as a productivity index for swards subjected to different treatments within a common environment (Hernández Garay et al., <span>1999</span>; Figure 7.2 of Matthew et al., <span>2000</span>).</p><p>One particularly interesting observation from visualizing sward size–density data in this way was that size–density compensation trajectories for perennial ryegrass and white clover within the same swards ran in opposite directions, across four different grazing intensities defined by postgrazing herbage mass targets (kg DM ha<sup>−1</sup>) (Figure 1a,b; Yu et al., <span>2008</span>). We interpret this as indicating that white clover occupies a light interception niche that perennial ryegrass cannot exploit, due to limited carbohydrate availability for new tiller initiation under defoliation pressure (Figure 1c).</p><p>Scaling theory also provides an intuitively logical basis for key principles of grazing optimization theory and conceptualization of environmental carrying capacity. We see from Figure 1c that heavy grazing pressure limits a sward's ability to generate leaf area, a condition likely linked to reduced carbohydrate reserves and overall plant vigour (Fulkerson & Donaghy, <span>2001</span>). In the middle range of the self-thinning diagram, herbage accumulation can occur with comparatively little shoot death. We also see from the self-thinning diagram or from Figure 7.2 of Matthew et al. (<span>2000</span>) that a grazing management regime that delivers a lower density of larger tillers may well be inherently better for leaf area development and sward productivity than one that delivers a higher density of smaller tillers. This point is not always appreciated by agronomists when formulating pasture management advice. It is tempting to believe that a higher tiller density must be better. Meanwhile, at the upper left end of the self-thinning line (beyond the range shown in Figure 1c), the transition to 1:1 scaling means the death rate of shoots and species diversity loss may be exacerbated in prolonged periods without grazing. These principles of ensuring sufficient leaf area for grassland vegetation to meet its energy needs on one hand while avoiding negative consequences of zero grazing on the other hand deserve to be incorporated into the formulation of management guidelines when determining the carrying capacity of environmentally sensitive areas and policies concerning their use in livestock farming.</p><p>Another point to note is that although <i>w</i> scales with <i>d</i> at −3/2 in the formulation of Yoda et al. (<span>1963</span>), sward herbage mass (i.e., biomass, <i>b</i>) scales with <i>d</i> at −½, since <i>b</i> = <i>w. d</i> and therefore the <i>b:d</i> slope is less than the <i>w:d</i> slope by exactly 1.0. This implies that whenever a turf or pasture manager deliberately varies mowing or defoliation height or interval—such as when pasture is stockpiled at high herbage mass in autumn for winter feeding—substantial shifts in sward shoot population must occur within the stand to maintain equilibrium along the boundary line. An increase from low to high herbage mass—starting from a high shoot density at the lower mass—suggests that a supra-optimal LAI may develop, leading to leaf loss through senescence and a reduction in shoot population. Subsequent release of the stored herbage may create a situation where there is no longer sufficient shoot density to fully exploit the environmental potential LAI. The interplay under a <i>b:d</i> boundary line of slope −½ for sward shoot <i>w:d</i> coordinates for turf plots cut at different time intervals and heights is visualized in fig. 7.2 of Matthew et al. (<span>2000</span>). In that data set when plots are cut at 14-day intervals, coordinates for 75- and 100-mm cutting heights plot near the −½ <i>b:d</i> boundary line, but plots cut at 50 and 25 mm height plot progressively further below it. In a grazing context, Parmenter and Boswell (<span>1983</span>) found that compared with four or five grazings during winter, which would have avoided reduction in tiller density, spring herbage production on plots grazed only twice in winter and expected to have reduced tiller density as a result, was reduced by 12.3% on average over 3 years.</p><p>It would be nice if the story could end there, but it doesn't. In 1998, Enquist et al. published an analysis of self-thinning in plant stands, based on total aboveground dry weight (<i>w</i>) and stand density (<i>d</i>) data compiled from various studies, spanning approximately 11 orders of magnitude in plant size. Based on theoretical considerations from fractal geometry of vascular distribution networks (as in the stems and branches of trees and shrubs), they derived and present data in their fig. 2 that appear to demonstrate a −4/3 <i>w:d</i> scaling relationship, similar to 4/3 body weight:energy requirement relationships that are well established for animals. This −4/3 relationship in plants is predicted by Enquist et al. (<span>1998</span>) to be invariant with respect to body size and is said to “highlight the universality of the 3/4-power scaling of resource use and the related ¼-power scaling of other structural, functional and ecological attributes.” The analysis of Enquist et al. (<span>1998</span>) was questioned (Kozłowski & Konarzewski, <span>2004</span>), defended by Brown et al. (<span>2005</span>), and questioned again (Kozłowski & Konarzewski, <span>2005</span>) but is now widely regarded as definitive. Enquiry has moved on to exploring the implications at the ecosystem level, leading to the elucidation of the worldwide leaf economic spectrum (Wright et al., <span>2004</span>). Questions about finding a grand unifying theory have been asked (Niklas, <span>2004</span>) but not convincingly answered.</p><p>The writer is currently working on a manuscript exploring a data set of 795 observations, spanning eight orders of magnitude, collected at Technische Universität München in 2003 and designed to provide an empirical test of the conclusions of Enquist et al. (<span>1998</span>). Subject to input from co-authors and peer reviewers, we hope to publish the results of this study in Grassland Research later in the year. Suffice it to say, this is a second instance where the significance lies in the exponent, and the exponent must be calculated precisely. Based on our own results, we propose that an alternative interpretation of the data presented by Enquist et al. (<span>1998</span>) in their Figure 2 would be to fit a scaling line of slope 1:1 to plants weighing less than 10⁰ g (after discarding two outliers of 10⁻⁴ g), while for larger plants, the scaling slope can be either ⅔ or ¾, depending on whether measures of leaf area or plant mass are scaled. It goes without saying that when a single fit-line is placed across a two-phase relationship, the joint fit line will be an average of the two phases. It is not illogical to question whether plants with a dry weight of less than 1.0 g require allometry governed by a vascular distribution network in the same way as trees and shrubs or to ask why plant allometry should be defined solely by resource distribution constraints, with no consideration of resource capture. Our data show that −3⁄2 scaling of certain plant body dimensions—reflecting optimisation of light capture by leaves—and −4⁄3 scaling of mass-related dimensions of other organs, consistent with vascular distribution theory, can be reconciled through phenomena such as the non-1:1 scaling of leaf mass to leaf area. The next step is to integrate these insights with the global leaf economics spectrum (Wright et al., <span>2004</span>). Herein lies another lesson for all researchers: it is a great travesty that the data for our empirical test has remained unpublished on a computer hard drive for over 20 years. PhD graduates transitioning into their first job, along with others holding unpublished datasets, are strongly encouraged to think creatively about how to make time to publish their findings, so they may be considered by the wider scientific community.</p>","PeriodicalId":100593,"journal":{"name":"Grassland Research","volume":"4 2","pages":"89-92"},"PeriodicalIF":0.0000,"publicationDate":"2025-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/glr2.70020","citationCount":"0","resultStr":"{\"title\":\"On plant scaling\",\"authors\":\"Cory Matthew\",\"doi\":\"10.1002/glr2.70020\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>This editorial revisits the topic of plant allometry. This topic is the subject of a large volume of literature, so coverage here is necessarily selective, focusing on points of interest for grassland research. In my final year of undergraduate study (1983), three different courses I took included a module based on Yoda's 1963 study, “Self-thinning in overcrowded pure stands” (Yoda et al., <span>1963</span>). Principles elucidated in that paper were seen as fundamental to the theoretical understanding of crop-specific husbandry recommendations for yield optimization. Meanwhile, Hutchings (<span>1983</span>) published an article “Ecology's law in search of a theory,” indicating a lack of consensus among researchers of that era as to what ecological drivers were operating to produce the plant behaviour patterns Yoda and colleagues had described.</p><p>Briefly, the self-thinning rule (Yoda et al., <span>1963</span>) states that when values for single plant mean dry weight (<i>w</i>) for plants in a crowded stand are plotted against stand density on a log–log scale, the points for plants of different species or plants of the same species at different ages will fall along a line of slope −3/2, which became known as the “−3/2 boundary line.” As a stand approaches the boundary line, for example through an increase in plant size over time or through increased planting density, some plants will be lost from the population so that size/density (i.e., <i>w:d</i>) trajectories over time or across planting densities follow the boundary line. The intensity of competition increases and plant allocation between body parts changes as the boundary line is approached. This also is important in crop husbandry. For example, height or leaf accumulation may be favoured at the expense of reproductive yield or bulb development.</p><p>Data from such studies suggest that an effective tactical approach for fodder beet production involves planting at 8 plants per m<sup>2</sup>, allowing approximately 60 days for leaf area development, followed by 90 days for carbohydrate translocation to support bulb fill. At this plant density, bulbs are comparatively large (which is desirable), and during the bulb-fill growth stage, the crop accumulates bulb dry weight at rates that can exceed 350 kg DM ha<sup>−1</sup> day<sup>−1</sup>. During the leaf area development phase, there is opportunity for weeds to colonize bare soil, and weed control—often requiring a costly herbicide combination—is critical (Matthew et al., <span>2011</span>). For maize, experimental data from Wisconsin showed that the optimal plant density for grain production was approximately 6000 plants m<sup>−2</sup> lower than that for silage production. This occurred because for silage the forage biomass gains from a higher planting density of around 80 000 plants ha<sup>−1</sup> outweighed the competition-induced loss in grain yield above 75 000 plants ha<sup>−1</sup> (Cusicanqui & Lauer, <span>1999</span>). In oil palm plantations, the optimal tree spacing in a triangular planting pattern represents a compromise between higher tree density for increased fruit yield and wider spacing to slow height growth and extend the period during which manual harvesting will remain feasible. In one study, a decrease in planting distance from 9.5 to 7.5 m increased tree height by 50 cm from 3.5 to 4.0 m after 14 years (Bonneau & Impens, <span>2022</span>). In forestry science, the self-thinning line has been parameterized on a species-specific basis (Pretzsch & Biber, <span>2005</span>). This is because it is an important reference line for density control in the context of thinning. If the species-specific lines are known, suboptimal densities and the resulting production losses can be avoided.</p><p>Davies (<span>1988</span>) in her Figure 3.6 observed that data for tiller weight and density from perennial ryegrass swards fit the −3/2 boundary rule, citing other research in support. However, on closer scrutiny, this conclusion is an oversimplification as data for larger tillers sit above the trend line, while data for smaller tillers sit below it. The actual slope for Davies' data set is more like −5/2, rather than −3/2. This discrepancy between presumed and actual slope calls to mind a comment of Mrad et al. (<span>2020</span>) who reviewed eight potential mechanisms to explain the −3/2 power rule and in their closing comments cited Russian physicist Lev Landau: “Money is in the exponent, and the exponent needs to be calculated precisely.” On reflection, it is evident that the operation of the self-thinning rule will be a far more complex process for a grass sward than for a forest tree stand. Forest trees are perennial and must “average” their population density across seasons. In most tree species and forests, the population density is fixed across time after establishment, or recruitment is slow. A few tree species can produce new shoots from roots, meaning the population is dynamic. Grass tillers by contrast often have a life of less than 1 year and may have a dormant period during winter cold or summer drought, when leaves have senesced and the shoot apical meristem positioned near the ground level survives as a quiescent bud cocooned in undeveloped leaf primordia or sheaths of mature dead leaves. They can also readily generate new shoots from axillary buds or crown buds in the case of alfalfa. Hence, shoot population in grass swards and in forage crops such as alfalfa can fluctuate dynamically through a season or through a regrowth cycle following defoliation.</p><p>Matthew et al. (<span>1995</span>), in their analysis of tiller and shoot density data for ryegrass swards and alfalfa stands, conceptualized the −3/2 boundary line as a constant leaf area line for different shoot size–density combinations, representing the maximum leaf area index (LAI) that a given environment can support. For swards with coordinates plotting above the boundary line, the rate of leaf senescence would exceed that of leaf formation, whereas for those below the line, leaf formation rate would exceed senescence—thus defining the boundary line, or positions along it, as points of equilibrium. These authors proposed four phases of size–density dynamics during the defoliation and regrowth cycle of a grass sward: (i) initiation of new shoots in early regrowth to accelerate LAI recovery, with population density and LAI increasing; (ii) near −5/2 self-thinning, where LAI continues to rise while smaller or younger shoots die due to basal shading by larger, expanding shoots; (iii) size–density compensation along the −3/2 self-thinning line at constant sward LAI, with herbage mass increasing from pseudostem accumulation to support leaves as surviving shoots increase in height; and (iv) a phase of constant herbage mass, representing a “ceiling herbage mass” characteristic of a specific vegetation type. Phase (iv), by mathematical necessity, requires 1:1 self-thinning if there is any increase over time in mean shoot size. Using this model, the authors developed a slope correction to account for the “steeper than −3/2” self-thinning linked to LAI increase in phase (ii) and a plant shape correction to account for the impact of change in plant shape during regrowth on the population required to provide the environmentally sustainable LAI. The plant shape correction was based on a simple dimensionless parameter: m<sup>2</sup> leaf per (m<sup>3</sup> volume)<sup>2/3</sup>, which was designated “<i>R</i>” and seemed effective in understanding the impact of treatments such as shading on plant behaviour. For example, in alfalfa, shading reduced <i>R</i> by suppressing branching (unpublished data). We also proposed that the distance of a point, defined by tiller size–density coordinates, from an arbitrarily positioned self-thinning line, could be used as a productivity index for swards subjected to different treatments within a common environment (Hernández Garay et al., <span>1999</span>; Figure 7.2 of Matthew et al., <span>2000</span>).</p><p>One particularly interesting observation from visualizing sward size–density data in this way was that size–density compensation trajectories for perennial ryegrass and white clover within the same swards ran in opposite directions, across four different grazing intensities defined by postgrazing herbage mass targets (kg DM ha<sup>−1</sup>) (Figure 1a,b; Yu et al., <span>2008</span>). We interpret this as indicating that white clover occupies a light interception niche that perennial ryegrass cannot exploit, due to limited carbohydrate availability for new tiller initiation under defoliation pressure (Figure 1c).</p><p>Scaling theory also provides an intuitively logical basis for key principles of grazing optimization theory and conceptualization of environmental carrying capacity. We see from Figure 1c that heavy grazing pressure limits a sward's ability to generate leaf area, a condition likely linked to reduced carbohydrate reserves and overall plant vigour (Fulkerson & Donaghy, <span>2001</span>). In the middle range of the self-thinning diagram, herbage accumulation can occur with comparatively little shoot death. We also see from the self-thinning diagram or from Figure 7.2 of Matthew et al. (<span>2000</span>) that a grazing management regime that delivers a lower density of larger tillers may well be inherently better for leaf area development and sward productivity than one that delivers a higher density of smaller tillers. This point is not always appreciated by agronomists when formulating pasture management advice. It is tempting to believe that a higher tiller density must be better. Meanwhile, at the upper left end of the self-thinning line (beyond the range shown in Figure 1c), the transition to 1:1 scaling means the death rate of shoots and species diversity loss may be exacerbated in prolonged periods without grazing. These principles of ensuring sufficient leaf area for grassland vegetation to meet its energy needs on one hand while avoiding negative consequences of zero grazing on the other hand deserve to be incorporated into the formulation of management guidelines when determining the carrying capacity of environmentally sensitive areas and policies concerning their use in livestock farming.</p><p>Another point to note is that although <i>w</i> scales with <i>d</i> at −3/2 in the formulation of Yoda et al. (<span>1963</span>), sward herbage mass (i.e., biomass, <i>b</i>) scales with <i>d</i> at −½, since <i>b</i> = <i>w. d</i> and therefore the <i>b:d</i> slope is less than the <i>w:d</i> slope by exactly 1.0. This implies that whenever a turf or pasture manager deliberately varies mowing or defoliation height or interval—such as when pasture is stockpiled at high herbage mass in autumn for winter feeding—substantial shifts in sward shoot population must occur within the stand to maintain equilibrium along the boundary line. An increase from low to high herbage mass—starting from a high shoot density at the lower mass—suggests that a supra-optimal LAI may develop, leading to leaf loss through senescence and a reduction in shoot population. Subsequent release of the stored herbage may create a situation where there is no longer sufficient shoot density to fully exploit the environmental potential LAI. The interplay under a <i>b:d</i> boundary line of slope −½ for sward shoot <i>w:d</i> coordinates for turf plots cut at different time intervals and heights is visualized in fig. 7.2 of Matthew et al. (<span>2000</span>). In that data set when plots are cut at 14-day intervals, coordinates for 75- and 100-mm cutting heights plot near the −½ <i>b:d</i> boundary line, but plots cut at 50 and 25 mm height plot progressively further below it. In a grazing context, Parmenter and Boswell (<span>1983</span>) found that compared with four or five grazings during winter, which would have avoided reduction in tiller density, spring herbage production on plots grazed only twice in winter and expected to have reduced tiller density as a result, was reduced by 12.3% on average over 3 years.</p><p>It would be nice if the story could end there, but it doesn't. In 1998, Enquist et al. published an analysis of self-thinning in plant stands, based on total aboveground dry weight (<i>w</i>) and stand density (<i>d</i>) data compiled from various studies, spanning approximately 11 orders of magnitude in plant size. Based on theoretical considerations from fractal geometry of vascular distribution networks (as in the stems and branches of trees and shrubs), they derived and present data in their fig. 2 that appear to demonstrate a −4/3 <i>w:d</i> scaling relationship, similar to 4/3 body weight:energy requirement relationships that are well established for animals. This −4/3 relationship in plants is predicted by Enquist et al. (<span>1998</span>) to be invariant with respect to body size and is said to “highlight the universality of the 3/4-power scaling of resource use and the related ¼-power scaling of other structural, functional and ecological attributes.” The analysis of Enquist et al. (<span>1998</span>) was questioned (Kozłowski & Konarzewski, <span>2004</span>), defended by Brown et al. (<span>2005</span>), and questioned again (Kozłowski & Konarzewski, <span>2005</span>) but is now widely regarded as definitive. Enquiry has moved on to exploring the implications at the ecosystem level, leading to the elucidation of the worldwide leaf economic spectrum (Wright et al., <span>2004</span>). Questions about finding a grand unifying theory have been asked (Niklas, <span>2004</span>) but not convincingly answered.</p><p>The writer is currently working on a manuscript exploring a data set of 795 observations, spanning eight orders of magnitude, collected at Technische Universität München in 2003 and designed to provide an empirical test of the conclusions of Enquist et al. (<span>1998</span>). Subject to input from co-authors and peer reviewers, we hope to publish the results of this study in Grassland Research later in the year. Suffice it to say, this is a second instance where the significance lies in the exponent, and the exponent must be calculated precisely. Based on our own results, we propose that an alternative interpretation of the data presented by Enquist et al. (<span>1998</span>) in their Figure 2 would be to fit a scaling line of slope 1:1 to plants weighing less than 10⁰ g (after discarding two outliers of 10⁻⁴ g), while for larger plants, the scaling slope can be either ⅔ or ¾, depending on whether measures of leaf area or plant mass are scaled. It goes without saying that when a single fit-line is placed across a two-phase relationship, the joint fit line will be an average of the two phases. It is not illogical to question whether plants with a dry weight of less than 1.0 g require allometry governed by a vascular distribution network in the same way as trees and shrubs or to ask why plant allometry should be defined solely by resource distribution constraints, with no consideration of resource capture. Our data show that −3⁄2 scaling of certain plant body dimensions—reflecting optimisation of light capture by leaves—and −4⁄3 scaling of mass-related dimensions of other organs, consistent with vascular distribution theory, can be reconciled through phenomena such as the non-1:1 scaling of leaf mass to leaf area. The next step is to integrate these insights with the global leaf economics spectrum (Wright et al., <span>2004</span>). Herein lies another lesson for all researchers: it is a great travesty that the data for our empirical test has remained unpublished on a computer hard drive for over 20 years. PhD graduates transitioning into their first job, along with others holding unpublished datasets, are strongly encouraged to think creatively about how to make time to publish their findings, so they may be considered by the wider scientific community.</p>\",\"PeriodicalId\":100593,\"journal\":{\"name\":\"Grassland Research\",\"volume\":\"4 2\",\"pages\":\"89-92\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2025-07-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/glr2.70020\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Grassland Research\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/glr2.70020\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Grassland Research","FirstCategoryId":"1085","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/glr2.70020","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
这篇社论回顾了植物异速生长的主题。这是一个大量文献的主题,所以这里的报道必然是有选择性的,集中在草原研究的兴趣点上。在我本科学习的最后一年(1983年),我修了三门不同的课程,其中一个模块是基于尤达1963年的研究,“过度拥挤的纯净森林中的自我稀疏”(尤达等人,1963年)。该论文阐明的原则被视为对作物特定畜牧业产量优化建议的理论理解的基础。与此同时,Hutchings(1983)发表了一篇题为《寻找理论的生态学法则》的文章,指出那个时代的研究人员对产生尤达及其同事所描述的植物行为模式的生态驱动因素缺乏共识。简而言之,自疏规则(Yoda et al., 1963)指出,当在对数对数尺度上绘制拥挤林分中植物的单株平均干重(w)值与林分密度时,不同物种的植物或不同年龄的同一物种的植物的点将沿着斜率为- 3/2的直线下降,这被称为“- 3/2边界线”。当林分接近边界线时,例如随着时间的推移植物大小增加或通过增加种植密度,一些植物将从种群中消失,因此大小/密度(即w:d)随时间或跨种植密度的轨迹将遵循边界线。随着边界线的接近,竞争强度增加,各身体部位之间的植物分配也发生了变化。这在农作物养殖中也很重要。例如,以牺牲繁殖产量或鳞茎发育为代价,可能有利于植株的高度或叶片积累。来自这些研究的数据表明,饲料用甜菜生产的有效策略是每平方米种植8株,允许大约60天的叶面积发育,然后用90天的碳水化合物转运来支持球茎填充。在这种植物密度下,鳞茎相对较大(这是理想的),在鳞茎填充生长阶段,作物鳞茎干重的积累速度可超过350 kg DM ha - 1 day - 1。在叶面积发育阶段,杂草有机会在裸露的土壤上繁殖,而杂草控制——通常需要昂贵的除草剂组合——是至关重要的(Matthew et al., 2011)。对于玉米,威斯康辛州的试验数据表明,谷物生产的最佳种植密度约为6000株m−2,比青贮生产低。这是因为对于青贮来说,较高的种植密度(约8万株/公顷)所带来的饲料生物量收益超过了超过7.5万株/公顷(Cusicanqui &;劳尔,1999)。在油棕种植园中,三角形种植模式下的最佳树间距代表了在较高的树密度以提高果实产量和较宽的树间距以减缓高度生长和延长人工采伐仍然可行的时间之间的折衷。在一项研究中,种植距离从9.5米减少到7.5米,14年后树高从3.5米增加到4.0米,增加了50厘米(Bonneau &;Impens, 2022)。在林业科学中,自疏线已在特定物种的基础上参数化(Pretzsch &;Biber, 2005)。这是因为在疏化的情况下,它是密度控制的重要参考线。如果已知特定品种,则可以避免次优密度和由此导致的生产损失。Davies(1988)在她的图3.6中观察到多年生黑麦草的分蘖重量和密度数据符合- 3/2边界规则,并引用了其他支持的研究。然而,仔细观察就会发现,这个结论过于简单化了,因为大型分蘖机的数据位于趋势线上方,而小型分蘖机的数据位于趋势线下方。Davies数据集的实际斜率更像是- 5/2,而不是- 3/2。假设斜率和实际斜率之间的差异让人想起了Mrad等人(2020)的评论,他们回顾了8种解释- 3/2幂法则的潜在机制,并在他们的结语中引用了俄罗斯物理学家Lev Landau:“钱在指数中,指数需要精确计算。”经过反思,很明显,自疏规则的操作对于草地来说将是一个比森林林分复杂得多的过程。森林树木是多年生的,必须在不同季节“平均”它们的人口密度。在大多数树种和森林中,种群密度在建立后的一段时间内是固定的,或者是缓慢的。一些树种可以从根长出新枝,这意味着种群是动态的。 相比之下,禾本科分蘖植物的寿命通常不到1年,在冬季寒冷或夏季干旱时可能会有一个休眠期,此时叶片已经衰老,位于地面附近的茎尖分生组织以静止的芽的形式存在,包裹在未发育的叶原基或成熟死叶的鞘中。它们也可以很容易地从苜蓿的腋芽或冠芽中产生新芽。因此,草地和苜蓿等饲料作物的茎枝数量可以在一个季节或在落叶后的再生周期中动态波动。Matthew et al.(1995)在分析黑麦草和紫花苜蓿草地分蘖和芽密度数据时,将- 3/2边界线定义为不同芽大小-密度组合的叶面积常数线,代表给定环境所能支持的最大叶面积指数(LAI)。对于坐标在边界线以上的树,叶片衰老的速度将超过叶片形成的速度,而对于坐标在边界线以下的树,叶片形成的速度将超过衰老的速度,从而将边界线或沿边界线的位置定义为平衡点。作者提出了草地落叶再生周期中大小-密度动态的四个阶段:(1)随着种群密度和LAI的增加,再生早期新芽的萌发加速了LAI的恢复;(ii)在−5/2附近自疏,LAI继续上升,而较小或较年轻的枝条由于基部被较大的、不断扩大的枝条遮阴而死亡;(iii)在一定的草地LAI条件下,沿−3/2自疏线的尺寸-密度补偿,随着活枝高度的增加,牧草质量从假茎积累到支撑叶逐渐增加;(iv)牧草质量恒定阶段,代表特定植被类型的“天花板牧草质量”特征。阶段(iv),根据数学上的需要,如果随着时间的推移,平均枝梢尺寸有任何增加,则需要1:1的自薄。利用该模型,作者开发了一个斜率校正,以解释第(ii)阶段与LAI增加相关的“大于−3/2”的自疏,以及一个植物形状校正,以解释再生期间植物形状变化对提供环境可持续LAI所需的种群的影响。植物形状校正基于一个简单的无量纲参数:m2叶片/ (m3体积)2/3,该参数被指定为“R”,似乎可以有效地理解遮阳等处理对植物行为的影响。例如,在紫花苜蓿中,遮荫通过抑制分支来降低R(未发表的数据)。我们还提出,由分蘖大小-密度坐标定义的点与任意定位的自疏线之间的距离,可以用作在共同环境中遭受不同处理的禾草的生产力指数(Hernández Garay et al., 1999;图7.2 (Matthew et al., 2000)。以这种方式可视化草地大小-密度数据的一个特别有趣的观察结果是,同一草地内多年生黑麦草和白三叶草的大小-密度补偿轨迹在相反的方向上运行,跨越四种不同的放牧强度,这些放牧强度由放牧后牧草质量目标(kg DM ha - 1)定义(图1a,b;Yu et al., 2008)。我们认为这表明白三叶占据了多年生黑麦草无法利用的光截获生态位,因为在落叶压力下,新分蘖形成所需的碳水化合物有限(图1c)。尺度理论也为放牧优化理论的关键原理和环境承载力的概念化提供了直观的逻辑基础。从图1c中我们可以看到,过度放牧限制了草地产生叶面积的能力,这种情况可能与碳水化合物储备减少和植物整体活力有关(Fulkerson &;多纳吉,2001)。在自疏图的中间范围,牧草积累发生,茎部死亡相对较少。我们还从Matthew et al.(2000)的自疏图或图7.2中看到,相对于小蘖密度较高的放牧管理制度而言,大蘖密度较低的放牧管理制度更有利于叶面积发展和草地生产力。在制定牧场管理建议时,农学家并不总是欣赏这一点。人们很容易相信分蘖密度越高越好。同时,在自疏线的左上端(超出图1c所示范围),过渡到1:1的尺度意味着长时间不放牧可能会加剧嫩枝死亡率和物种多样性的丧失。 这些原则一方面确保草地植被有足够的叶面积以满足其能量需求,另一方面避免零放牧的负面后果,在确定环境敏感地区的承载能力和在畜牧业中使用环境敏感地区的政策时,应纳入制定管理准则。另一点需要注意的是,虽然在尤达等人(1963)的公式中,w的尺度是d为- 3/2,但由于b = w,草叶牧草质量(即生物量,b)的尺度是d为- 1 /2。因此b: D的斜率比w: D的斜率小1.0。这意味着,当草皮或牧场管理者故意改变刈割或落叶的高度或间隔时——比如在秋季为冬季取食而大量储存牧草时——林分内的草枝数量必须发生实质性的变化,以保持边界线上的平衡。牧草质量由低到高——从较低质量的高茎密度开始——表明可能出现超优LAI,导致叶片衰老损失和茎枝数量减少。随后释放储存的牧草可能会造成不再有足够的梢密度来充分利用环境潜力LAI的情况。马修等人(2000)的图7.2显示了不同时间间隔和高度切割草坪地块的斜率为- 1 / 2的b:d边界线下的相互作用。在该数据集中,当地块每隔14天切割一次时,75和100毫米切割高度的坐标在−½b:d边界线附近绘制,但在50和25毫米高度切割的地块逐渐在其下方绘制。在放牧的背景下,Parmenter和Boswell(1983)发现,与冬季放牧4或5次可以避免分蘖密度减少的情况相比,在冬季只放牧两次并预计会减少分蘖密度的地块上,春季牧草产量在3年内平均减少了12.3%。如果故事能就此结束就好了,但事实并非如此。1998年,Enquist等人发表了一篇关于植物林分自疏的分析,该分析基于从各种研究中收集的总地上干重(w)和林分密度(d)数据,涵盖了植物大小的大约11个数量级。基于维管分布网络(如树木和灌木的茎和枝)的分形几何理论考虑,他们得出并在图2中展示了数据,这些数据似乎证明了- 4/3 w:d比例关系,类似于动物体重:能量需求的4/3关系。Enquist等人(1998)预测,植物的- 4/3关系在体型方面是不变的,据说“突出了资源利用的3/4次方比例的普遍性,以及其他结构、功能和生态属性的相关¼次方比例。”Enquist等人(1998)的分析受到质疑(Kozłowski &;Konarzewski, 2004),被Brown等人(2005)辩护,并再次受到质疑(Kozłowski &;Konarzewski, 2005),但现在被广泛认为是决定性的。调查已经转移到探索生态系统层面的影响,从而阐明了全球叶片经济光谱(Wright et al., 2004)。关于寻找大统一理论的问题已经被提出(Niklas, 2004),但没有令人信服的答案。作者目前正在撰写一份手稿,探索2003年在Technische Universität m<e:1> nchen收集的795个观测数据集,跨越8个数量级,旨在为Enquist et al.(1998)的结论提供经验检验。根据共同作者和同行审稿人的意见,我们希望在今年晚些时候在《草地研究》上发表这项研究的结果。我只想说,这是第二个指数有意义的例子,而指数必须精确地计算。根据我们自己的结果,我们建议对Enquist et al.(1998)在他们的图2中提供的数据进行另一种解释,即将斜率为1:1的缩放线拟合到重量小于10⁰g的植物上(在丢弃两个10⁻⁴的例外值之后),而对于较大的植物,缩放斜率可以是2 / 3或3 / 4,这取决于叶片面积或植物质量的测量是否被缩放。不言而喻,当在两相关系上放置一条拟合线时,联合拟合线将是两相的平均值。问干重小于1.0 g的植物是否需要像树木和灌木一样由维管分布网络控制的异速生长,或者问为什么植物异速生长应该仅仅由资源分布约束来定义,而不考虑资源捕获,这并不是不合逻辑的。 我们的数据表明,某些植物体尺寸的- 3 / 2缩放(反映了叶片对光捕获的优化)和其他器官质量相关尺寸的- 4 / 3缩放(符合维管分布理论)可以通过叶质量与叶面积的非1:1缩放等现象进行协调。下一步是将这些见解与全球叶片经济学光谱相结合(Wright et al., 2004)。对所有研究人员来说,这是另一个教训:我们的实证测试数据在电脑硬盘上保存了20多年而未发表,这是一个极大的讽刺。正过渡到第一份工作的博士毕业生,以及其他持有未发表数据集的人,强烈鼓励他们创造性地思考如何抽出时间发表他们的发现,这样他们就可以被更广泛的科学界所考虑。
This editorial revisits the topic of plant allometry. This topic is the subject of a large volume of literature, so coverage here is necessarily selective, focusing on points of interest for grassland research. In my final year of undergraduate study (1983), three different courses I took included a module based on Yoda's 1963 study, “Self-thinning in overcrowded pure stands” (Yoda et al., 1963). Principles elucidated in that paper were seen as fundamental to the theoretical understanding of crop-specific husbandry recommendations for yield optimization. Meanwhile, Hutchings (1983) published an article “Ecology's law in search of a theory,” indicating a lack of consensus among researchers of that era as to what ecological drivers were operating to produce the plant behaviour patterns Yoda and colleagues had described.
Briefly, the self-thinning rule (Yoda et al., 1963) states that when values for single plant mean dry weight (w) for plants in a crowded stand are plotted against stand density on a log–log scale, the points for plants of different species or plants of the same species at different ages will fall along a line of slope −3/2, which became known as the “−3/2 boundary line.” As a stand approaches the boundary line, for example through an increase in plant size over time or through increased planting density, some plants will be lost from the population so that size/density (i.e., w:d) trajectories over time or across planting densities follow the boundary line. The intensity of competition increases and plant allocation between body parts changes as the boundary line is approached. This also is important in crop husbandry. For example, height or leaf accumulation may be favoured at the expense of reproductive yield or bulb development.
Data from such studies suggest that an effective tactical approach for fodder beet production involves planting at 8 plants per m2, allowing approximately 60 days for leaf area development, followed by 90 days for carbohydrate translocation to support bulb fill. At this plant density, bulbs are comparatively large (which is desirable), and during the bulb-fill growth stage, the crop accumulates bulb dry weight at rates that can exceed 350 kg DM ha−1 day−1. During the leaf area development phase, there is opportunity for weeds to colonize bare soil, and weed control—often requiring a costly herbicide combination—is critical (Matthew et al., 2011). For maize, experimental data from Wisconsin showed that the optimal plant density for grain production was approximately 6000 plants m−2 lower than that for silage production. This occurred because for silage the forage biomass gains from a higher planting density of around 80 000 plants ha−1 outweighed the competition-induced loss in grain yield above 75 000 plants ha−1 (Cusicanqui & Lauer, 1999). In oil palm plantations, the optimal tree spacing in a triangular planting pattern represents a compromise between higher tree density for increased fruit yield and wider spacing to slow height growth and extend the period during which manual harvesting will remain feasible. In one study, a decrease in planting distance from 9.5 to 7.5 m increased tree height by 50 cm from 3.5 to 4.0 m after 14 years (Bonneau & Impens, 2022). In forestry science, the self-thinning line has been parameterized on a species-specific basis (Pretzsch & Biber, 2005). This is because it is an important reference line for density control in the context of thinning. If the species-specific lines are known, suboptimal densities and the resulting production losses can be avoided.
Davies (1988) in her Figure 3.6 observed that data for tiller weight and density from perennial ryegrass swards fit the −3/2 boundary rule, citing other research in support. However, on closer scrutiny, this conclusion is an oversimplification as data for larger tillers sit above the trend line, while data for smaller tillers sit below it. The actual slope for Davies' data set is more like −5/2, rather than −3/2. This discrepancy between presumed and actual slope calls to mind a comment of Mrad et al. (2020) who reviewed eight potential mechanisms to explain the −3/2 power rule and in their closing comments cited Russian physicist Lev Landau: “Money is in the exponent, and the exponent needs to be calculated precisely.” On reflection, it is evident that the operation of the self-thinning rule will be a far more complex process for a grass sward than for a forest tree stand. Forest trees are perennial and must “average” their population density across seasons. In most tree species and forests, the population density is fixed across time after establishment, or recruitment is slow. A few tree species can produce new shoots from roots, meaning the population is dynamic. Grass tillers by contrast often have a life of less than 1 year and may have a dormant period during winter cold or summer drought, when leaves have senesced and the shoot apical meristem positioned near the ground level survives as a quiescent bud cocooned in undeveloped leaf primordia or sheaths of mature dead leaves. They can also readily generate new shoots from axillary buds or crown buds in the case of alfalfa. Hence, shoot population in grass swards and in forage crops such as alfalfa can fluctuate dynamically through a season or through a regrowth cycle following defoliation.
Matthew et al. (1995), in their analysis of tiller and shoot density data for ryegrass swards and alfalfa stands, conceptualized the −3/2 boundary line as a constant leaf area line for different shoot size–density combinations, representing the maximum leaf area index (LAI) that a given environment can support. For swards with coordinates plotting above the boundary line, the rate of leaf senescence would exceed that of leaf formation, whereas for those below the line, leaf formation rate would exceed senescence—thus defining the boundary line, or positions along it, as points of equilibrium. These authors proposed four phases of size–density dynamics during the defoliation and regrowth cycle of a grass sward: (i) initiation of new shoots in early regrowth to accelerate LAI recovery, with population density and LAI increasing; (ii) near −5/2 self-thinning, where LAI continues to rise while smaller or younger shoots die due to basal shading by larger, expanding shoots; (iii) size–density compensation along the −3/2 self-thinning line at constant sward LAI, with herbage mass increasing from pseudostem accumulation to support leaves as surviving shoots increase in height; and (iv) a phase of constant herbage mass, representing a “ceiling herbage mass” characteristic of a specific vegetation type. Phase (iv), by mathematical necessity, requires 1:1 self-thinning if there is any increase over time in mean shoot size. Using this model, the authors developed a slope correction to account for the “steeper than −3/2” self-thinning linked to LAI increase in phase (ii) and a plant shape correction to account for the impact of change in plant shape during regrowth on the population required to provide the environmentally sustainable LAI. The plant shape correction was based on a simple dimensionless parameter: m2 leaf per (m3 volume)2/3, which was designated “R” and seemed effective in understanding the impact of treatments such as shading on plant behaviour. For example, in alfalfa, shading reduced R by suppressing branching (unpublished data). We also proposed that the distance of a point, defined by tiller size–density coordinates, from an arbitrarily positioned self-thinning line, could be used as a productivity index for swards subjected to different treatments within a common environment (Hernández Garay et al., 1999; Figure 7.2 of Matthew et al., 2000).
One particularly interesting observation from visualizing sward size–density data in this way was that size–density compensation trajectories for perennial ryegrass and white clover within the same swards ran in opposite directions, across four different grazing intensities defined by postgrazing herbage mass targets (kg DM ha−1) (Figure 1a,b; Yu et al., 2008). We interpret this as indicating that white clover occupies a light interception niche that perennial ryegrass cannot exploit, due to limited carbohydrate availability for new tiller initiation under defoliation pressure (Figure 1c).
Scaling theory also provides an intuitively logical basis for key principles of grazing optimization theory and conceptualization of environmental carrying capacity. We see from Figure 1c that heavy grazing pressure limits a sward's ability to generate leaf area, a condition likely linked to reduced carbohydrate reserves and overall plant vigour (Fulkerson & Donaghy, 2001). In the middle range of the self-thinning diagram, herbage accumulation can occur with comparatively little shoot death. We also see from the self-thinning diagram or from Figure 7.2 of Matthew et al. (2000) that a grazing management regime that delivers a lower density of larger tillers may well be inherently better for leaf area development and sward productivity than one that delivers a higher density of smaller tillers. This point is not always appreciated by agronomists when formulating pasture management advice. It is tempting to believe that a higher tiller density must be better. Meanwhile, at the upper left end of the self-thinning line (beyond the range shown in Figure 1c), the transition to 1:1 scaling means the death rate of shoots and species diversity loss may be exacerbated in prolonged periods without grazing. These principles of ensuring sufficient leaf area for grassland vegetation to meet its energy needs on one hand while avoiding negative consequences of zero grazing on the other hand deserve to be incorporated into the formulation of management guidelines when determining the carrying capacity of environmentally sensitive areas and policies concerning their use in livestock farming.
Another point to note is that although w scales with d at −3/2 in the formulation of Yoda et al. (1963), sward herbage mass (i.e., biomass, b) scales with d at −½, since b = w. d and therefore the b:d slope is less than the w:d slope by exactly 1.0. This implies that whenever a turf or pasture manager deliberately varies mowing or defoliation height or interval—such as when pasture is stockpiled at high herbage mass in autumn for winter feeding—substantial shifts in sward shoot population must occur within the stand to maintain equilibrium along the boundary line. An increase from low to high herbage mass—starting from a high shoot density at the lower mass—suggests that a supra-optimal LAI may develop, leading to leaf loss through senescence and a reduction in shoot population. Subsequent release of the stored herbage may create a situation where there is no longer sufficient shoot density to fully exploit the environmental potential LAI. The interplay under a b:d boundary line of slope −½ for sward shoot w:d coordinates for turf plots cut at different time intervals and heights is visualized in fig. 7.2 of Matthew et al. (2000). In that data set when plots are cut at 14-day intervals, coordinates for 75- and 100-mm cutting heights plot near the −½ b:d boundary line, but plots cut at 50 and 25 mm height plot progressively further below it. In a grazing context, Parmenter and Boswell (1983) found that compared with four or five grazings during winter, which would have avoided reduction in tiller density, spring herbage production on plots grazed only twice in winter and expected to have reduced tiller density as a result, was reduced by 12.3% on average over 3 years.
It would be nice if the story could end there, but it doesn't. In 1998, Enquist et al. published an analysis of self-thinning in plant stands, based on total aboveground dry weight (w) and stand density (d) data compiled from various studies, spanning approximately 11 orders of magnitude in plant size. Based on theoretical considerations from fractal geometry of vascular distribution networks (as in the stems and branches of trees and shrubs), they derived and present data in their fig. 2 that appear to demonstrate a −4/3 w:d scaling relationship, similar to 4/3 body weight:energy requirement relationships that are well established for animals. This −4/3 relationship in plants is predicted by Enquist et al. (1998) to be invariant with respect to body size and is said to “highlight the universality of the 3/4-power scaling of resource use and the related ¼-power scaling of other structural, functional and ecological attributes.” The analysis of Enquist et al. (1998) was questioned (Kozłowski & Konarzewski, 2004), defended by Brown et al. (2005), and questioned again (Kozłowski & Konarzewski, 2005) but is now widely regarded as definitive. Enquiry has moved on to exploring the implications at the ecosystem level, leading to the elucidation of the worldwide leaf economic spectrum (Wright et al., 2004). Questions about finding a grand unifying theory have been asked (Niklas, 2004) but not convincingly answered.
The writer is currently working on a manuscript exploring a data set of 795 observations, spanning eight orders of magnitude, collected at Technische Universität München in 2003 and designed to provide an empirical test of the conclusions of Enquist et al. (1998). Subject to input from co-authors and peer reviewers, we hope to publish the results of this study in Grassland Research later in the year. Suffice it to say, this is a second instance where the significance lies in the exponent, and the exponent must be calculated precisely. Based on our own results, we propose that an alternative interpretation of the data presented by Enquist et al. (1998) in their Figure 2 would be to fit a scaling line of slope 1:1 to plants weighing less than 10⁰ g (after discarding two outliers of 10⁻⁴ g), while for larger plants, the scaling slope can be either ⅔ or ¾, depending on whether measures of leaf area or plant mass are scaled. It goes without saying that when a single fit-line is placed across a two-phase relationship, the joint fit line will be an average of the two phases. It is not illogical to question whether plants with a dry weight of less than 1.0 g require allometry governed by a vascular distribution network in the same way as trees and shrubs or to ask why plant allometry should be defined solely by resource distribution constraints, with no consideration of resource capture. Our data show that −3⁄2 scaling of certain plant body dimensions—reflecting optimisation of light capture by leaves—and −4⁄3 scaling of mass-related dimensions of other organs, consistent with vascular distribution theory, can be reconciled through phenomena such as the non-1:1 scaling of leaf mass to leaf area. The next step is to integrate these insights with the global leaf economics spectrum (Wright et al., 2004). Herein lies another lesson for all researchers: it is a great travesty that the data for our empirical test has remained unpublished on a computer hard drive for over 20 years. PhD graduates transitioning into their first job, along with others holding unpublished datasets, are strongly encouraged to think creatively about how to make time to publish their findings, so they may be considered by the wider scientific community.