On plant scaling

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.