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# Insights into plant size-density relationships from models and agricultural crops

Contributed by James H. Brown, April 6, 2012 (sent for review January 31, 2012)

## Abstract

There is general agreement that competition for resources results in a tradeoff between plant mass, *M*, and density, but the mathematical form of the resulting thinning relationship and the mechanisms that generate it are debated. Here, we evaluate two complementary models, one based on the space-filling properties of canopy geometry and the other on the metabolic basis of resource use. For densely packed stands, both models predict that density scales as *M*^{−3/4}, energy use as *M*^{0}, and total biomass as *M*^{1/4}. Compilation and analysis of data from 183 populations of herbaceous crop species, 473 stands of managed tree plantations, and 13 populations of bamboo gave four major results: (*i*) At low initial planting densities, crops grew at similar rates, did not come into contact, and attained similar mature sizes; (*ii*) at higher initial densities, crops grew until neighboring plants came into contact, growth ceased as a result of competition for limited resources, and a tradeoff between density and size resulted in critical density scaling as *M*^{−0.78}, total resource use as *M*^{−0.02}, and total biomass as *M*^{0.22}; (*iii*) these scaling exponents are very close to the predicted values of *M*^{−3/4}, *M*^{0}, and *M*^{1/4}, respectively, and significantly different from the exponents suggested by some earlier studies; and (*iv*) our data extend previously documented scaling relationships for trees in natural forests to small herbaceous annual crops. These results provide a quantitative, predictive framework with important implications for the basic and applied plant sciences.

The structure and dynamics of plant populations and communities often reflect the interacting consequences of three fundamental processes: (*i*) competition for resources, (*ii*) the effect of body size on resource use, and (*iii*) the effect of plant density on growth and mortality (1⇓⇓–4). Two approaches traditionally have been used to study these interactions. One focuses on theoretical models and empirical measurements of abundance, spacing, survival, mortality, and recruitment as functions of plant size in relatively undisturbed natural populations and communities, especially forests (4⇓⇓⇓⇓⇓⇓–11), where the thinning process is complicated by effects of shading and other factors on asymmetries in resource supply and resulting growth and mortality rates (11⇓⇓⇓⇓–16). The second approach focuses on the structure and dynamics of plants in agricultural settings (17⇓⇓⇓⇓–22), where plants of nearly identical age grow under controlled conditions. Studies of such simplified agricultural systems have led to theoretical and empirical self-thinning relationships that characterize the temporal trajectory of decreasing population density as a function of increasing plant size as stands develop under conditions of resource limitation and competition (17⇓–19). These exhibit a characteristic phenomenology in which plants grow with minimal mortality until they reach a size-dependent critical density, *N _{crit}*, where all available resources are used; thus, further growth of some individuals is only possible if resources are made available by mortality of other individuals. In a resource-limited environment, the scaling of resource use with plant size typically results in a tradeoff between density and plant size that takes the formwhere

*N*is the density of maximally packed individuals with average mass , is a normalization constant, and α is a scaling exponent. When plotted on logarithmic axes, this equation gives a linear relationship, where indexes the elevation or log-log

_{crit}*y*-intercept and α is the log-log slope (Fig. 1).

Often, the initial density, such as the seeding density in agricultural fields, is less than *N _{crit}*. In these cases, plants typically grow with minimal mortality until they start to experience resource limitation and competition, after which they follow a thinning line, with some individuals dying and freeing up resources that surviving individuals use to grow (17). If the initial density is sufficiently low, plants never reach a critical thinning density and they mature or are harvested at some constant maximal size (Fig. 1).

For decades, plant scientists have debated the empirical form of the thinning line and the mechanisms that give rise to it (18, 21⇓⇓⇓⇓⇓–27). Much of the debate has revolved around the value of the exponent, α. The classic studies of Yoda et al. (18) and Hutchings (28), based on the geometry of canopies, suggest that α = −2/3, whereas more recent theoretical studies, based on the allometric scaling of metabolic rate and resource use, suggest that α = −3/4 (7, 9, 29). It should be noted in this context that earlier studies traditionally treated *N _{crit}* as the independent variable and as the dependent variable, such that the exponent of the “thinning law” would be either −3/2 or −4/3 rather than −2/3 or −3/4. Here, we follow standard practice in allometric analysis and treat density and mass as the dependent and independent variables, respectively, plotting on the

*y*axis and on the

*x*axis. Note, however, that there are no strictly “independent” and “dependent” variables. Both density and size are emergent outcomes of the competitive process.

Here, we show that two very different theoretical models, one based on biophysical constraints on how plants occupy space when “closely packed” and another based on metabolism and energy use, predict the same density-size scaling relationship, with α = −3/4, which can be extended to predict scaling of whole-stand energy use and biomass. We then use data from crop plants to evaluate the assumptions and predictions of these models.

## Models

We present two models that predict how the critical density scales with plant size. Both models assume that (*i*) monocultures of annual crop plants start from seeds planted with regular spacing in a homogeneous environment; (*ii*) seeds germinate simultaneously, seedlings are initially the same size, and all plants grow at the same maximal rate until canopies intersect; and (*iii*) growth stops and plants mature and are harvested when canopies overlap maximally so that all resources are used.

### Biophysical Packing Model.

The relationship between critical plant density, *N _{crit}*, and plant mass,

*M*, can be modeled by considering how plants occupy a limited space and how this packing gives the relationship between plant density and canopy radius,

*r*, for any monoculture of plants of the same size. When plants are small in comparison to the ground area, canopies of neighboring plants do not make contact, resources go unused, and there is no competition (Fig. 2

*A*). When the size of plants is large in comparison to the planting area, canopies of neighboring plants touch and intersect as plants grow in size (Fig. 2

*B*). The

*N*is defined as the density associated with the “critical plant size” when canopy coverage is maximally packed, such thatwhere

_{crit}*n*and

_{A}*n*are the number of plants in the orthogonal dimensions of the planted field (Fig. 2

_{B}*B*). Biophysical theory and empirical data indicate that plant mass is proportional to the 8/3 power of any reference linear dimension of plant size, such as plant height or canopy radius (i.e.,

*M*∝

*r*

^{8/3}) (30, 31), from which it follows that

*r*

^{2}∝

*M*

^{3/4}and

It should be noted that the *M* ∝ *r*^{8/3} relationship emerges as a consequence of the growth and light-harvesting requirements of plants with self-supporting stems and that these requirements are consistent with the elastic requirements for mechanical stability (31). In general, the area of any canopy is , where *l* is the length of one side of the packing arrangement, such that , and *f* is some fraction of canopy radius that depends on the planting pattern. It follows that the critical density is proportional to *l*^{2}, such that . These geometric packing relationships can be extended into the third dimension of canopy height to model the extent to which the canopies of neighboring plants intersect as a function of the critical density, *N _{crit}*, and a linear dimension of plant size, such as canopy radius or height (

*SI Methods: Analysis of Canopy Competition*).

### Metabolic Model.

The second model is based on three assumptions about resource supply and resource use: (*i*) a limiting resource, here assumed to be light, is supplied to an area of ground at a fixed energy rate, *E*, (*ii*) when plants are densely packed and competing, all resources are used, such that the rate of resource use by all plants, *R*, equals the rate of resource supply, , and (*iii*) the rate of resource use per plant or metabolic rate, *B*, scales as the 3/4 power of body mass, such that . We assume an exponent of 3/4 here, based on theoretical analyses and empirical studies of energy use as a function of plant size in natural stands subject to competitive thinning (7, 9, 29, 32–34). Because the rate of resource use per ground area by all plants is simply the total rate of resource use per plant times the number of plants per area (i.e., ), and because the critical density occurs when the total rate of resource use equals the rate of resource supply, it follows thatand because *E* is a parameter of the environment, and therefore independent of plant size, this means that , such thatthus, , which is the same as Eq. **3**.

A caveat is that direct empirical measurements of metabolic rate as a function of plant size suggest scaling exponents between 3/4 and 1 (34⇓–36). We suggest that the steeper exponents may reflect small plants growing in uncrowded, noncompetitive conditions. Keeping just assumptions (*i*) and (*ii*), Eqs. **4** and **5** can be modified for the general case in which and . It necessarily follows that , such that γ = −α. Once the scaling of and the value of α have been determined empirically (*Empirical Tests*), the assumption that γ = 3/4 can be reevaluated.

### Models of Whole-Stand Energy Use and Biomass.

These models can be extended to predict scaling of energy use and biomass of entire stands. Eq. **5** gives the condition of “energy equivalence,” where the rate of energy use by a stand is constant, independent of plant size, because energy use per plant and plant density scale with reciprocal exponents (7, 37, 38). Moreover, by definition, total biomass per unit ground area, *W*, is the product of density times average plant size. Therefore, the *M*^{−3/4} scaling of density leads straightforwardly to the prediction that

Our two thinning models are consistent at a number of levels. Both rely on established allometric relationships, for biomechanics and metabolism, respectively, to predict how competition affects packing of individuals, and they then use this to predict how density, energy use, and biomass of stands scale with plant size (Eqs. **3**, **5**, and **6**). The two models are based on different constraints and processes. The packing model invokes well-established biophysical and geometric constraints, whereas the metabolic model is based on scaling of resource use. The packing and metabolic models are based explicitly on competition for space and light, respectively, but they are equivalent because all light is effectively used when the canopies are maximally packed and all space is filled. Of course, the structural and functional requirements for mechanical support and metabolism must be integrated at the whole-plant level. Thus, it is encouraging that the two models, based on different constraints on traits of individual plants, make the same predictions.

## Empirical Tests

We tested the model predictions using newly acquired data from a set of field experiments with six species of herbaceous crops planted at different densities. For comparison, we also compiled and analyzed existing data for 473 stands of trees in plantations, 13 populations of bamboo, and 1,338 natural forests undergoing secondary succession following a disturbance event.

### Density-Size Scaling.

Data on scaling of density as a function of average mass, , in the several datasets are presented in Table 1 and Fig. 3 (Dataset S1). Fig. 3*A* shows the data for annual crops, where some plantings started with such low seed densities that they did not reach *N _{crit}* before the plants attained mature size and were harvested. The data for these crops strongly support the prediction that all plants grow at nearly the same maximal rate and experience virtually no mortality until they begin to compete for resources. The critical density,

*N*, is determined by the size at which growth stops as a result of competition (Fig. 1). Within the same set of experiments using the same species, plantings at initial densities lower than

_{crit}*N*matured with nearly identical plant sizes, whereas plantings at initial densities greater than

_{crit}*N*experienced competitive inhibition and matured with plant sizes aligned along the thinning line. For example,

_{crit}*N*for well-watered clover (

_{crit}*Trifolium repens*) was ∼10

^{3}m

^{−2}; plantings at lower densities all matured with a similar average plant size somewhat greater than 1 g, whereas plantings at higher densities clustered around the common thinning line (Fig. 3

*A*). Across the different experiments with different species and varying environmental conditions, however,

*N*and size at maturity varied by several orders of magnitude. Thus, the data for crops suggest the thinning described by Eq.

_{crit}**1**and Fig. 1, with a common exponent close to −3/4.

Exponents estimated for the entire dataset and different subsamples are given in Fig. 3*B* and Table 1. Fig. 3*B* plots all the data, including the newly acquired data on crops, bamboo, and plantations from this study and previously published data for natural forests undergoing succession. Fitted slopes for log *N _{crit}* as a function of log were all between −0.72 and −0.80, and all confidence intervals included the predicted −3/4 (Table 1). Because

*N*scales with plant mass with an exponent very close to −3/4, this necessarily implies that metabolic rate, or energy use per individual scales with an exponent very close to 3/4, supporting assumption (

_{crit}*iii*) of the metabolic model. Correcting the above relationships for variation in average annual precipitation and temperature had little effect on the estimated scaling relationships (Table S1). Although water availability and temperature may affect absolute rates of plant growth, thinning, and succession, they do not appear to alter the scaling exponents, at least in areas where forests were the historic vegetation type.

### Scaling of Whole-Stand Energy Use and Biomass.

Data used to estimate scaling of energy use and total biomass per unit ground area as a function of average plant size for entire stands are presented in Table 1 and Fig. 4. The newly acquired data for crops, bamboo, and plantations fit well with the previously published data for natural forests but extend the scaling relationships to much smaller plant sizes. Exponents calculated by substitution using Eqs. **3**, **5**, and **6** are reasonably close to exponents estimated by fitting regressions (Table 1 and *Methods*). Regression slopes for biomass of the different plant types, except for bamboo, are steeper than the theoretically predicted exponent of 1/4. We are not sure how much stock to put in this possible discrepancy, especially because the calculated slopes for these groups (0.20–0.28) and both the calculated and regression slopes for all plants (0.22–0.26) are similar and very close to 1/4. Analyses of the combined data show that plant size alone accounts for most of the variation in density, energy use, and biomass: Plant size varies over 10 orders of magnitude from small herbs to large trees, but the variation in *N _{crit}*,

*E*, and

*W*among plants of a given size is only about 1 order of magnitude.

## Discussion

We have presented two mutually complementary models that predict the same density-size scaling relationship for plants competing for limiting resources, and we have shown that the predictions of these models are statistically congruent, with large empirical datasets that span a wide range of managed and natural plant populations and communities, from monocultures of monocot and dicot crops to diverse, complex natural forests. Therefore, the density-size relationships previously documented for forests (7⇓–9) can be generalized to a much wider range of plant sizes, growth habits, and agricultural as well as undisturbed ecosystems. Moreover, the self-thinning scaling relationship that emerges as a consequence of the tradeoff between growth and mortality in forests appears to be identical to the scaling of *N _{crit}* with plant size in crops, where there is very little mortality. It is as if artificial selection to grow and to mature synchronously at a size that maximizes crop yield “freezes” the thinning process at the stage at which all resources are used.

Across all the data used in this study, critical plant densities scale as the −0.78 exponent of average plant body mass. This numerical value is closer to the −3/4 exponent predicted by our packing and metabolic models than to either the −2/3 exponent predicted by previous geometric models of thinning (18, 28) or the −1.0 exponent that would be predicted if rates of metabolism, resource use, and growth scaled linearly with mass (35, 36). Additionally, the observed scaling exponents for whole-stand energy use and biomass are very close to predicted values of 0 and 1/4, respectively.

These results offer a different perspective from many studies that have emphasized how competitive and successional processes vary across plant taxa, life forms, and environmental conditions (21, 22, 24–30, 39–42). They suggest to us that scaling exponents, when measured across orders of magnitude variation in plant size, are relatively independent of phylogenetic affiliation, growth habit, and abiotic environmental features. Nevertheless, the substantial variation that remains is undoubtedly biologically important. The standard practice of using log-log plots in allometric analyses exposes both the strengths and weaknesses of the approach taken here. On the one hand, it highlights general trends that emerge as a consequence of the interplay of natural or artificial selection and geometric, physical, and biological constraints and that can be captured in relatively simple mathematical expressions. On the other hand, it deemphasizes biologically important variation that has important consequences when the focus shifts to particular species or habitats.

Additionally, our models and data raise but do not answer two important questions about the thinning phenomenon. First, they show that the *M*^{−3/4} scaling of *N _{crit}* is consistent with the

*M*

^{3/4}scaling of metabolic rate assumed by the metabolic model. This does not explain, however, how such

*M*

^{3/4}scaling can be reconciled with the linear scaling of metabolic rates reported for small, mostly herbaceous plants (35). We speculate that the change from

*M*to

*M*

^{3/4}scaling with increasing plant size (34, 36) reflects an ontogenetic transition from linear scaling when small, young plants are composed mostly of primary tissues to

*M*

^{3/4}scaling as larger plants accumulate secondary tissues (36). This speculation leads to the testable hypothesis that metabolic rates of small herbaceous plants when growing in isolation scale linearly with mass, whereas those of the same plants when competing at maximal density scale as approximately the 3/4 power.

The models and data presented here also do not explain how density and competition affect mortality. Annual crops typically experience little mortality before harvesting despite wide variation in initial planting density. This presumably reflects artificial selection for plants of the same variety in the same planting to grow at similar rates and to mature at similar size and age under crowded but highly managed conditions. Somewhat similar considerations apply to plantation forests, where trees are harvested periodically as they grow in a manner that substitutes managed density reduction for natural mortality. The agricultural stands used in our study differ in this important respect from natural forests, grasslands, and other plant communities, where mortality plays a major role in generating emergent age and size distributions. More complex, spatially explicit models are required to address such unmanaged systems, where competition leads to mortality and the death of some plants frees up resources so that others can continue to grow (1–4, 11–15, 42).

Our models make simplifying assumptions so that the parameters and functions can be expressed in mathematical equations. Basically, the models assume that plants with the same idealized structural and functional constraints grow under ideal homogeneous conditions. All these assumptions are violated to some extent by real plants growing in real environments. There is always variation in both intrinsic plant traits (e.g., size, age, architecture) and environmental conditions (spacing and soil heterogeneity). By controlling planting time, initial density, spacing, genetic and phenotypic variation, and environmental heterogeneity, agricultural stands effectively eliminate much of the variation in the structure and dynamics of natural stands attributable to these factors. In a very important sense, agricultural plantings provide empirical experimental model systems that have been largely underappreciated by theorists. The intentionally simplified conditions can be captured in the intentionally simplified assumptions of mathematical models (1). The most basic mechanisms through which density and competition affect resource use, growth, and mortality can be studied without the complicating effects of the multiplicity of factors that cause variation in uncontrolled natural populations and communities.

## Methods

### Experimental Dataset.

We collected and analyzed newly acquired data on agricultural crops growing under controlled conditions. Experiments with broomcorn (*Sorghum vulgare Pers*.), corn (*Zea mays L*.), sunflower (*Helianthus annuus*), soybean (*Glycine max*), caraway (*Coriandrum sativum*), and barley (*Hordeum vulgare L*.) were conducted at the Yuzhong experimental station of Lanzhou University, China, from April to July 2008. The initial planting densities were designed to cause different levels of competition and result in different final plant densities, *N _{crit}*, and plant sizes, . Specifically, we used 1, 3, 10, 100, 500, 1 × 10

^{3}, and 2 × 10

^{3}seeds m

^{−2}for corn and soybean; 1, 3, 10, 100, 500, 1 × 10

^{3}, 2 × 10

^{3}, and 3 × 10

^{3}seeds m

^{−2}for broomcorn and caraway; and 1, 3, 10, 100, 1 × 10

^{3}, 2 × 10

^{3}, and 6 × 10

^{3}seeds m

^{−2}for barley, with three replicates for each density. Plot areas were 1 m × 1 m, with 0.3- m-wide buffer zones to avoid edge effects. The local growing conditions were sufficient to ensure vigorous plant growth without water or nutrient stress (21, 22). Numbers of stems were counted and above-ground dry plant biomass was measured after harvesting reproductively mature plants on each plot. Leaf, stem, and reproductive tissue was oven-dried at 115 °C for 30 min and then at 65 °C for 48 h, and was weighed (27). Because the crops were harvested and weighed after the plants had matured, the temporal trajectories during the experiments cannot be shown.

### Expanded Datasets.

We also compiled existing data for 473 stands of trees in plantations and 13 populations of bamboo. Both water availability and temperature are known to affect the rate of plant growth, density, community succession, and total plant biomass. To focus on the effects of plant size, we tried to minimize extreme effects of water and temperature by including only crops, bamboo, and plantations in historically forested landscapes. For this limited dataset, we analyzed whether variation in mean annual precipitation or temperature affected the exponents of the empirically determined scaling relations (Table S1).

To compare our observed thinning relationships with others previously reported in the literature (43, 44), we compiled and analyzed existing data on 1,338 forests undergoing secondary succession following a disturbance event (45). The objective was to amass a dataset for *N _{crit}* as a function of for as many species, life forms, and habitats as possible. The only restriction was that data were from stands of known age, so as to avoid errors in estimating when plants of different ages varied widely in size.

### Statistical Analyses.

To quantify scaling relationships, we excluded the data from crops planted at such low densities that plants matured without coming into contact and competition (Fig. 3*B*, green triangles). We used the remaining data for crops and all data for bamboo, plantations, and natural forests to estimate the scaling exponents for critical density (*N _{crit}*), energy (

*E*), and biomass (

*W*) as a function of average plant size in four ways. First, we quantified the scaling of density by fitting data for log

*N*and log using reduced major axis (RMA) regression. RMA regression is appropriate in this case because neither density nor mass is an dependent or independent variable; both have measurement variation, and the correlation between them is high (46) (Table 1). Second, we quantified scaling of energy use by fitting ordinary least squares (OLS) regressions to log-log plots of

_{crit}*E*vs. (where

*E*was calculated as

*N*

_{crit}_{}). OLS is the appropriate regression method in this case because RMA regression is not accurate when regression slopes are low (46). Third, we quantified scaling of biomass by fitting both OLS and RMA regressions to log-log plots of

*W*vs. (where

*W*was calculated as

*N*

_{crit}_{}). Fourth, we also estimated the scaling of energy use and biomass by substitution. Having obtained the empirical exponent for the scaling of

*N*as a function of by regression, we substituted this value into Eqs.

_{crit}**5**and

**6**to estimate the scaling of

*E*and

*W*. This approach is appropriate, because the slope of

*N*vs. is the only independent parameter estimated directly from the data. Additionally, substitution avoids problems of estimating slopes by regression when observed and expected values are close to zero (energy use) or when the same parameter, , is included on both axes (biomass) (46). Note that the slopes of the OLS regressions for

_{crit}*E*and

*W*as a function of were very similar to the exponents calculated by substitution (Table 1).

We tested for effects of water availability and temperature on the scaling exponents by residual analysis (Tables S1 and S2 and *SI Methods: Residual Analysis of Environmental Factors*).

## Acknowledgments

We thank B. J. Enquist, C. P. Kempes, S. Mori, and E. P. White for their constructive comments and helpful suggestions on the manuscript. This study and J.D. were supported by the Natural Science Foundation of China (Grants 31000286, 30430560, and 30730020), J.D. was also supported by the Program for New Century Excellent Talents in University, and W.Z. and J.H.B. were supported by a Howard Hughes Medical Institute-National Institute Biomedical Imaging and Bioengineering Interfaces grant.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. E-mail: liujq{at}nwipb.ac.cn or jhbrown{at}unm.edu.

Author contributions: J.D., J.L., and J.H.B. designed research; J.D., Z.W., Z.F., M.J., G.W., J.R., C.Z., J.L., K.J.N., and J.H.B. performed research; W.Z. and J.R. contributed new reagents/analytic tools; J.D., W.Z., Z.W., Z.F., M.J., G.W., and C.Z. analyzed data; and J.D., W.Z., J.R., J.L., K.J.N., S.T.H., and J.H.B. wrote the paper.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1205663109/-/DCSupplemental.

Freely available online through the PNAS open access option.

## References

- ↵
- Thornley JHM,
- Johnson IR

- ↵
- Adler FR

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Enquist BJ,
- West GB,
- Brown JH

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Chu CJ,
- et al.

- ↵
- Kira T,
- Ogawa H,
- Sakazaki N

- ↵
- Yoda K,
- Kira T,
- Ogawa H,
- Hozumi H

- ↵
- ↵
- ↵
- ↵
- ↵
- Zeide B

- ↵
- ↵
- ↵
- ↵
- ↵
- Hutchings MJ

- ↵
- ↵
- Niklas KJ

- ↵
- Niklas KJ,
- Spatz H-C

- ↵
- West GB,
- Brown JH,
- Enquist BJ

- ↵
- ↵
- ↵
- Mori S,
- et al.

- ↵
- ↵
- ↵
- ↵
- West GB,
- Enquist BJ,
- Brown JH

- ↵
- Enquist BJ,
- Niklas KJ

- ↵
- Niklas KJ,
- Midgley JJ,
- Enquist BJ

- ↵
- Cannell MGR

- ↵

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