We introduce a theoretical framework that predicts the optimum planting density and maximal yield for an annual crop plant. Two critical parameters determine the trajectory of plant growth and the optimal density, N opt , where canopies of growing plants just come into contact, and competition: (i) maximal size at maturity, M max , which differs among varieties due to artificial selection for different usable products; and (ii) intrinsic growth rate, g, which may vary with variety and environmental conditions. The model predicts (i) when planting density is less than N opt , all plants of a crop mature at the same maximal size, M max , and biomass yield per area increases linearly with density; and (ii) when planting density is greater than N opt , size at maturity and yield decrease with −4/3 and −1/3 powers of density, respectively. Field data from China show that most annual crops, regardless of variety and life form, exhibit similar scaling relations, with maximal size at maturity, M max , accounting for most of the variation in optimal density, maximal yield, and energy use per area. Crops provide elegantly simple empirical model systems to study basic processes that determine the performance of plants in agricultural and less managed ecosystems. E fficiency of agriculture will need to increase to feed the growing human population as arable land, water, and fertilizers become increasingly limited (1, 2). A relevant question is, What is the optimal density to plant seeds of an annual crop? The answer should be of interest to applied plant scientists who want to predict planting densities that maximize yields and to basic plant scientists who want to better understand the fundamental processes of growth and competition.Here we develop and test analytical models that predict the optimal seeding density that maximizes yield for annual crop plants. These models were inspired by theories and data on plant scaling relations (3-10). We modify the theories to model the growth and maturation of annual crops as a function of density and mature plant size. We evaluate the models using data from agricultural crops in controlled experiments in China. Empirical and Conceptual BackgroundThere is an intermediate seeding density for an annual crop that maximizes yield at harvest. When seeds are planted at lower density, yields are reduced because the plants grow to mature size without using all available resources. When seeds are planted at higher density, plants compete for resources and mature at smaller sizes; total yield declines because mature size per individual decreases faster than number of individuals per area increases.The dynamics of crop production can be modeled as the outcome of four interacting processes. First, the growth of an individual annual plant from germination to maturity traces a sigmoidal trajectory that reflects allocation of energy and biomass to new tissue as a function of plant size. Second, size at maturity depends on density: Initially all plants grow at nearmaximal rates, but if individuals ...

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.allometric scaling | energy equivalence | plant energetics | self-thinning T he 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)(18)(19)(20)(21)(22), where plants of nearly identical age grow under controlled conditions. Studies of such simplified agricultural systems have led to theoretical and empirical selfthinning 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)(18)(19). These exhibit a characteristic phenomenology in which plants grow with minimal mortality until they reach a size-dependent critical density, ...

Simultaneous and accurate measurements of whole-plant instantaneous carbon-use efficiency (ICUE) and annual total carbon-use efficiency (TCUE) are difficult to make, especially for trees. One usually estimates ICUE based on the net photosynthetic rate or the assumed proportional relationship between growth efficiency and ICUE. However, thus far, protocols for easily estimating annual TCUE remain problematic. Here, we present a theoretical framework (based on the metabolic scaling theory) to predict whole-plant annual TCUE by directly measuring instantaneous net photosynthetic and respiratory rates. This framework makes four predictions, which were evaluated empirically using seedlings of nine Picea taxa: (i) the flux rates of CO(2) and energy will scale isometrically as a function of plant size, (ii) whole-plant net and gross photosynthetic rates and the net primary productivity will scale isometrically with respect to total leaf mass, (iii) these scaling relationships will be independent of ambient temperature and humidity fluctuations (as measured within an experimental chamber) regardless of the instantaneous net photosynthetic rate or dark respiratory rate, or overall growth rate and (iv) TCUE will scale isometrically with respect to instantaneous efficiency of carbon use (i.e., the latter can be used to predict the former) across diverse species. These predictions were experimentally verified. We also found that the ranking of the nine taxa based on net photosynthetic rates differed from ranking based on either ICUE or TCUE. In addition, the absolute values of ICUE and TCUE significantly differed among the nine taxa, with both ICUE and temperature-corrected ICUE being highest for Picea abies and lowest for Picea schrenkiana. Nevertheless, the data are consistent with the predictions of our general theoretical framework, which can be used to access annual carbon-use efficiency of different species at the level of an individual plant based on simple, direct measurements. Moreover, we believe that our approach provides a way to cope with the complexities of different ecosystems, provided that sufficient measurements are taken to calibrate our approach to that of the system being studied.

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