Models and tests of optimal density and maximal yield for crop plants

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.