Experimental Results on the Effects of Nonregular Spatial Patterns of Plants on Yield per Area

Based on several simplifying assumptions, a stochastic approach was developed which allows an estimation of the effects of nonregular spatial patterns of the distribution of individual plants on yield per area (F). In this approach, two random variables were attached to each plant: single plant yield (E) and individual space per plant (A). The latter was estimated by the area of Thiessen polygons. Yield per area was calculated theoretically by the expectation of the ratio E / A. Appropriate approximations of this expectation depend on the means (ē and ā), coefficients of variation (vE and vA) of E and A and their correlation (rEA). Yield per area can be decomposed into two additive terms: the first term gives the commonly used estimate ē/ā— or h(ā)/ā if a functional relationship between E and A is assumed: E = h(A). In this study, the two relationships E = k1 + k2 · ln A and E = A/(k3 + k4A) were used (with appropriately chosen constants k1, k2, k3, and k4). The second term in the decomposition of F can be interpreted as the effect of variable individual plant spaces on yield per area. In this paper, all theoretical concepts and results were applied to 17 experimental data sets of three cultivars of winter oilseed rape (Brassica napus L.). Single plant yields (E) and individual plant areas (A) were positively correlated with correlation coefficients from 0.64 up to 0.91. The ranges for both coefficients of variation were similar: 0.27 ≤ vE ≤ 0.65 and 0.28 ≤ vA ≤ 0.59. One obtains no significant differences in the goodness-of-fit for both tested relationships between E and A although the logarithmic relationship seems to be slightly superior. For only three data sets one obtains negative values for the percentage of the second term in the decomposition of F. This indicates an overestimation of yield per area by the commonly used estimates h(ā)/ā and ē/ā, respectively. These overestimations, however, are less than 5 %. In all other cases with positive values for the second term the yield per area is underestimated by the common estimates. For almost all data sets, however, the percentages of F which are explained by the common estimates are much larger than 90 %.     

Note on the Effect of Non-regular Spatial Patterns of Plants on Yield per Area Based on a Logarithmic Relationship between Single Plant Yield and Individual Area

Based on several simplifying assumptions, a previously developed stochastic approach allows an estimation of the effects of non-regular spatial patterns of the distribution of individual plants on yield per area (F). In this approach, two random variables were attached to each plant: single plant yield (E) and individual space per plant (A). The latter was estimated by the area of Thiessen-polygons. Yield per area was calculated theoretically by the expectation of the ratio E/A. Based on a logarithmic relationship between E and A the expectation of E/A can be expressed by an approximation which depends on the mean and on the variance of the individual plant spaces. An improvement of this approximation can be easily obtained by including skewness and kurtosis of the distribution of individual plant areas. Finally, all theoretical concepts and results were applied to an experimental data set of winter oilseed rape ( Brassica napus L.).     

Effects of Nonregular Spatial Distribution of Plants on Yield per Area: A Theoretical Approach with Applications to Winter Oilseed Rape (Brassica napus L.)

In this study a stochastic approach, based on several simplifying assumptions, is developed that allows an estimation of the effects of nonregular spatial patterns of the distribution of individual plants on yield per area. In this approach, two random variables were attached to each individual plant: single plant yield and individual space per plant. The latter can be estimated, for example, by the area of Thiessen polygons. Yield per area is calculated theoretically by the expectation (=mean) of the ratio between individual yield and area. Based on the logarithmic relationship between single plant yield and individual space per plant, yield per area can be broken down into three additive terms: the first term depends only on the mean of individual plant areas, while the second depends on their mean and variance simultaneously. This second term is the product of the variance and a factor which depends only on the mean. The third term is a function of the mean and of higher order (≥ 3) central moments of a fractional linear transformation of individual plant area. Finally, these theoretical concepts were applied to 17 experimental data sets of three cultivars of winter oilseed rape (Brassica napus L.) with measurements for single plant yields and individual areas for the Thiessen polygon tesselations. Since yield per area is theoretically defined by the expectation of the ratio between individual yield and area, it is estimated by the arithmetic mean of the individual yield/area‐ratios. The agreement between this estimate and the sum of the estimated terms from the aforementioned additive decomposition of yield per area is particularly good. For the 17 data sets, percentages for these additive terms of approximately 71.4 % up to 98.4 % (mean: 89.0 %) for the first term, 1.6 % up to 20.0 % (mean: 9.8 %) for the second term and 0 % up to 8.7 % (mean: 1.2 %) for the third term are obtained. As a consequence it may be concluded, that yield per area can be explained mainly by dependencies on the means of individual plant areas while the variance of individual plant spaces is of only minor importance. The effect of the third term is insignificant. These results clearly indicate an answer to the main issue raised in the paper, namely the importance of seeding density as opposed to seeding accuracy/uniformity: nonuniformity is of limited influence and seeding density is the main correlate of yield/area. Seed placement accuracy of seeding technology therefore plays an only minor role.     

氮钾配施对油菜产量及氮素利用的影响

实际生产中氮钾肥投入不平衡严重限制了氮肥肥效及作物的产量潜力。为了探明不同施氮量下钾肥施用对油菜产量及氮素利用的影响,于2016—2017年及2017—2018年在湖北省武穴市开展连续2年的田间试验,采用氮钾两因素完全试验设计,设氮0、90、180、270 kg N hm-2和钾0、60、120、180 kg K2O hm-2各4个水平。在油菜成熟期取样测定产量、地上部氮钾积累量以及氮肥利用率。结果表明,在钾供应不足时(K0和K60),冬油菜施用氮肥的平均增产率为113.7%,而在钾供应充足的条件下(K120和K180),施用氮肥的平均增产率高达172.9%;与K0处理相比, K120处理冬油菜氮肥回收利用率平均提高了16.6%,继续增施钾肥对不同施氮量下冬油菜氮肥回收利用率的进一步提高无显著影响;达到区域平均产量时,钾供应充足较低钾(K60)投入平均降低33.9%的氮肥用量。综上所述,氮钾配施显著提高了冬油菜产量和氮肥利用率,在冬油菜实际生产中除了重视氮肥施用外,应增加钾肥投入,通过优化氮钾肥配施比例可进一步提高油菜产量,实现冬油菜高产和养分高效。     

Ein allgemeiner Ansatz zur Quantifizierung des Einflusses der Güte der Sätechnik auf den Flächenertrag

A general approach to determine the effect of accuracy of sowing technique on yield per area
Caused by nonperfect seed placement accuracy of sowing machines as well as by many other abiotic and biotic factors the resulting plant stands exhibit nonregular plant distributions. Based on several simplifying assumptions a stochastic approach provides a quantitative determination of the effects of accuracy of sowing technique on yield per area. In this approach, two random variables are assigned to each individual plant: single plant yield E and single plant area A, which is calculated in this paper by the construction and quadrature of the so-called Thiessen-polygons. The yield per area F is calculated as the expectation of the ratio E/A. By assuming a deterministic mathematical relationship between E and A, the calculation of F reduces to the calculation of the expectation of a function of only one random variable A. A simple approximation with sufficient accuracy for many applications only depends on the mean and on the variance of the areas of the individual plants. For demonstration purposes, the theoretical approaches and results have been finally applied to three data sets for drilled seeds of winter oilseed rape (plant density: 60 plants/m² and distance between rows: 10 cm). These data sets exhibit different accuracies of the longitudinal distributions within rows which have been quantitatively measured by the coefficient of variation for the distances between plants within rows: Yield depression increases with an increasing variability of plant distances within rows.