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 %.