Non-regular spatial patterns of plants and their effect on several agronomic traits per area

Based on several simplifying assumptions, a stochastic approach is developed that allows an estimation of the effects of non-regular spatial patterns of the distribution of individual plants on the mean value (K) of trait per area. In this approach, two random variables are attached to each plant: single plant trait measurement (E) and individual space per plant (A). The latter is estimated by the area of Thiessen polygons. K is calculated theoretically by the expectation of the ratio E/A. Appropriate approximations of this expectation depend on the means (E and A), coefficients of variation (vE and vA) of E and A and their correlation (rEA). K can be decomposed into two additive terms: the first term gives the commonly used estimate E/A. If a functional relationship, E=h(A), between E and A is assumed, this first term is h(A)/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 K can be interpreted as the effect of variable individual plant spaces on K. In this paper, these theoretical concepts were applied to 17 experimental data sets of three cultivars of winter oilseed rape (Brassica napus L.) with single plant measurements for the traits grain yield, number of pods, grain yield per pod, total dry matter, harvest index, 1000-grain weight, number of seeds and number of seeds per pod. The means, standard deviations and coefficients of variation of the individual plant areas exhibit a large variability. The differences within cultivars are larger than the differences between cultivars. The correlation coefficients between E and A can be positive and large (for grain yield, number of seeds and number of pods), small (for 1000-grain weight and number of seeds per pod) or intermediate (for total dry matter, harvest index and grain yield per pod). There were no significant differences in the goodness-of-fit for either of the tested relationships between E and A, although the logarithmic relationship seems to be slightly superior. There were only a few data sets where negative values were found for the percentage (K=100%) of the second term in the decomposition of K. This indicates an overestimation of K by the commonly used estimates E/A and h(A)/A, respectively. These overestimations, however, are less than 5.2%. In all other cases with positive values for the second term, K is underestimated by the common estimates with values from 0 up to 40%. With regard to the numerical amount of the second terms, the eight traits can be clearly partitioned into two distinct groups: group 1 = (grain yield, total dry matter, number of pods, number of seeds) with small percentages for the second term and group 2=(1000-grain weight, harvest index, grain yield per pod, number of seeds per pod) with large percentages for the second term. The Spearman rank correlation coefficients of second term percentages (based on replications=data sets) for pairs of traits belonging to the same group are positive and large with larger values for group 2 than for group 1. The correlations between traits belonging to different groups, however, are intermediate or small.