Crop Science 43:421-425 (2003)
© 2003 Crop Science Society of America
NOTES
Estimating yield depression caused by nonuniformity of spatial plant patterns
Manfred Hühn*
Inst. of Crop Science and Plant Breeding, Christian-Albrechts Univ., Am Botanischen Garten 1-9, D-24118 Kiel, Germany
* Corresponding author (mhuehn{at}plantbreeding.uni-kiel.de)
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ABSTRACT
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Agricultural plant stands at harvest are characterized by more or less irregular spatial distributions of their individual plants. The objective of this study was to quantify the yield depression caused by nonuniformity of spatial plant patterns. On the basis of several assumptions, a stochastic approach is proposed that allows an estimation of the effects of irregular spatial patterns of the distribution of individual plants on yield (Y). The variables, single plant yield (S) and individual area per plant (A) estimated by the area of Thiessen polygons, were evaluated. Yield was calculated theoretically by the expectation of the ratio S/A. On the basis of a nonlinear relationship between single plant yield and individual area per plant, yield can be represented by two additive terms. The first term depends on the mean of individual plant areas. The second term is negative, depends on the mean and variance simultaneously, is proportional to the variance of individual plant areas, and can be interpreted as the effect of variable individual plant areas on yield. This provides the amount of yield depression caused by nonuniformity of spatial distribution of plants across the area. Theoretical concepts and results were applied to an experimental yield data set of winter oilseed rape. The amount of yield depression was 5.3%.
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INTRODUCTION
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PREVIOUS PUBLICATIONS (Mülle and Heege, 1981) demonstrated that the yield of cereals and other crops, such as oilseed rape, increases with increasing uniformity of the spatial distribution of the individual plants across an area. The idealized condition under which the yield is maximized is the state where all plants yield the same because of the lack of plant-to-plant interference, with the equal sharing of growth resources (minimized competition) (Fasoula and Fasoula, 2000). Different areas of the single plants will lead to existing competitive effects caused by such significant plant-to-plant interferences. The variability of the available areas per individual plant is, therefore, of major interest. Nonperfect seed placement by planters and many other uncontrollable abiotic and biotic factors are responsible for the resulting irregular spatial distributions of agricultural plant stands. The amount of nonuniformity at harvest can be quite different and the same might be expected for its effect on yield.
In previous studies (Hühn 1998, 1999a, 1999b, 2000a, 2000b), a stochastic approach was proposed and examined, which allows modeling and estimation of the effects of nonregular spatial patterns of the distribution of individual plants on yield. In this approach, two random variables were used: single plant yield (S) and individual area per plant (A). In these studies, the individual area per plant was estimated using the area of Thiessen polygons that are defined as the smallest polygons that can be obtained by erecting perpendicular bisectors to the horizontal lines joining the center of a plant to the centers of its neighboring competitors. The polygon around a plant includes all points in the plane that are closer to that plant than to any other plant. These polygons are mutually exclusive and collectively exhaustive of the total area (Fig. 1).
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Fig. 1. Irregular spatial pattern of rapeseed plants and resulting Thiessen polygons used to estimate yield depression.
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For the construction of Thiessen polygons and the calculation of their areas, efficient algorithms and computer programs are available (Green and Sibson, 1978; Byars et al., 1998; Guibas et al., 1990; Waupotitsch, 1992). In this modeling approach, the polygon area represents the area potentially available for plant growth. Its numerical magnitude simulates the availability of growth resources and essential environmental factors (e.g., light, water, nutrients, physical growing space) for each individual plant. These Thiessen polygon areas have been widely used in the construction of many individual based competition indices that were developed to predict the growth of individual plants as a function of interference from a certain subset of other plants in the neighborhood (Mead, 1966; Rhynsburger, 1973; Liddle et al., 1982; Mithen et al., 1984; Fasoulas and Fasoula, 1995). The biological processes involved in population growth and formation of yield are, of course, much more complex than can be described by such purely spatial considerations. But, Thiessen polygons are at least a reasonable measure of two-dimensional area available to comparable individuals and may provide useful independent variables for predicting plant growth in agricultural plant stands.
The effects of nonuniformity of spatial patterns of plants on yield and the amount of heterogeneity of the spatial distribution of seeds or plants across an area can be quantitatively expressed by the variance of Thiessen polygon areas. This parameter, however, strongly depends on the accuracy of planting technique and precision of seed placement by planters. The seed spacing quality of planters influences field emergence, plant development, and crop yield. To assess seed spacing quality, the horizontal distribution across an area is of major importance. The seed distribution across the area depends on the quality of longitudinal distribution within rows and on row width as well as on seeding rate. Row width and seeding rate are considered to be given and fixed. By this approach, the quality of horizontal seed distribution across the area and, therefore, the available single areas for growth, development, and yield of each individual plant, are mainly determined by the longitudinal distribution of seeds within rows (Griepentrog, 1999). The latter characteristic, however, is mainly determined by the seed spacing accuracy. The objective of this study was to quantify the yield depression caused by nonuniformity of the spatial distribution of plants.
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MATERIALS AND METHODS
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Theory
A stochastic approach was used with random variables single plant yield (S) and individual area per plant (A) to estimate the effects of nonregular spatial patterns of individual plants on yield (Y). Y was calculated theoretically by the expectation (= mean) of the ratio (single plant yield)/(individual area per plant). If one denotes the expectation of a random variable X by E[X], one obtains Y = E[S/A].
On the basis of Taylor-series techniques, approximations for Y can easily be derived by the method of statistical differentials (Johnson and Kotz, 1969). For normally distributed random variables, S and A, one obtains by Taylor series expansion including terms up to the fourth order (Hühn, 1990):
| [1] |
where 
= mean of S, 
= mean of A, 
S = standard deviation of S, A = standard deviation of A, and rSA = correlation coefficient between S and A.
By Eq. [1], Y depends on five parameters: mean and standard deviation of single plant yield, mean and standard deviation of individual area per plant, and the correlation between both traits. These five parameters, however, are not independent of one another. Interrelationships among the parameters are obvious; for example, changing variability of individual areas will cause a corresponding variability of single plant yields. The exact patterns of these interrelationships are unknown. No single parameter can be changed without changing the others.
Eq. [1] can be rewritten as
| [2] |
where vA = A/ = coefficient of variation of A and vS = S/ = coefficient of variation of S. (In this paper, all coefficients of variation are expressed as proportions and not as percentages).
Some numerical values of the second term in Eq. [2] [expressed as a percentage of Y, that is, Y = (first term + second term) = 100%] for different vA, vS, and rSA can be found in a table (Table 1). These percentages were simply calculated by inserting numerical values for vA, vS, and rSA into the formula of the second term in Eq. [2]. If one is only interested in percentages, the second term must be related to the sum of the first and the second term and the term / is cancelled out in the numerator and in the denominator of this ratio, that is, the percentages in Table 1 were calculated by the formula
 | [3] |
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Table 1. Potential yield loss or yield gain (expressed as a percentage of yield) caused by nonuniformity of plant spatial patterns for different coefficients of variation of area per plant (vA), yield per plant (vS), and different correlations between both traits (rSA).
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For small vA (vA 
0.20) and small or intermediate vS (vS 0.40), the second terms from Eq. [2] are <5% (for each correlation, rSA). For very small vA (vA 0.10) and moderately small vS (vS 0.20), the second terms are even smaller than 1% (for each correlation, rSA). Thus, we may conclude that for many situations in the field of agronomic applications, the second term in Eq. [2] is sufficiently small, so that one obtains a reasonable approximation for Y by the ratio of and .
 | [4] |
By this approach, Y is underestimated if vA/vS > rSA and Y is overestimated if vA/vS < rSA. Essential improvements can be obtained if a functional relationship between S and A is available. In this study, we use
 | [5] |
where k1 and k2 are appropriately chosen constants that must be estimated from the data. This basic type of yield-density relationship has been frequently applied and verified in the literature (Kira et al., 1953; Shinozaki and Kira, 1956, 1961; Baker and Briggs, 1983; Spitters, 1983).
The nonlinear relationship between S and A in Eq. [5] can be transformed into a linear relationship if one uses the variables S-1 and A-1. By this simple transformation, linear regression techniques can be applied. This nonlinear relationship in Eq. [5], of course, is not generally valid for arbitrary values of its variables. All investigations in this paper are restricted to the range of values for A and S where this relationship holds.
By the method of statistical differentials (Johnson and Kotz, 1969), one obtains
 | [6] |
where h
= 
/
= value of h
for A = , and h'' = -2k1k2/3 = second derivative of h for A = .
Combining Eq. [4] with Eq. [6] leads to the final result
| [7] |
By Eq. [7], Y depends on the mean 
and variance 
of individual plant areas. Yield per area is composed of two additive terms where the first term only depends on the mean of individual plant areas, whereas the second term depends on the mean and variance 2A simultaneously. The second term is proportional to the variance of individual plant areas, that is, the second term tends to zero for 2A 
0. The second term in Eq. [7] can be interpreted as the effect of variable individual plant areas on yield, that is, the second term estimates the amount of yield depression caused by nonuniformity of spatial distribution of plants across the area. The interpretation of relationship (Eq. [7]) can be further simplified: The total area of the plant population (Atotal) is considered to be given and fixed. Atotal is completely covered by the individual Thiessen polygons with individual areas, Ai, where i = 1,2,..., n, where n is the number of plants in the total area:
For a certain given total area and a predetermined number of plants, n, (given plant density of the stand), can be considered to be constant. In this interpretation, Eq. [7] presents an explicit relationship for Y dependent on the variability of A. By Eq. [7], yield decreases with increasing variability of A. Eq. [7] provides a quantitative estimation of the decrease of yield caused by nonuniformity of spatial patterns of plants. The second term on the right-hand side of Eq. [7] expresses this yield decrease. With decreasing 2A (= increasing uniformity of spatial pattern), yield increases and tends to 1/
. The second term on the right-hand side of Eq. [7], therefore, provides an estimate of the potential of possible yield increases that can be realized by improving the uniformity of the spatial distribution of plants across an area.
Field Tests
Three German cultivars (Ceres, Falcon, Liberator) of winter oilseed rape (Brassica napus L.) were grown in Hohenschulen, a location near Kiel, Schleswig-Holstein (northern part of Germany) on a sandy loam soil in August 1990. In this paper, we are mainly interested in the theoretical and methodological aspects of an analysis of nonuniformity of spatial plant patterns. For demonstration purposes, therefore, we only use one data set [Plot 4 of Falcon from Hühn (1999b)(2000b) with n = 34 plants]. For a description of experimental details of the field experiment (e.g., site, weather conditions, sowing technique, cultivars, crop management, husbandry details, disease assessment, design, plot size, row distance, density, Thiessen polygon tessellation, trait measurement), refer to Hühn (1999b)(2000b). The units for the trait measurements are g for S and mm2 for A (Table 2). The postulated relationship between S and A in Eq. [5] contains two unknown parameters, k1 and k2, which must be estimated from the data. This was done by applying linear regression analysis techniques with S-1 = dependent variable and A-1 = independent variable (Fig. 2).
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Fig. 2. Linear regression of inverse of single plant yield on inverse of individual area per plant for the rapeseed data set used to estimate yield depression.
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RESULTS AND DISCUSSION
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The parameter estimates for the data set (Table 2) are: A = 11 940 mm2; S = 8.10 g; = 26189 mm2; 
= 15.08 g; k1 = 1112; k2 = 28.87 x 10-3; n = 34; rSA = 0.91; vA = 0.4559; and vS = 0.5374. The coefficient of determination for the linear regression of S-1 on A-1 (Fig. 2) is 67.74 (%). For the decomposition of yield (expressed in g mm-2) into two terms according to Eq. [1], one obtains
The estimate of yield by Eq. [1] is Y = 56.18 x 10-5 g mm-2. Expressed as a percentage of Y, the second term is -2.5%. It is moderately small as expected from the theoretical results (Table 1). If one estimates Y from the approximation in Eq. [7], one obtains
The estimate of Y from Eq. [7] is 50.85 x 10-5 g mm-2. Expressed as a percentage of Y, the second term is -5.3%. The amount of yield depression caused by nonuniformity of spatial distribution of plants across the area is 
5%.
The estimate of Y from Eq. [7] deviates considerably from the previous estimate based on Eq. [1]. This discrepancy, however, is not surprising because the approximation in Eq. [7] is derived from Eq. [1] by neglecting its second term. Furthermore, Eq. [7] contains still another approximation: the approximation for using Eq. [6]. The approximation in Eq. [7], therefore, must be considered to be a very rough estimate of Y. Improvements can be obtained by means of several approaches: (i) Equation [1] is based on the assumption of normal distribution of variables S and A. Improvements may be obtained for arbitrarily distributed variables. (ii) The theoretical results of this paper are based on several approximations that have been derived by techniques of Taylor series expansion. These approximations can be improved by including higher order terms from the Taylor series, so that the accuracy of the approximations will increase. (iii) The functional relationship between S and A may be improved; for example, by the assumption of different relationships for different stages of plant growth and development. (iv) In the modeling approach of this paper, the areas of the Thiessen polygons have been used as indicators of the availability of growth resources and essential environmental factors. The biological processes involved in plant growth and formation of yield are, of course, much more complex than can be analyzed by such simple spatial considerations. Other definitions of available area and modified measures of the degree to which growth resources may be limited by the number, size, and proximity of neighboring competitors may be better than the simple Thiessen polygon areas. Such advanced developments and improvements may include concepts of three-dimensional available spaces per plant. Furthermore, the introduction of dynamic measures and models of plant interferences and plant community dynamic processes instead of only static descriptions with representations of the state of a dynamic system at one point in time seems to be necessary. System dynamics cannot be represented by one-time measurements. The analysis, therefore, must consider temporal changes and time trends. Time-dependent models and measures are needed.
The elaboration of these generalizations and improvements [(i), (ii), (iii), and (iv)] are beyond the scope of this paper. This paper presents the first step of an improved analysis of the effects of nonuniform spatial distributions of plants across an area. The results show that improved uniformity of the spatial distribution of plants across an area increases yield by helping to reduce the plant-to-plant interference with the equal sharing of resources. The results also indicate that an ideal spatial pattern is the equilateral hexagonal pattern that reduces to zero the variance of Thiessen polygons and optimizes uniformity. The fit between theoretically expected and experimentally obtained values of yield from large-scale production operations must be expected to be particularly poor. Numerous reasons may be responsible: sampling effects, experimental error, lack of accuracy of the approximations, and so on. But, the main reason is that the actually harvested yield is a result of applying common agricultural practices from practical agronomy (for example, large-scale harvesting techniques) whereas the theoretically expected yield is based on the measurements of carefully harvested individual plants. Application of common harvesting techniques from practical agronomy, however, must lead to losses in grain yield compared with individually harvested single plants. As a consequence, the theoretically expected Y-value must be much larger than the experimentally obtained Y-value. The theoretically expected Y-value can be interpreted as the maximum possible yield potential under idealized conditions.
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ACKNOWLEDGMENTS
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The author gratefully acknowledges the cooperation and help provided by Dr. K. Sieling (empirical data set) and the technical assistance by Mrs. H. Jensen, and Mrs. M. Wolfram.
Received for publication February 4, 2002.
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