Directory UMM :Data Elmu:jurnal:E:European Journal of Agronomy:Vol12.Issue1.Jan2000:

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www.elsevier.com/locate/eja

Non-regular spatial patterns of plants and their e

_ff

ect

on several agronomic traits per area

M. Hu¨hn *

Institute of Crop Science and Plant Breeding, Christian-Albrechts-University Kiel, Olshausenstr 40, D-24118 Kiel, Germany Accepted 4 August 1999

Abstract

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. Kis 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 (v

EandvA) ofEandAand their correlation (rEA).Kcan be decomposed into two additive terms: the first term gives the commonly used estimateE:/A:. If a functional relationship,E=_{h(A), between}_E_and_A

is assumed, this first term ish(A:)/A:. In this study, the two relationshipsE=_k

1+k2lnAandE=A/(k3+k4A) were used (with appropriately chosen constantsk

1,k2,k3 and k4). The second term in the decomposition of K can be interpreted as the effect of variable individual plant spaces onK.

In this paper, these theoretical concepts were applied to 17 experimental data sets of three cultivars of winter oilseed rape (Brassica napusL.) 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 betweenE andAcan 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 betweenEandA, 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}

ofKby the commonly used estimatesE:/A: andh(A:)/A:, respectively. These overestimations, however, are less than 5.2%. In all other cases with positive values for the second term,Kis 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. © 2000 Elsevier Science B.V. All rights reserved.

Keywords:Non-regular spatial patterns; Single plant traits; Thiessen polygons; Winter oilseed rape (Brassica napus L.); Yield/density relationships

* Corresponding author. Tel.:+49-431-880-2578; fax:+49-431-880-2566. E-mail address:[email protected] (M. Hu¨hn)

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1. Introduction

The spatial distributions of the individual mem-bers of plant populations at harvest exhibit more or less non-regular patterns. This may be due to inaccurate seed placement of sowing machines as well as to many other uncontrollable biotic and abiotic factors. Non-regular spatial distributions of the individual plants, however, are commonly considered as disadvantageous. In several previous publications (e.g. Mu¨lle and Heege, 1981), it was demonstrated that the yield of cereals (and func-tionally similar crops such as oilseed rape) increases with increasing uniformity of the spatial distribution of the individual plants over the area. The variability of the available spaces of the indivi-dual plants is, therefore, of major interest.

In previous papers (Hu¨hn, 1998, 1999a,b), a stochastic approach was developed that allows an estimation of the effects of non-regular spatial patterns of the distribution of individual plants on yield per area. In the present paper, this approach has been extended to some further agronomic traits of economic interest.

2. Theory

Following Hu¨hn (1998, 1999a,b), two variables were considered for each individual plant: trait measurement (E) and individual space (A). The latter was estimated by the area of Thiessen poly-gons: the smallest polygon that can be obtained by erecting perpendicular bisectors to the hori-zontal lines joining the center of the plant to the centers of its neighbours/competitors. The polygon around a plant includes all points in the plane that are closer to that plant than to any other. These polygons are, of course, mutually exclusive and collectively exhaustive of the total area. A diagram-matic representation of this area round a plant

Fig. 1. Non-regular spatial pattern of plants and resulting

and some of its neighbours is given in Fig. 1. _{Thiessen polygons.}

Efficient algorithms and computer programs are available (Hu¨hn, 1999b) for the construction of

Thiessen polygons and for the calculation of is considered to represent the availability of

resources for growth ( light, water, nutrients, physi-their areas.

In this modelling approach, the polygon area is cal growing space, etc.) for each individual plant. In the proposed stochastic scheme (Hu¨hn, 1998, the area potentially available for plant growth. It

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1999a,b), the individual measurements E

i, i=1, all interesting terms relative to_{the relative amount of the second term on the}f(A:) (=100%). For 2,…,n, and the individual spacesA

i,i=1, 2,…,n,

are considered as realizations of two random vari- right-hand side of Eq. (1), one obtains: ablesE and A. The mean value (K) for trait per

Second term of Eq. (1)(%) area is calculated theoretically by the expectation

of the ratio E/A. In Hu¨hn (1998, 1999a,b), two

=50s_2Af ◊₍_A₉₎

f(A₉) [relative tof(A9)]. (2a) approaches were applied:

1. Assumption of a functional mathematical

rela-This quantity can be interpreted as the effect of

tionship between E and A: E=_h₍_A_{). This}

variable individual plant spaces on the mean value approach implies a reduction of the

investiga-(K) of trait per area. This term is expressed relative tions to only one remaining random variableA.

tof(A:), which means as a percentage of the value 2. Explicit inclusion of both random variables E

for the ideal situation of constant individual andAand description of their interdependence

plant spaces. by covariances or correlations.

The second term in Eq. (1) can be also expressed For approach 1, the mean value for trait per

relative to the totalK[f(A:)+1/2s_2Af◊(A:)=100%]:

area (K) can be approximated (Hu¨hn, 1998,

1999a,b):

Second term of Eq. (1)(%) K$f(A9)+

1 2 s2Af

◊₍_A₉_), ₍₁₎ ₌_50s₂

A f◊₍_A₉₎

K (relative toK). (2b)

This approach leads to a slightly modified inter-with f(A)=h(A)/A; A:=mean of A; s_2A=variance

pretation of the second term in Eq. (1): It quanti-ofA;f◊=_{second derivative of}_f_{. Eq. (1) is obtained}

fies the effect of variable individual plant spaces by expanding f(A) in a Taylor series about A:,

as a percentage of the actually realized total mean taking expected values of both sides and assuming

value (K) of trait per area. Both percentages

that higher-order terms (≥_{3) are negligible. In this}

[relative to f(A:) and relative to K], however, are approximation Eq. (1), the mean value for trait

similar to each other. They are smaller than 0.9% per area depends on the mean and on the variance

for second-term percentages [by Eq. (2a)] up to of individual plant spaces.

10%. In this paper, we only use the second Eq. (1) provides a deeper insight into the

rela-approach with K=100% _{because of its higher}

tionships between the mean value for trait per area

practical agronomic relevance. (K) and uniformity of the spatial distribution of

For the relationship between E and A, the

the individual plants over the area (measured by

following two expressions have been frequently s_2A):

applied in the literature:

K-increase with increasing uniformity [for

f◊(A:)<0];

E=_h(A)=_k

1+k2lnA (3)

K-decrease with increasing uniformity [for

f◊(A:)>0]

E=_h(A)= A

k 3+k4A

, (4)

(assumption: constant meanA:).

Applied to yield per area, these theoretical

considerations provide an explicit condition for with appropriately chosen constants k

1,k2,k3,

and k

4 ( Kira et al., 1953; Shinozaki and Kira, the well-known agronomic fact that the yield

increases with increasing uniformity of the spatial 1956, 1961; Bleasdale and Nelder, 1960; de Wit,

1960; Risser, 1969; Willey and Heath, 1969; distribution of the individual plants. The condition

for this fact can be reduced to the validity of Harper, 1977; Baker and Briggs, 1983; Spitters,

1983; Griepentrog, 1995). inequalities:f◊₍_A_:₎<_{0 or}_f◊₍_A_:₎>_0.

For constant individual areas (s2_A=_0), _K _{For approach 2, no explicit functional}

relation-ship between E and A is used. Both random

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variables are included in the analyses. The mean of resistances, strong branching ability (Liberator). The cultivars were randomly arranged on the test value for trait per area (K) can be approximated

site. Several problems arose with Phoma lingam

by (Hu¨hn, 1998, 1999a,b):

and Verticilium dahliae. Herbicide, fungicide and

insecticide treatments as well as plant nutrition

E₉ A9

+E9 A9 (vA

−_r

EAvE) (vA+3v3A), (5) _{regimes were applied according to standard}

cul-tural practices. The 13 plots used for this study

with E: and A:=means of E and A; v

E and were selected from a large long-term field

experi-v

A=coefficients of variation of E and A; ment used for testing agronomic practices for

EA=correlation coefficient betweenEandA. monocultures of winter oilseed rape. The plots

For constant individual plant areas with

were selected according to a differentiation of the v

A=0, the mean value for trait per area reduces actual plant population densities at harvest, i.e.

to the commonly used estimate E:/A:. In analogy

they were chosen to obtain a range of mean to the discussion of approach 1, it seems

reason-population densities rather than to obtain a range able to express all interesting terms relative to the _{of within-plot variabilities. As a consequence of} ratio E:/A:(=100%). For the relative amount of

these requirements, the plot numbers were not

the second term on the right-hand side of Eq. (5), _{constant ( Table 1). Dinter (1991) describes the}

one obtains: _{experimental details (site, design, crop}

manage-ment, weather conditions, cultivars, husbandry Second term of Eq. (5) (%)

details, disease assessment, etc.). =₁₀₀₍_v

A−rEAvE) (vA+3v3A) (relative toE9/A9). Separately for each individual plot, a system of Cartesian coordinates was introduced with axes at (6)

the left and lower borders of the plots. Cartesian

This term can be interpreted as the effect of _{coordinates were determined at harvest for each}

variable individual plant spaces on the mean value _{individual plant in the plot. In addition, the} coordi-of trait per area — expressed relative to E:/A:. _{nates of the adjacent neighbouring plants from}

out-Again, the second term in Eq. (5) can also be _{side were determined. Based on these coordinates,}

expressed relative to the totalK. _{the Thiessen polygons were constructed, and their}

The main objective of this study is to obtain a _{areas were calculated geometrically (unit: cm}₂_).

deeper insight into the relationships between the _{The numbers of plots per cultivar and the}

mean value of trait per area and the uniformity of _{numbers of plants per plot were not constant}

the spatial distribution of the individual plants _{(Table 1). In addition to the separate analyses for} over the area for agronomic traits of economic _{each individual plot, the di}_ff_{erent plots of the same}

interest. _{cultivar were collected into one data set and}

ana-lysed. Finally, the analysis was also carried out for the collected total plant material (over plots and

3. Materials and methods over cultivars). Analyses, therefore, were

per-formed for 17 data sets.

Three German cultivars (Ceres, Falcon, At harvest (August 1991), each plant was

har-Liberator) of winter oilseed rape (Brassica napus vested individually. The following traits were

mea-L.) were grown in Hohenschulen, a location near sured on a per plant basis: grain yield (g), number

Kiel, Schleswig-Holstein (northern part of of pods, total dry matter (g) and 1000-grain weight

Germany) on a sandy loam soil in August 1990 (g). The traits grain yield per pod (g), harvest index

(plot size: 1 m₂; row distance: 16 cm; density: 60 (=_{ratio of grain yield and total dry matter), number} seeds per m2). Some of the characteristics of the of seeds and number of seeds per pod were derived

cultivars worth mentioning are: high and stable from the measurements. To calculate the 1000-grain

yield, regeneration ability after leaf losses during weight, a sample of 1000 seeds was counted and

winter (Ceres), yield stability, dry stress tolerance weighted. Some methodological information on

moisture processing may be of interest. The above-(Falcon) and high oil and seed yield, broad range

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Table 1

Numbers of plots per cultivar, numbers of plants per plot and parameters (means, standard deviations, coefficients of variation) of the frequency distributions of individual plant areas

Cultivar Plot Number of Frequency distribution of individual plant areas plants

Mean (A:) Standard deviation (s

A) Coefficient of variation (%) (vA)

Ceres 1 22 454.34 200.04 44.03

2 43 220.10 100.16 45.51

3 34 305.23 128.33 42.04

Falcon 1 40 255.05 131.84 51.69

2 25 355.14 190.55 53.65

3 34 246.43 120.22 48.79

4 34 261.89 119.40 45.59

5 21 420.79 204.71 48.65

6 9 484.34 217.43 44.89

7 13 660.53 188.42 28.53

Liberator 1 31 294.35 119.09 40.46

2 32 291.21 99.99 34.34

3 28 300.02 138.73 46.24

Ceres All plots 99 301.39 163.02 54.09

Falcon All plots 176 330.37 193.94 58.70

Liberator All plots 91 294.99 118.14 40.05

Total All plots 366 313.74 169.92 54.16

ground parts of the plants were separated into seeds (null hypothesis: zero correlation) and for a

com-parison of the goodness-of-fit for the two

and remaining parts after drying (72 h, 35°_{C ). Seed}

approaches based on Eqs. (3) and (4) have been weight and moisture content were measured, and

carried out by elementary statistical procedures grain yield and 1000-grain weight were calculated

(Sachs, 1969) for error probabilities a=5% _and by eliminating the moisture content. Total dry

a=1%, respectively. matter was measured after drying (105°_{C ) until}

weight constancy. The plants were harvested at rapeseed growth stage No. 87 (according to the

4. Results and discussion

classification of the German ‘Biologische

Bundesanstalt’, which is based on the

Eucarpia-The spatial characteristics of the Thiessen

poly-code) where stage of development No. 87 means _{gon tesselations for the 17 data sets can be specified}

that most seeds are half-sided black. This date of _{by some parameters of the frequency distribution}

harvest guarantees (1) that no special precautions _{of individual plant areas: means (}_A_:_{), standard}

were needed because of the range of maturities of _{deviations (s}

A), and coefficients of variation (vA) seed and (2) that no checks were necessary to verify _{(Table 1). The 17 data sets are somewhat di}_ff_erent

that no over-ripe seed had been lost. _{in their means (range: 220.10–660.53 cm}₂_),

stan-Each of the two relationships betweenEandA _{dard deviations (range: 99.99–217.43 cm}₂_{), and}

contains two unknown parameters [k

1and k2 for coefficients of variation (range: 28.53–58.70)

Eq. (3) andk

3andk4for Eq. (4)], which must be (Table 1). This considerable variability between

estimated from the data. This was carried out by the data sets can be hardly explained by existing

transforming Eqs. (3) and (4) and applying linear differences between cultivars: the differences

regression analysis techniques. The goodness-of-fit between plots within cultivars are greater than the

for both relationships was calculated and com- differences between cultivars ( Table 1). This

con-pared by using residual variances. clusion is true forA:,s

AandvA. The means (A:) of the three cultivars are more or less similar: 301.39 Tests of significance for correlation coefficients

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Fig. 2. Experimental results for the relationships between single plant trait values and individual spaces for one data set (Ceres, plot 1) and eight traits.

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(Ceres), 330.37 ( Falcon), and 294.99 (Liberator). dual plant areas for eight traits and 17 data sets. These correlation coe_fficients are positive (with The standard deviations (s

A), however, are more

differentiated: 163.02 (Ceres), 193.94 (Falcon), and some slight exceptions for the trait 1000-grain weight) ( Table 2). The numerical magnitude of the 118.14 (Liberator). The same is true for the

coefficients of variation (v

A): 54.09 (Ceres), 58.70 correlations, however, are quite di_di fferent for the fferent traits: largest for grain yield, number of (Falcon), and 40.05 (Liberator). The variability of

individual plant areas (s

A) is considerably large seeds and number of pods and lowest for_{1000-grain weight and number of seeds per pod.} (for all data sets). This result may be partly

explained as an outcome of the sowing technique The correlation coefficients for the traits total dry matter, harvest index and grain yield per pod are applied. However, many other biotic and abiotic

effects during stand development may be responsi- intermediate in their numerical magnitude. It may also be of interest to study the relation-ble. Largers

A values, therefore, can be obtained

even in the case of accurate sowing techniques and ships between these correlation coefficients from Table 2 and the mean plant areas ( Table 1) where perfect crop management practices.

Some experimental results on the relationships data sets are considered as replications. For total dry matter, for example, one obtains no signifi-between single plant trait value and individual

space are presented in Fig. 2. For demonstration cance of the correlation coe_fficients from Table 2 in any case where the mean space is more than purposes, the results for only one plot of one

cultivar (Ceres, plot 1) are given. A graphical 400. The Spearman rank correlation coe_fficients

between mean plant areas (A:) and the correlation representation of the collected data sets of all the

plots for each cultivar seems to be inappropriate coefficients from Table 2, however, are not signifi-cantly different from zero for all the eight traits because of the heterogeneity between plots.

Table 2 contains the correlation coefficients (for an error probability of 1%). The largest

rank correlation coefficients were obtained for between trait measurements (per plant) and

indivi-Table 2

Correlation coefficients between trait measurements per plant and individual areas for the eight traits and the 17 data setsa

Cultivar Plot Traits (per plant)

Grain Total dry Number Number 1000-grain Harvest Grain yield Number of yield (g) matter (g) of pods of seeds weight (g) index per pod (g) seeds per pod

Ceres 1 0.75** 0.40 0.73** 0.73** 0.24 0.66** 0.57** 0.46*

2 0.79** 0.35* 0.79** 0.80** −0.02 0.57** 0.18 0.24

3 0.70** 0.28 0.67** 0.67** 0.14 0.41* 0.47** 0.24

Falcon 1 0.79** 0.30 0.77** 0.78** −0.18 0.60** 0.28 0.36*

2 0.87** 0.64** 0.89** 0.86** −0.04 0.33 0.27 0.21

3 0.78** 0.35* 0.83** 0.79** 0.03 0.69** 0.07 0.04

4 0.91** 0.74** 0.89** 0.89** 0.07 0.22 0.38* 0.31

5 0.66** 0.31 0.67** 0.62** 0.20 0.17 0.37 0.13

6 0.89** 0.47 0.88** 0.90** 0.27 0.56 0.33 0.05

7 0.68* 0.52 0.66* 0.71** −0.01 0.25 0.41 0.23

Liberator 1 0.86** 0.42* 0.72** 0.84** 0.10 0.40* 0.62** 0.61**

2 0.64** 0.38* 0.42* 0.64** −0.06 0.03 0.51** 0.49**

3 0.72** 0.31 0.68** 0.72** 0.12 0.52** 0.30 0.28

Ceres All plots 0.82** 0.52** 0.81** 0.81** 0.20* 0.54** 0.48** 0.33** Falcon All plots 0.86** 0.65** 0.86** 0.86** 0.23** 0.39** 0.44** 0.22** Liberator All plots 0.69** 0.33** 0.60** 0.68** 0.05 0.31** 0.45** 0.43** Total All plots 0.83** 0.57** 0.81** 0.83** 0.16** 0.39** 0.43** 0.28**

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1000-grain weight (+0.58), number of seeds per between the goodness-of-fit for both relationships [Eqs. (3) and (4)] for any of the 136 comparisons

pod (−0.31) and total dry matter (+0.28).

However, a detailed discussion of these relation- (17 data sets×eight traits). However, in most

cases, the logarithmic relationship Eq. (3) seems ships is beyond the scope of this paper.

The experimental results from Table 2 are in to be slightly superior to the hyperbolic

relation-ship Eq. (4) (exception: 1000-grain weight). The good agreement with the expected relationships.

In the Thiessen-polygon approach of this paper, numerical results (estimates of the four parameters

1,k2,k3andk4; residual variances; test statistics variable environmental conditions are modelled by

variable areas of the individual plants. These areas for goodness-of-fit comparisons) are not presented in this paper. The main conclusion is that both reflect different availabilities of environmental

resources (water, light, nutrients, physical growing relationships betweenEandAare equally appro-priate to describe the experimental data.

space, etc.). For traits that sensitively respond to

environmental fluctuations ( low heritability), the The commonly used estimate for mean value

for trait per area (K) is the first term in the correlation between trait measurements (per plant)

and individual areas must be expected to be large. theoretical decomposition of K into two additive

terms in Eq. (1) (for approach 1) and in Eq. (5) The correlation for environmentally less sensitive

traits ( large heritability), however, must be (for approach 2). These estimates are f(A:) and

E:/A:, respectively. The second term can be interpre-expected to be low. As has been demonstrated in

many experimental studies, the most environmen- ted as the e_ffect of variable individual plant spaces on the mean value for trait per area. In this paper, tally sensitive traits are grain yield, number of

seeds and number of pods with low heritabilities. these terms are expressed relative to the total mean value for trait per area (K=100%). These percent-The largest heritabilities are obtained for the

1000-grain weight and number of seeds per pod. ages are presented in Table 3 [ for Eq. (1) with

relationship in Eq. (3)], Table 4 [ for Eq. (1) with These expectations are confirmed by the

experi-mental results from Table 2. relationship in Eq. (4)], and Table 5 [for Eq. (5)].

Only in a few exceptional cases (plots 2, 4, 6, 7 There are no statistically significant differences

Table 3

Relative contribution (%) of the second term in Eq. (1) with relationship Eq. (3) for the eight traits and the 17 data sets Cultivar Plot Traits (per plant)

Grain Total dry Number Number 1000-grain Harvest Grain yield Number of yield (g) matter (g) of pods of seeds weight (g) index per pod (g) seeds per pod

Ceres 1 4.48 8.92 6.52 5.05 15.43 12.26 13.65 14.42

2 6.65 12.51 7.79 6.60 17.34 13.03 16.20 15.92

3 1.97 8.54 4.48 3.17 14.57 10.62 12.60 13.25

Falcon 1 3.71 14.49 5.37 2.96 21.86 14.98 20.02 19.25

2 0.05 6.66 1.77 −0.06 22.43 18.36 21.55 21.60

3 2.70 11.91 2.92 3.19 19.40 13.14 19.12 18.99

4 −4.24 −2.77 −1.62 −4.08 16.75 15.50 15.66 15.89

5 5.14 10.77 7.58 6.53 17.83 16.24 17.14 18.43

6 −1.39 6.69 −0.63 −0.96 16.03 11.88 16.08 16.74

7 −2.33 −2.59 −1.06 −2.06 7.62 5.64 5.78 6.62

Liberator 1 1.30 7.78 6.98 1.57 13.67 9.89 8.91 9.14

2 3.96 3.78 6.84 3.80 10.49 9.85 7.81 7.73

3 3.16 9.85 5.26 3.36 16.97 12.74 16.08 16.65

Ceres All plots 0.10 9.04 4.05 1.52 21.84 17.45 19.86 20.68

Falcon All plots −5.17 1.79 −1.28 −2.93 24.64 21.06 23.48 24.44 Liberator All plots 2.43 6.90 5.88 2.49 13.44 10.64 10.85 11.13

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Table 4

Relative contribution (%) of the second term in Eq. (1) with relationship Eq. (4) for the eight traits and the 17 data sets Cultivar Plot Traits (per plant)

Grain Total dry Number Number 1000-grain Harvest Grain yield Number of yield (g) matter (g) of pods of seeds weight (g) index per pod (g) seeds per pod

Ceres 1 3.38 9.20 5.81 4.23 15.54 13.26 14.31 15.08

2 7.11 11.63 7.69 6.70 17.52 14.02 16.30 16.02

3 1.40 6.02 3.51 1.09 14.76 11.58 12.02 11.99

Falcon 1 7.18 14.76 8.14 6.38 21.88 15.83 20.07 19.23

2 3.31 10.27 3.07 3.69 22.20 17.52 22.12 22.25

3 5.85 13.29 5.99 4.60 19.67 14.97 19.09 18.57

4 3.29 4.10 3.90 3.68 17.00 15.93 15.87 16.28

5 8.03 13.65 9.87 9.82 17.51 15.55 17.12 18.75

6 2.15 8.65 2.62 2.74 15.97 11.01 16.07 16.84

7 0.15 1.49 0.11 0.01 7.64 5.64 5.39 6.37

Liberator 1 3.15 9.46 8.48 3.59 13.69 9.25 8.29 8.80

2 5.74 8.59 8.54 6.27 10.19 9.44 7.59 8.06

3 7.73 12.91 8.22 8.53 17.03 14.82 16.71 17.36

Ceres All plots 6.19 11.95 8.14 6.34 22.16 19.13 20.39 20.88

Falcon All plots 5.10 11.81 6.36 5.18 25.06 21.83 24.00 24.56

Liberator All plots 5.57 10.66 8.22 6.26 13.32 10.78 10.89 11.48

Total All plots 5.47 11.94 7.13 5.96 22.04 18.96 20.56 21.22

Table 5

Relative contribution (%) of the second term in Eq. (5) for the eight traits and the 17 data sets Cultivar Plot Traits (per plant)

Grain Total dry Number Number 1000-grain Harvest Grain yield Number of yield (g) matter (g) of pods of seeds weight (g) index per pod (g) seeds per pod

Ceres 1 7.37 14.37 10.61 8.09 15.56 18.59 20.43 21.13

2 13.69 20.61 15.08 13.78 25.99 20.70 24.12 23.93

3 8.14 15.61 10.89 9.72 20.65 17.03 19.14 19.95

Falcon 1 13.87 26.35 16.45 13.19 37.63 26.85 31.26 30.70

2 9.75 15.96 13.15 9.70 35.11 32.25 33.72 33.68

3 10.08 22.52 10.87 11.05 28.85 22.38 28.65 28.80

4 −2.43 −1.43 1.95 −2.35 24.87 23.51 23.48 23.67

5 16.60 21.32 18.61 17.63 20.35 27.07 11.18 28.17

6 1.65 12.43 2.69 1.98 17.79 19.39 17.05 23.19

7 0.88 0.17 1.93 1.16 9.41 7.84 7.81 2.05

Liberator 1 6.05 12.96 12.10 6.40 16.37 15.91 14.92 15.11

2 7.11 6.57 9.82 6.68 15.80 13.50 11.36 11.03

3 9.07 18.24 12.18 9.08 20.94 20.51 24.16 24.55

Ceres All plots 7.36 20.72 13.57 9.69 25.91 30.28 32.70 33.62

Falcon All plots 3.67 12.80 10.57 7.85 40.23 37.71 39.11 40.14

Liberator All plots 6.64 11.72 10.61 6.49 17.53 16.31 16.30 16.37

Total All plots 6.05 15.18 12.41 8.11 34.84 31.76 33.07 33.84

of Falcon; Falcon over all plots; total material ) trait per area if one applies the commonly used

estimates f(A:) and E:/A:. These overestimations, are negative values found for the second term.

(10)

Eq. (1) with the relationship in Eq. (4) (Table 4), dual plant area will be responded by a corresponding change in individual trait value of no negative percentage for the second term was

obtained (for no data set and for no trait). The this plant. With respect to the ratio E/A, the

changes inE andA, therefore, can be considered hyperbolic relationship between E and A,

there-fore, leads to no overestimation of mean value for as compensatory effects. For the mean value of

E/A, i.e. forK, changes inA(measured bys2_A) are trait per area if the common estimatef(A:) is used.

In all cases with positive values for the second of no particular importance. The second term,

however, is proportional tos_2A. We can conclude, term, the mean value for trait per area is

underesti-mated by the commonly used estimates f(A:) and therefore, that the second term is of no major

relevance for the traits of group 1. Their percent-E:/A:, respectively ( Tables 3–5). The magnitude of

this underestimation exhibits a large range of ages are expected to be small. For the traits of

group 2, the reverse conclusions are true. percentages for the different data sets, for the

different traits and for the different approaches: To make further conclusions on the second

terms from Tables 3–5, their relationships can be from 0%(data set: Falcon plot 7; trait: number of

seeds) ( Table 4) up to 40% (data set: Falcon over quantitatively described by Spearman rank

corre-lation coefficients. By this procedure, the 17 data all plots; trait: 1000-grain weight) ( Table 5). With

regard to the numerical amount of the second sets are considered as replications. The Spearman

rank correlation coe_fficients between traits are terms from Tables 3–5, no clear and unique

pat-terns and relationships can be observed summarized in Tables 6 and 7, while those between

approaches are presented in Table 8. With regard 1. between traits (for the same approach and the

same data set), to the correlation results from Tables 6–8, the

eight traits can be clearly divided into two groups 2. between approaches (for the same trait and the

same data set), and so that the results within groups are similar, and

the results between groups are somewhat different. 3. between data sets (for the same trait and the

same approach). These groups are groups 1 and 2 as before.

The correlations for pairs of traits belonging to Nevertheless, several general and less sophisticated

conclusions can be made: the same group are positive and large with larger

values for group 2 than for group 1. The correla-The most evident result is a classification of the

eight traits into two distinct groups ( Tables 3–5): tions between traits belonging to different groups, however, are intermediate or small ( Tables 6 and 7).

As has been explained before, the numerical value of the second terms from Tables 3–5 can be grain yield

total dry matter number of pods number of seeds

H

group 1 (with small percentages

for the second term) interpreted as the e_{plant spaces on the mean value for trait per area.}ffects of variable individual The numerical amount of the Spearman rank correlation coefficients from Tables 6 and 7 (based on replications=data sets) are, therefore, indica-tive of similar or dissimilar reactions of the 1000-grain weight

harvest index grain yield per pod number of seeds per pod

H

group 2 (with large percentages

for the second term)

different traits to variations in individual plant spaces. Thus, we can conclude that the traits of group 1 exhibit similar reactions to fluctuating plant areas. The same is true for the traits of group 2. Traits from different groups, however, show This classification is true for each approach.

Group 1 contains the most important compo- dissimilar reactions. The traits of group 1 are

environmentally more sensitive, while the traits of nents for formation of yield per area. These traits

are characterized by a large environmental sensitiv- group 2 are less sensitive. In this paper, the parti-tion of the eight traits into the two distinct groups ity, which implies that each fluctuation in

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indivi-Table 6

Spearman rank correlation coefficients between pairs of traits of the second-term-percentages (right upper part for percentages of Table 3; left lower part for percentages of Table 4)a

Grain Total dry Number Number 1000-grain Harvest Grain yield Number of yield (g) matter (g) of pods of seeds weight (g) index per pod (g) seeds per pod

Grain yield 0.76** 0.89** 0.98** −0.15 −0.24 −0.16 −0.22

Total dry matter 0.84** 0.63** 0.73** 0.24 0.07 0.23 0.18

Number of pods 0.71** 0.52* 0.87** −0.26 −0.32 −0.27 −0.34

Number of seeds 0.97** 0.76** 0.75** −0.15 −0.23 −0.16 −0.22

1000-grain weight 0.40 0.65** −0.04 0.37 0.94** 0.99** 0.98**

Harvest index 0.37 0.54* −0.02 0.36 0.93** 0.93** 0.93**

Grain yield per pod 0.42 0.68** −0.03 0.38 0.99** 0.92** 0.99**

Number of seeds per pod 0.38 0.65** −0.04 0.35 0.96** 0.93** 0.99**

a*, ** Significance for error probabilities of 5 and 1%, respectively. Table 7

Spearman rank correlation coefficients between pairs of traits of the second-term percentages (for percentages of Table 5)a

Grain Total dry Number Number 1000-grain Harvest Grain yield Number of yield (g) matter (g) of pods of seeds weight (g) index per pod (g) seeds per pod

Grain yield 0.87** 0.80** 0.94** 0.31 0.25 0.16 0.31

Total dry matter 0.86** 0.92** 0.52* 0.47 0.41 0.57*

Number of pods 0.85** 0.47 0.49* 0.34 0.52*

Number of seeds 0.49* 0.44 0.33 0.50*

1000-grain weight 0.86** 0.89** 0.90**

Harvest index 0.81** 0.97**

Grain yield per pod 0.88**

a*, ** Significance for error probabilities of 5 and 1%, respectively

1 and 2 have been obtained by two different

criteria: numerical amount of the second-term

per-Table 8 _{centages and Spearman rank correlation coe}

ffi

-Spearman rank correlation coefficients between pairs of

cients between these second-term percentages. The

approaches of the second-term percentages from Tables 3–5a

classification of traits

Trait Comparison _{1. based on environmental sensitivity,}

2. based on rank correlations between

second-Tables 3 Tables 3 Tables 4

term-percentages, and

and 4 and 5 and 5

3. based on numerical amount of

second-term-Grain yield 0.64** 0.79** 0.70** _percentages

Total dry matter 0.70** 0.88** 0.83** _{is coincident.} Number of pods 0.72** 0.53* 0.47

The Spearman rank correlation coe_fficients

Number of seeds 0.58* 0.65** 0.60*

between pairs of approaches (for the same trait)

1000-grain weight 0.99** 0.92** 0.91**

Harvest index 0.97** 0.99** 0.97** of the second-term percentages from Tables 3–5

Grain yield per pod 0.99** 0.86** 0.88** _{are positive and large ( Table 8). The correlations} Number of seeds per pod 0.99** 0.98** 0.99**

for the traits of group 2 are larger than those for the traits of group 1. With regard to the numerical

a*, ** Significance for error probabilities of 5 and 1%,

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di_fferent approaches lead to similar results for the G. Rave (analysis of the empirical data sets) and the technical assistance by Mrs H. Jensen, and rank orders of the data sets. This is particularly

true for the traits of group 2 ( Table 8). Mrs B. Pallasch.

The clear classification of the eight traits into two distinct groups provides a cogent

interpreta-tion of the main results of this paper: The mean _References

value of ‘trait per area’ can be decomposed into

two additive terms where the first term depends _{Baker, R.J., Briggs, K.G., 1983. Relationships between plant} on the mean of individual plant spaces, while the density and yield in barley. Crop Sci. 23, 590–592.

Bleasdale, J.K.A., Nelder, J.A., 1960. Plant population and

second term simultaneously depends on their

vari-crop yield. Nature 188, 342

ance and mean. This second term is proportional

de Wit, C.T., 1960. On competition ( Versl. landbouwk.

to the variance of individual plant areas, i.e. second

Onderz.), Pudoc Wageningen. Agric. Res. Rep. 66 (8)

term=_{0 for variance}=_{0. The second term can be}

82 pp.

interpreted as the effect of variable individual plant Dinter, R.G., 1991. Ertragsbildung und Ertragsstruktur von Winterraps in Abha¨ngigkeit von der Standfla¨che der

Ein-spaces on the mean value of the ‘trait per area’.

zelpflanze. Masters thesis, Faculty of Agriculture,

Univer-The traits grain yield, total dry matter, number of

sity Kiel.

pods and number of seeds with a high

environmen-Griepentrog, H.W., 1995. La¨ngsverteilung von Sa¨maschinen

tal sensitivity are characterized by small percent- _{und ihre Wirkung auf Standfla¨che und Ertrag bei Raps.}

ages for the second term. This can be explained Agrartechnische Forschung 1, 129–136.

Harper, J.L., 1977. Population Biology of Plants. Academic

because changes in individual trait values and

Press, London.

changes in individual plant spaces may be

consid-Hu¨hn, M., 1998. Ein allgemeiner Ansatz zur Quantifizierung

ered — at least partially — as compensatory

des Einflusses der Gu¨te der Sa¨technik auf den Fla¨chenertrag.

effects. With regard to the mean of the ratios _{J. Agron. Crop Sci. 181, 249–255.}

(individual trait value)/(individual plant space) _{Hu¨hn, M., 1999a. Theoretical results on the eff}ects of nonregu-lar spatial patterns of plants on yield per area. J. Agron.

changes in plant spaces are of no particular

impor-Crop Sci. 182, 1–7.

tance. For the traits 1000-grain weight, harvest

Hu¨hn, M., 1999b. Experimental results on the effects of

nonreg-index, grain yield per pod and number of seeds

ular spatial patterns of plants on yield per area. J. Agron.

per pod with lesser environmental sensitivity, the _{Crop Sci. 182, 89–97.}

reverse conclusions are true. Kira, T., Ogawa, H., Sakazaki, N., 1953. Intraspecific

competi-tion among higher plants. I. Competicompeti-tion–yield–density

If one considers the traits, which were measured

interrelationships in regularly dispersed populations. J. Inst.

on a per plant basis, as ratio traits (trait

Polytech Osaka, D 4, 1–16.

value)/(plant space) the following conclusions can

Mu¨lle, G., Heege, H.J., 1981. Kornverteilung u¨ber die Fla¨che

be made. For grain yield, total dry matter, number _{und Ertrag bei Getreide. Z. Acker Pflanzenbau 150, 97–}

of pods and number of seeds, non-regular spatial _112.

Risser, P.G., 1969. Competitive relationships among

herba-distributions of the individual plants over the area

ceous grassland plants. Bot. Rev. 35, 251–284.

must not necessarily be considered as

disadvanta-Sachs, L., 1969. Statistische Auswertungsmethoden. Zweite

geous (small percentages for the second term). For

neubearb. und erw Aufl., Springer, Berlin.

the traits 1000-grain weight, harvest-index, grain _{Shinozaki, K., Kira, T., 1956. Intraspecific competition among} yield per pod and number of seeds per pod, higher plants. VII. Logistic theory of the C–D effect. J. Inst.

Polytech Osaka D7, 35–72.

however, the effects of non-regular spatial

distribu-Shinozaki, K., Kira, T., 1961. Intraspecific competition among

tions of the individual plants are substantial ( large

higher plants. X. The C–D rule its theory and practical uses.

percentages for the second term).

J. Biol. Osaka Cy. Univ. 12, 69–82.

Spitters, C.J.T., 1983. An alternative approach to the analysis of mixed cropping experiments I and II. Neth. J. Agric. Sci.

Acknowledgements _{31 pp. 1–11, 143–155.}

Willey, R.W., Heath, S.B., 1969. The quantitative relationships

The author gratefully acknowledges the cooper- between plant population and crop yield. Adv. Agron. 21, 281–321.

(1)

(Ceres), 330.37 ( Falcon), and 294.99 (Liberator). dual plant areas for eight traits and 17 data sets. These correlation coe_fficients are positive (with The standard deviations (s

A), however, are more

differentiated: 163.02 (Ceres), 193.94 (Falcon), and some slight exceptions for the trait 1000-grain

weight) ( Table 2). The numerical magnitude of the 118.14 (Liberator). The same is true for the

coefficients of variation (v

A): 54.09 (Ceres), 58.70 correlations, however, are quite di_di fferent for the

fferent traits: largest for grain yield, number of (Falcon), and 40.05 (Liberator). The variability of

individual plant areas (s

A) is considerably large seeds and number of pods and lowest for_{1000-grain weight and number of seeds per pod.} (for all data sets). This result may be partly

explained as an outcome of the sowing technique The correlation coefficients for the traits total dry

matter, harvest index and grain yield per pod are applied. However, many other biotic and abiotic

effects during stand development may be responsi- intermediate in their numerical magnitude.

It may also be of interest to study the relation-ble. Largers

A values, therefore, can be obtained

even in the case of accurate sowing techniques and ships between these correlation coefficients from

Table 2 and the mean plant areas ( Table 1) where perfect crop management practices.

Some experimental results on the relationships data sets are considered as replications. For total

dry matter, for example, one obtains no signifi-between single plant trait value and individual

space are presented in Fig. 2. For demonstration cance of the correlation coe_fficients from Table 2

in any case where the mean space is more than purposes, the results for only one plot of one

cultivar (Ceres, plot 1) are given. A graphical 400. The Spearman rank correlation coe_fficients

between mean plant areas (A:) and the correlation representation of the collected data sets of all the

plots for each cultivar seems to be inappropriate coefficients from Table 2, however, are not signifi-cantly different from zero for all the eight traits because of the heterogeneity between plots.

Table 2 contains the correlation coefficients (for an error probability of 1%). The largest

rank correlation coefficients were obtained for between trait measurements (per plant) and

indivi-Table 2

Correlation coefficients between trait measurements per plant and individual areas for the eight traits and the 17 data setsa Cultivar Plot Traits (per plant)

Grain Total dry Number Number 1000-grain Harvest Grain yield Number of yield (g) matter (g) of pods of seeds weight (g) index per pod (g) seeds per pod

Ceres 1 0.75** 0.40 0.73** 0.73** 0.24 0.66** 0.57** 0.46*

2 0.79** 0.35* 0.79** 0.80** −0.02 0.57** 0.18 0.24

3 0.70** 0.28 0.67** 0.67** 0.14 0.41* 0.47** 0.24

Falcon 1 0.79** 0.30 0.77** 0.78** −0.18 0.60** 0.28 0.36*

2 0.87** 0.64** 0.89** 0.86** −0.04 0.33 0.27 0.21

3 0.78** 0.35* 0.83** 0.79** 0.03 0.69** 0.07 0.04

4 0.91** 0.74** 0.89** 0.89** 0.07 0.22 0.38* 0.31

5 0.66** 0.31 0.67** 0.62** 0.20 0.17 0.37 0.13

6 0.89** 0.47 0.88** 0.90** 0.27 0.56 0.33 0.05

7 0.68* 0.52 0.66* 0.71** −0.01 0.25 0.41 0.23

Liberator 1 0.86** 0.42* 0.72** 0.84** 0.10 0.40* 0.62** 0.61**

2 0.64** 0.38* 0.42* 0.64** −0.06 0.03 0.51** 0.49**

3 0.72** 0.31 0.68** 0.72** 0.12 0.52** 0.30 0.28

(2)

1000-grain weight (+0.58), number of seeds per between the goodness-of-fit for both relationships [Eqs. (3) and (4)] for any of the 136 comparisons

pod (−0.31) and total dry matter (+0.28).

However, a detailed discussion of these relation- (17 data sets×eight traits). However, in most

cases, the logarithmic relationship Eq. (3) seems ships is beyond the scope of this paper.

The experimental results from Table 2 are in to be slightly superior to the hyperbolic

relation-ship Eq. (4) (exception: 1000-grain weight). The good agreement with the expected relationships.

In the Thiessen-polygon approach of this paper, numerical results (estimates of the four parameters

1,k2,k3andk4; residual variances; test statistics variable environmental conditions are modelled by

variable areas of the individual plants. These areas for goodness-of-fit comparisons) are not presented

in this paper. The main conclusion is that both reflect different availabilities of environmental

resources (water, light, nutrients, physical growing relationships betweenEandAare equally

appro-priate to describe the experimental data. space, etc.). For traits that sensitively respond to

environmental fluctuations ( low heritability), the The commonly used estimate for mean value

for trait per area (K) is the first term in the correlation between trait measurements (per plant)

and individual areas must be expected to be large. theoretical decomposition of K into two additive

terms in Eq. (1) (for approach 1) and in Eq. (5) The correlation for environmentally less sensitive

traits ( large heritability), however, must be (for approach 2). These estimates are f(A:) and

E:/A:, respectively. The second term can be interpre-expected to be low. As has been demonstrated in

many experimental studies, the most environmen- ted as the e_ffect of variable individual plant spaces

on the mean value for trait per area. In this paper, tally sensitive traits are grain yield, number of

seeds and number of pods with low heritabilities. these terms are expressed relative to the total mean

value for trait per area (K=100%). These

percent-The largest heritabilities are obtained for the

1000-grain weight and number of seeds per pod. ages are presented in Table 3 [ for Eq. (1) with

relationship in Eq. (3)], Table 4 [ for Eq. (1) with These expectations are confirmed by the

experi-mental results from Table 2. relationship in Eq. (4)], and Table 5 [for Eq. (5)].

Only in a few exceptional cases (plots 2, 4, 6, 7 There are no statistically significant differences

Table 3

Relative contribution (%) of the second term in Eq. (1) with relationship Eq. (3) for the eight traits and the 17 data sets Cultivar Plot Traits (per plant)

Grain Total dry Number Number 1000-grain Harvest Grain yield Number of yield (g) matter (g) of pods of seeds weight (g) index per pod (g) seeds per pod

Ceres 1 4.48 8.92 6.52 5.05 15.43 12.26 13.65 14.42

2 6.65 12.51 7.79 6.60 17.34 13.03 16.20 15.92

3 1.97 8.54 4.48 3.17 14.57 10.62 12.60 13.25

Falcon 1 3.71 14.49 5.37 2.96 21.86 14.98 20.02 19.25

2 0.05 6.66 1.77 −0.06 22.43 18.36 21.55 21.60

3 2.70 11.91 2.92 3.19 19.40 13.14 19.12 18.99

4 −4.24 −2.77 −1.62 −4.08 16.75 15.50 15.66 15.89

5 5.14 10.77 7.58 6.53 17.83 16.24 17.14 18.43

6 −1.39 6.69 −0.63 −0.96 16.03 11.88 16.08 16.74

7 −2.33 −2.59 −1.06 −2.06 7.62 5.64 5.78 6.62

Liberator 1 1.30 7.78 6.98 1.57 13.67 9.89 8.91 9.14

2 3.96 3.78 6.84 3.80 10.49 9.85 7.81 7.73

3 3.16 9.85 5.26 3.36 16.97 12.74 16.08 16.65

Ceres All plots 0.10 9.04 4.05 1.52 21.84 17.45 19.86 20.68

Falcon All plots −5.17 1.79 −1.28 −2.93 24.64 21.06 23.48 24.44 Liberator All plots 2.43 6.90 5.88 2.49 13.44 10.64 10.85 11.13

(3)

Table 4

Relative contribution (%) of the second term in Eq. (1) with relationship Eq. (4) for the eight traits and the 17 data sets Cultivar Plot Traits (per plant)

Grain Total dry Number Number 1000-grain Harvest Grain yield Number of yield (g) matter (g) of pods of seeds weight (g) index per pod (g) seeds per pod

Ceres 1 3.38 9.20 5.81 4.23 15.54 13.26 14.31 15.08

2 7.11 11.63 7.69 6.70 17.52 14.02 16.30 16.02

3 1.40 6.02 3.51 1.09 14.76 11.58 12.02 11.99

Falcon 1 7.18 14.76 8.14 6.38 21.88 15.83 20.07 19.23

2 3.31 10.27 3.07 3.69 22.20 17.52 22.12 22.25

3 5.85 13.29 5.99 4.60 19.67 14.97 19.09 18.57

4 3.29 4.10 3.90 3.68 17.00 15.93 15.87 16.28

5 8.03 13.65 9.87 9.82 17.51 15.55 17.12 18.75

6 2.15 8.65 2.62 2.74 15.97 11.01 16.07 16.84

7 0.15 1.49 0.11 0.01 7.64 5.64 5.39 6.37

Liberator 1 3.15 9.46 8.48 3.59 13.69 9.25 8.29 8.80

2 5.74 8.59 8.54 6.27 10.19 9.44 7.59 8.06

3 7.73 12.91 8.22 8.53 17.03 14.82 16.71 17.36

Ceres All plots 6.19 11.95 8.14 6.34 22.16 19.13 20.39 20.88

Falcon All plots 5.10 11.81 6.36 5.18 25.06 21.83 24.00 24.56

Liberator All plots 5.57 10.66 8.22 6.26 13.32 10.78 10.89 11.48

Total All plots 5.47 11.94 7.13 5.96 22.04 18.96 20.56 21.22

Table 5

Relative contribution (%) of the second term in Eq. (5) for the eight traits and the 17 data sets Cultivar Plot Traits (per plant)

Grain Total dry Number Number 1000-grain Harvest Grain yield Number of yield (g) matter (g) of pods of seeds weight (g) index per pod (g) seeds per pod

Ceres 1 7.37 14.37 10.61 8.09 15.56 18.59 20.43 21.13

2 13.69 20.61 15.08 13.78 25.99 20.70 24.12 23.93

3 8.14 15.61 10.89 9.72 20.65 17.03 19.14 19.95

Falcon 1 13.87 26.35 16.45 13.19 37.63 26.85 31.26 30.70

2 9.75 15.96 13.15 9.70 35.11 32.25 33.72 33.68

3 10.08 22.52 10.87 11.05 28.85 22.38 28.65 28.80

4 −2.43 −1.43 1.95 −2.35 24.87 23.51 23.48 23.67

5 16.60 21.32 18.61 17.63 20.35 27.07 11.18 28.17

6 1.65 12.43 2.69 1.98 17.79 19.39 17.05 23.19

7 0.88 0.17 1.93 1.16 9.41 7.84 7.81 2.05

Liberator 1 6.05 12.96 12.10 6.40 16.37 15.91 14.92 15.11

2 7.11 6.57 9.82 6.68 15.80 13.50 11.36 11.03

3 9.07 18.24 12.18 9.08 20.94 20.51 24.16 24.55

Ceres All plots 7.36 20.72 13.57 9.69 25.91 30.28 32.70 33.62

Falcon All plots 3.67 12.80 10.57 7.85 40.23 37.71 39.11 40.14

Liberator All plots 6.64 11.72 10.61 6.49 17.53 16.31 16.30 16.37

Total All plots 6.05 15.18 12.41 8.11 34.84 31.76 33.07 33.84

of Falcon; Falcon over all plots; total material ) trait per area if one applies the commonly used

estimates f(A:) and E:/A:. These overestimations, are negative values found for the second term.

(4)

Eq. (1) with the relationship in Eq. (4) (Table 4), dual plant area will be responded by a corresponding change in individual trait value of no negative percentage for the second term was

obtained (for no data set and for no trait). The this plant. With respect to the ratio E/A, the

changes inE andA, therefore, can be considered

hyperbolic relationship between E and A,

there-fore, leads to no overestimation of mean value for as compensatory effects. For the mean value of

E/A, i.e. forK, changes inA(measured bys2_A) are trait per area if the common estimatef(A:) is used.

In all cases with positive values for the second of no particular importance. The second term,

however, is proportional tos_2A. We can conclude, term, the mean value for trait per area is

underesti-mated by the commonly used estimates f(A:) and therefore, that the second term is of no major

relevance for the traits of group 1. Their

percent-E:/A:, respectively ( Tables 3–5). The magnitude of