2 M. Hu¨hn European Journal of Agronomy 12 2000 1–12
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 neighbourscompetitors. 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
3 M. Hu¨hn European Journal of Agronomy 12 2000 1–12
1999a,b, the individual measurements E i
, i=1, all interesting terms relative to fA
: =100 . For
the relative amount of the second term on the 2,…, n, and the individual spaces A
i , i=1, 2,…, n,
are considered as realizations of two random vari- right-hand side of Eq. 1, one obtains:
ables E and A. The mean value K for trait per Second term of Eq. 1
area is calculated theoretically by the expectation of the ratio EA. In Hu¨hn 1998, 1999a,b, two
= 50s
2A f ◊A
9 f A
9 [relative to f A
9 ]. 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=hA. 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 variable A.
to fA : , which means as a percentage of the value
2. Explicit inclusion of both random variables E for the ideal situation of constant individual
and A and 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 total K [ fA
: +12s2 A
f ◊A : =100
]: area K can be approximated Hu¨hn, 1998,
1999a,b: Second term of Eq. 1
Kf A 9 +
1 2
s 2
A f ◊A
9 , 1
= 50s
2 A
f ◊A 9
K relative to K .
2b This approach leads to a slightly modified inter-
with f A=hAA; A : =mean of A; s2
A =
variance pretation of the second term in Eq. 1: It quanti-
of A; f ◊=second derivative of f. Eq. 1 is obtained fies the effect of variable individual plant spaces
by expanding fA 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 fA : 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=hA=k 1
+ k
2 ln A
3 K-decrease
with increasing
uniformity [for
f ◊A : 0]
E=hA= A
k 3
+ k
4 A
, 4
assumption: constant mean A : .
Applied to yield per area, these theoretical considerations provide an explicit condition for
with appropriately chosen constants k 1
, k 2
, k 3
, 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 s 2
A =
0, K For approach 2, no explicit functional relation-
ship between E and A is used. Both random reduces to KfA
: . It seems reasonable to express
4 M. Hu¨hn European Journal of Agronomy 12 2000 1–12
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
K E
9 A
9 +
E 9
A 9
v A
− r
EA v
E v
A +
3v 3A,
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 r
EA =
correlation coefficient between E and A. 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.. =
100v A
− r
EA v
E v
A +
3v 3A relative to E9A9.
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 total K.
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 different 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
Kategori