III. The Relationship between Plant Rectangularity and Crop Yield
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315
PLANT POPULATION A N D CROP YIELD
TABLE VI
COMPARISON
OF THE VARIATION
REMAINING
AFTER FITTING
EQS. (18)
Study
TSS
Donald ( 1954)
I . Wimmera ryegrass grown at 5
30.2
densities for seed
23.7
2. Subterranean clover grown at 5
densities for seed
Crawford (1 964)
Wheat var. HILGENDORF grown at 5
0.72
densities for grain
Pendleton and Dungan ( I 960)
Wheat grown at 6 densities for grain, 2.33
average over 4 nitrogen levels and
4 varieties
Puckridge and Donald ( 1967)
Wheat grown at 5 densities for grain 19.3
Bb
Equation ( I 8)
RSS/%'
AND
(20)O
Equation (20)
RSS
0.90
0.0149
0.00292
0.90
0.0394
0.0536
0.85
0.0000806
0.0000947
0.75
0.0007 1 I
0.000234
0.85
0.0171
0.040 1
RSS = Residual Sum Squares; TSS =Total Sum Squares about the mean.
*Value of B which reduced RSS/A2to a minimum for the particular set of data.
The extent to which rectangularity may effect the yield of a crop is
clearly dependent on the plasticity of the individual plant, which in turn
must be dependent on the plant species. However, the general pattern of
effects is illustrated by some winter wheat data of Harvey et al. ( 1 958)
reported in Table VII. The treatments of Harvey etal. were not extreme,
yet it can be seen that as rectangularity increases, either by increasing
seed rate or increasing row width, yield per area gradually declines.
Similar effects have been shown by Wiggans (1939) for soybeans,
Reynolds ( 1950) for peas, Pendleton and Seif ( I 96 1) for maize, Bleasdale (1963) for peas, and Weber et al. (1966) for soybeans. Reynolds
(1 950) also showed that as rectangularity increases the optimum density
may decrease (Fig. 14).
It would therefore seem desirable that equations describing the
relationships between plant population and crop yield should be able to
describe the effects of rectangularity as well as those of density. This
can be particularly important because in the many population studies
where different populations have been established on constant row width,
rectangularity is not constant but increases with increase in density.
Goodall ( 1 960) attempted to fit the model
316
R. W. WILLEY AND S. B. HEATH
TABLE VII
THE EFFECTOF Row WIDTHA N D SEED RATE ON WINTERWHEATGRAINYIELDS
(cwt./acre)"
Seed rate, stoneslacre
Row width (inches)
5.5
II
17
4
8
43.9
43.0
41.6
43.9
42.5
41.4
43.6
41.4
38.0
12
UAfterHarvey et al. ( 1 958).
w = adpi dJ"
or
where dl is the intrarow spacing and d2 is the interrow spacing. Thus
d l d z is the space available per plant. Equation (28) is therefore an extension of Eq. (6).
Goodall fitted this model to some soybean data of Wiggans ( 1939) which
covered a range of densities and row widths. He found a significant
30
t
'Ot
o . - - o o
10
20
30
P
FIG. 14. The effects of rectangularity on the yield/density relationship in dried peas
(Reynolds, 1950): the three curves represent different row widths, 8, 16, and 24 inches:
y = cwt./acre, p = wnedacre.
PLANT POPULATION A N D CROP YIELD
317
difference between bl and b2; he suggested that this was due to row
orientation effects. Donald (1963) pointed out, however, that Eq. (28)
has the undesirable characteristic that, if either of the power terms is
greater than the other, then the optimum rectangularity at a given density
would be obtained where the distance between plants was increased in
one direction and decreased in the other. Berry ( 1967) criticized Goodall’s
fit to Wiggans’ data, not only on account of the poor fit of log w against
log d l , but also because the values of d , and dz were not overlapping, and
therefore different values of b1 and 6, could be expected.
Berry ( 1967) extended the simplified equation (Eq. 18) of Bleasdale
and Nelder to take into account plant rectangularity
(29)
Since dld2= s, this model has included an extra term proportional to the
square root of density. For a given density, w is greatest where dl = d,,
i.e., where recta’ngularity is 1 : 1 , since (l/dl) (I/d2) is at a minimum
value. This relationship gave a satisfactory fit to Wiggans’ soybean data.
Berry considered that for irregularly spaced crops, i.e., where the rectangularity is not constant, Eq. (29) might still be used as a first approximation from Bleasdale’s simplified equation. For example, it could be
used where plants are irregularly spaced within the row and rectangularity
is defined by the mean intrarow distance and the interrow distance.
+
IV. The Variation in Yield of the individual Plant
It was emphasized in the introduction that the variation in the yield of
the individual plant has seldom been examined in yield/density studies.
The analysis has been in terms of the mean yield per plant at a particular
density with no consideration of the variation about this mean. Yet this
variation can be of great importance wherever the size of the individual
plant is an attribute of yield. For example, in Fig. 2D the effect of size
grading on the marketable yield of parsnips can be seen at each plant
density although the latter has little effect on total yield.
Kira et al. (1 953), Hozumi et al. ( 1956), and Stern ( 1 965) attempted to
examine the effect of density and time on the variation in individual plant
weights by calculating the coefficients of variation at each density. Kira et
al. and Stern showed that the coefficients of variation increased with time,
but the evidence was not consistent as to whether density affected the
value of the coefficient of variation at any one time. However, Mead
318
R. W. WILLEY A N D S. B. HEATH
(1967) has stressed that no attempt was made in these studies to test
whether the shape of the distribution curves were constant, a necessary
condition before coefficients of variation can be compared. Koyama and
Kira ( 1956) considered the frequency distributions of pldnt weights at
different densities. They found that although -the distribution for seed
weight was normal, as the plants grew the distribution became more and
more skew. The development of skewness was greatest at high densities.
Kira et al. ( 1953) also tried using a correlation coefficient between the
weight of an individual and the mean weight of the six plants nearest to it
in a soybean experiment sown in a regular hexagonal arrangement.
Surprisingly, the correlation coefficients were low and proved positive,
apparently suggesting cooperation among plants rather than competition.
Mead ( 1967) considered that for small samples the use of this form of the
correlation coefficient is a biased estimator of the degree of competition
in the population. Mead proposed a measure of the weight relationship
between a plant and its six immediate neighbors, grown in a regular
hexagonal arrangement, termed the competition coefficient. This is a
general measure of the plant neighbor relationship for all the plants in the
community depending only on the size of the neighboring plants but
assuming regular arrangement. Mead ( 1968) analyzed the results of some
experiments laid down to examine this relationship and showed that, for
cabbage, carrots, and to a lesser degree, radishes, the competition coefficients are predominantly negative, showing competition between
plants rather than cooperation as was indicated by Kira’s results. Mead
therefore considered the competition function of more use than the
correlation coefficient.
Mead ( 1 966) investigated the importance of irregularity of spacing on
the variation of individual plant yield within a population. He did this by
examining the importance of the size and shape of the space available to
the plant in determining the yield of that plant. He suggested a model in
which the total ground area under an irregularly spaced crop was divided
into polygons, each one being allocated to a single plant. This was done by
allocating any given spot of ground to the nearest plant. The polygons
could then be characterized by three parameters, the area, the extent to
which the polygon was elliptical rather than circular (a measure of
rectangularity), and a measure of how far the plant was from the center
of the polygon. On examining the relationship between the three parameters and the root diameter of carrots grown at three densities and three
row widths, he found that the proportion of the total variation in plant
yield attributable to polygon variation increased with time, the largest
mean proportion at a final harvest being 20 percent, or for individual plots