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III. The Relationship between Plant Rectangularity and Crop Yield

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



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III. The Relationship between Plant Rectangularity and Crop Yield

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