Europ. J. Agronomy 21 (2004) 369–377
Plant spacing effect on the nitrogen concentration of a crop
Ido Seginer∗
Civil and Environmental Engineering, Technion, Haifa 32000, Israel
Received 17 January 2003; received in revised form 2 July 2003; accepted 13 October 2003
Abstract
The nitrogen concentration of plants decreases as they grow, the decrease being more pronounced for densely planted crops.
The common explanation to this is the decreasing ratio between the metabolically active biomass and the support biomass, the
former being associated with the sunlit leaves. Based on this view of the crop, two simple models, termed here ‘specific’ and
‘general’, have been developed and fitted to existing data. The ‘specific’ model fits exponential functions to data of isolated
plants and to each planting density separately. The ‘general’ model uses a single set of parameters for all planting densities. Both
fit the available data rather well, but more data are needed to test them for a range of densities. Simulations with the ‘general’
model also agree qualitatively with the Shinozaki–Akira density-dependent yield-prediction formula.
© 2003 Elsevier B.V. All rights reserved.
Keywords: Nitrogen in plants; Plant spacing; N-dilution; Yield model; Growth model
1. N-dilution curves
It is now well established that the critical nitrogen concentration of most (if not all) field crops,
decreases as the plants grow. The ‘critical’ nitrogen
concentration (on total dry matter basis), is defined
as the minimum concentration required for maximum
rate of growth (Ulrich, 1952). Experimental data have
been used to quantify the critical concentration for
common crops (Greenwood et al., 1990, for several
C3 and C4 crops; Justes et al., 1994, for winter wheat;
Sheehy et al., 1998, for rice; Colnenne et al., 1998,
for winter oilseed rape). Presumably, any concentration above critical indicates the occurrence of ‘luxury
consumption’.
∗
Tel.: +972-4-8292-874; fax: +972-4-8252-203.
E-mail address: [email protected] (I. Seginer).
Lemaire and Salette (1984) suggested an allometric relationship, which describes the critical, so called
‘dilution’ curve
C = αM −β ,
(1)
where M is dry mass of shoot per unit ground area,
C the total-nitrogen concentration in the shoot, on
dry mass basis, and α and β are constants. Eq. (1)
is dimensionally awkward, but will be retained
here nevertheless, to conform with tradition (see
Appendix A).
Greenwood et al. (1990), utilising an extensive data
set, found that for M ≥ 0.1 kg[DM]m−2 , β is approximately 0.5 and α is about a third higher for C3 plants
than for C4 plants. Below M = 0.1 kg[DM]m−2 , the
value of C is supposed to be constant. Others have
found different (usually lower) values of β, for instance 0.25 for oilseed rape (Colnenne et al., 1998),
1161-0301/$ – see front matter © 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.eja.2003.10.007
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I. Seginer / Europ. J. Agronomy 21 (2004) 369–377
10
N content, C, %N (g[N]/100g[DM])
A
B
1
0.1
1.0
10.0
100.0
Plant mass, m , g[DM]/plant
1000.0
Fig. 1. Total nitrogen concentration in isolated plants (circles) and in a dense (10 plant m−2 ) canopy (triangles) of sweet sorghum. After
Lemaire and Gastal, 1997. Break-point A (not shown in the original figure) indicates the onset of ‘self-shading’ and break-point B indicates
the onset of ‘mutual-shading’. The dashed lines are fitted to the isolated plants data and the solid line is fitted to the canopy data with
the ‘specific’ model (Eqs. (14) and (17)). The N concentration of metabolic and support compartments are Cm = 5.3%N and Cs = 0,
respectively.
around 0.33 for various grasses and 0.44 for winter
wheat (Justes et al., 1994).
The parameters α and β appear to depend not
only on species, but also on plant density (inverse of
plant spacing). For isolated plants of sweet sorghum
(Lemaire and Chartier, 1992; Lemaire and Gastal,
1997), Fig. 1 shows that β is only about 0.12, while
for a dense crop (10 plant/m2 ) it is about 0.38. In
this example, when the individual plants of the dense
crop reach a size of about m = 15 g[DM] per plant
(M = 0.15 kg[DM]m−2 ), the plants start to compete
(probably by mutual-shading), and an abrupt change
in slope occurs (break-point B in Fig. 1).
Why does the critical concentration decline as the
plants grow? The common view follows the suggestion of Caloin and Yu (1984), that a plant can be
modelled (approximated) as consisting of two compartments, one responsible for photosynthesis and
nutrients uptake (metabolically active), and the other
is a support system, mainly mechanical and vascular.
Hardwick (1987) associated the two compartments
with the ‘skin’ and ‘core’ biomass of plants, respectively. The metabolically active compartment was
postulated by Caloin and Yu (1984) to have a higher
nitrogen concentration than the support compartment,
and their analysis indicated that relative growth rate
(RGR) is proportional to the fraction of the metabolic
compartment in the plant. This implies that plants
maintain a constant nitrogen concentration while they
are in the exponential growth phase, and that the concentration declines as soon as self- and mutual-shading
(competition) starts, leading to a reduced RGR.
The concept of two nitrogen compartments was
used by Ågren and Ingestad (1987) and by Levin
et al. (1989) to produce theoretical nitrogen dilution
curves.
Values for the nitrogen concentration in the two
compartments were provided by Lemaire and Gastal
(1997) for wheat and maize. The concentration in
the metabolically active compartment, Cm , appears to
be about 5–6 times higher than in the support compartment, where the concentration is Cs ≈ 0.8%N
(0.8 g[N]/100 g[DM]). Fig. 1 indicates similar results:
Cm ≈ 6%N, and Cs ≈ 1%N.
Hirose and Werger (1987) and Chen et al. (1993)
associated the metabolically-active nitrogen compartment with the sunlit leaves, and the support compartment with the shaded leaves. As plants grow, the ratio
I. Seginer / Europ. J. Agronomy 21 (2004) 369–377
371
of sunlit- to total-leaf-area decreases. Nitrogen-rich
compounds, no longer required in the shaded leaves,
may be re-allocated to the younger leaves (Seligman,
1993). The result is a decline of nitrogen concentration with depth in the canopy (Grindlay et al.,
1995; Alt et al., 2000). This view of the plant also
explains why a densely planted crop, where for a
given size of plant the ratio of sunlit (metabolic)
to shaded (support) biomass is small, has a lower
nitrogen concentration than a sparsely planted crop
(Fig. 1). Another consequence of this view is the
decreasing root-to-shoot (Aung, 1974; Caloin and
Yu, 1984; Levin et al., 1989), leaf-to-stem (Colnenne
et al., 1998) and sunlit-to-total-leaf-area (Tei et al.,
1996) ratios with age. In particular, the decreasing
root-to-shoot ratio may be viewed as a consequence
of the contrast between the two-dimensional capture
of sunlight by the canopy and the three-dimensional
capture of water and nutrients by the roots.
While dilution curves of standard crops have been
derived on the basis of simple models (Ågren and
Ingestad, 1987; Levin et al., 1989), the effect of plant
density on the dilution curve (Fig. 1) has not yet
been modelled. This note is an attempt to develop a
simple model, which can mimic the basic features of
Fig. 1. Two simple variants, ‘specific’ and ‘general’,
are compared, and the predictions of the latter are
also compared to an often-used density-dependent
yield-prediction formula.
where n is plant density and η and ζ are constants.
The harvestable biomass, Mh , increases asymptotically with n, mainly because the canopy of a densely
planted crop closes sooner (more to the left in Fig. 1),
and captures overall more light than that of a sparsely
planted crop. The size of an individual plant, mh =
Mh /n, decreases however as n increases.
The Shinozaki–Akira formula, where each time-point
requires a different pair of η and ζ values, is not a
time-dependent model and cannot mimic the results
of Fig. 1. It could, however, be used as a check on
the dynamic model which is considered next.
2. Density dependent growth and nitrogen
concentration
mm = fi {m},
The information regarding N-concentration in field
crops, as outlined above, is mostly derived from experiments with crops at standard spacings. There is,
however, considerably more information regarding the
influence of plant spacing on the growth and yield of
crops. Based on that information, a relationship has
been developed between plant density and harvestable
yield (attributed to Shinozaki and Akira by Li et al.,
1996), namely,
n
,
η + ζn
The effect of plant density on the N-dilution curve
can be modelled by considering first isolated plants.
To a first approximation, a plant well supplied with
nutrients grows at a rate which is proportional to the
light it intercepts. Assuming (as Hirose and Werger,
1987; Chen et al., 1993) that the metabolically active
biomass is proportional to the light intercepting area,
then
dm
= kmm ,
(3)
dt
where m and mm are the dry mass of a single plant
and of its metabolically active biomass, respectively
(mm is part of m), t is time, and k is a growth coefficient, constant for a given environment (climate). For
given species (variety) and environment, mm may be
considered a function of m. In general
(4)
where the subscript i indicates isolated plants. There
are two commonly used forms of fi
2.1. The Shinozaki–Akira yield model
Mh =
2.2. Growth of isolated plants
(2)
fi {m} = pi mγi ,
0 < γi < 1
allometric
(5)
and
fi {m} = qi (1 − exp{−ai m}),
ai > 0
asymptotic
(6)
where γ i , ai (light-extinction coefficient), pi and qi are
constants. The two equations (models) have the same
number of parameters (two), but Eq. (5) turns out to
fit the data of Fig. 1 better than Eq. (6) (not shown).
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I. Seginer / Europ. J. Agronomy 21 (2004) 369–377
On the other hand, Eq. (6) (Van Keulen et al., 1982)
has an advantage over Eq. (5) when the plants are very
small, because at that stage Eq. (6) predicts exponential growth, and does not allow mm to be larger than m.
One simple formulation which retains the advantages
of both models is
mm = fi {m} = min{m, pi mγi },
(7)
which is the model of choice in this study. Initially the
whole biomass is sunlit and metabolically active, but
eventually, beyond
1/(1−γi )
m = pi
(8)
mm becomes smaller than m, indicating the onset of
‘self-shading’ (break-point A in Fig. 1).
Regarding the choice of γ i , Aikman and
Benjamin (1994) chose to follow the skin-core concept (Hardwick, 1987) and to assume self-similarity
of the growing plants. Hence, they prescribe γi = 2/3
in Eq. (5). Often, however, experimental results for
isolated plants indicate higher values, such as γi = 0.8
(Caloin and Yu, 1982) and γi = 0.85 (Lemaire and
Chartier, 1992). These larger values imply that an
isolated plant does not remain self-similar as it grows.
Rather, it seems to become more sparsely arranged
over time, which reduces self-shading.
Following Caloin and Yu (1984) and Greenwood
et al. (1991), the concentration of nitrogen in isolated
plants may be expressed as
Ci m = Cm mm + Cs (m − mm )
(9)
or
mm
,
(10)
m
where Ci is the overall nitrogen concentration of the
isolated plant. Using Eqs. (4) and (7) to substitute for
mm in Eq. (10), the result is
Ci = Cs + (Cm − Cs )
fi {m}
m
min{m, pi m␥i }
.
(11)
= Cs + (Cm − Cs )
m
For the range beyond the ‘self-shading’ break-point
(right of A in Fig. 1; m > mm ) Eq. (11) becomes
Ci = Cs + (Cm − Cs )
Ci = Cs + (Cm − Cs )pi mγi −1 ,
(12)
with four fitting parameters (Cs , Cm , pi and γ i ). A
simpler special case is obtainable if the data points lie
on a line, as in Fig. 1, which may be described by the
single-plant allometric analogue of Eq. (1), namely by
C = λm−β .
(13)
The result of equating Eqs. (12) and (13) is
Ci = Cs + (Cm − Cs )
pi mγi
= λi m−βi
m
(14)
leading to
Cs = 0,
Cm pi = λi
and
γi = 1 − β i ,
(15)
with just two fitting parameters (pi is redundant and
may be set e.g. to 1 kg␤ [DM] plant−␤ ).
The two alternative formulations, Eqs. (12)
and (14), developed above for isolated plants (before
the canopy closes), will be used in the following as
elements in the overall crop models. The essential
difference between the two alternatives is that in the
first, Cs remains free to be fitted, while in the second
it is set to zero.
2.3. Closed canopy
Normally, inter-plant competition is apparent in
agricultural fields at a relatively early stage of growth,
resulting in a reduction of the RGR and nitrogen concentration to below the levels of isolated plants (as in
Fig. 1 beyond break-point B). At higher planting densities the canopy closes sooner, shifting break-point
B to the left. The N-concentration is, therefore, a
function of the (time-dependent) size of the individual plants, as well as of the planting density, and in
general may be expressed, similarly to Eq. (11), by
Cc = Cs + (Cm − Cs )
fc {m, n}
,
m
(16)
where Cm and Cs are the same as for the isolated
plants, while fc {m, n}/m decreases with the increase
of both m and n.
Two simple alternatives to formulating fc {m,n}
come to mind. First, fitting an allometric equation to
the canopy data, as in Fig. 1, by equating Eq. (16)
with the allometric expression, results in
Cc = Cs + (Cm − Cs )
fc {m, n}
= λc,n m−βc,n ,
m
(17)
where the parameters λc,n and βc,n must be fitted
separately for each planting-density. Two of the
I. Seginer / Europ. J. Agronomy 21 (2004) 369–377
373
parameters, namely Cs (=0), and Cm (=αi /pi ), may be
taken from the fit of the isolated plants (Eq. (15)) while
3.1. Specific models for each planting density
λc,n m1−βc,n
fc {m, n} =
Cm
The ‘specific’ model, Eqs. (14) and (17), was fitted
to the data of Fig. 1, resulting in the following parameter values (for use with m in g[DM] per plant and C in
%N (g[N]/100 g[DM]): λi = 5.3, βi = 1 − γi = 0.12,
pi = 1.0, λc,10 = 10.4 and βc,10 = 1 − γc,10 = 0.38,
where the subscripts 10 indicate that these parameter
values are for a density of 10 plant m−2 . The associated nitrogen concentrations, for the metabolic and
support compartments respectively (of both isolated
and canopy plants), are Cm = 5.3%N and Cs = 0%N.
The straight lines in Fig. 1 are plotted with these parameters.
The values of the coefficients λi and λc and of the
exponents βi and βc , increase as the density increases
from isolated plants, i, to canopy, c. This suggests
that they are also likely to increase from sparse to
dense canopies. Increasing the coefficient λc shifts the
canopy-line to the left and increasing the exponent βc
increases the negative slope, resulting in non-parallel
canopy-lines for different planting densities. The shift
is intuitively the correct trend, but the available data
is insufficient to test the suggested change in slope.
(18)
where βc,n = 1 − γc,n as in Eq. (15), requires a new
fitting.
Second, a more general, yet simple form of Eq. (16)
can be created, which describes the response to all densities in a single model, if the following two assumptions are made: (1) that very shortly after the onset of
mutual-shading (break-point B in Fig. 1), the canopy
closes completely (intercepts all impinging light), and
(2) that the metabolically active biomass is proportional to the light intercepting surface. Under these
assumptions, the metabolically active biomass of a
closed canopy,
∗
Mm
= nmm ,
(19)
remains constant for the rest of the growing period
and is independent of planting density. This changes
Eq. (16) to
Cc = Cs + (Cm − Cs )
∗
Mm
nm
(20)
∗ is a constant for a given species. Plotting
where Mm
this relationship over the coordinates of Fig. 1 would
produce a family of downwards-convex curves for a
range of planting densities, n. Eq. (20) is a straight line
on a double-logarithmic plot only if Cs = 0, but then
the slope of the line is −1, incompatible with the slope
of the canopy-data of Fig. 1 (compare with the slope
of the best-fit with Eq. (17), where −βc,10 = −0.38).
Hence, a fit to Eq. (20) would require the simultaneous
adjustment of all three parameters, namely Cs (=0),
Cm and Mn∗ .
Eqs. (14) and (17), where Cs = 0 (allometric), will
be referred to as the ‘specific’ canopy model, while
Eqs. (11) and (20), where Cm > Cs > 0, will be
referred to as the ‘general’ canopy model.
3. Fitting models to the data
Numerical fitting of the available data to both models will now be attempted. The data, evaluated directly
from Fig. 1, are pairs of m and C coordinates.
3.2. General model
The simultaneous fit of Eq. (11) to the data for the
isolated plants and Eq. (20) to the canopy data (n = 10
plant m−2 ), is shown in Fig. 2. The parameter values
for this fit (for C in %N and m in g[DM] per plant) are
∗ = 100 g[DM]m−2 ,
Cm = 6.0%N, Cs = 1.2%N, Mm
βi
−βi
and βi = 1 − γi =
pi = 0.85 kg [DM] per plant
0.17. Curves for n = 2 and n = 50 plant m−2 (no
data available) are also plotted in Fig. 2, to show the
trend predicted by the general model.
4. Simulated harvestable yield
In the dynamic model, Eq. (3), the coefficient k depends on the climate and serves as a time scale for
the problem, but does not affect the composition (nitrogen to biomass ratio) of the plants, as presented
in Figs. 1 and 2. Simulating with Eq. (3), with mm
replaced by Eq. (7), and in conjunction with the fitted parameters of the ‘general’ model (previous paragraph), the dry mass of the plant, m, is incremented
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I. Seginer / Europ. J. Agronomy 21 (2004) 369–377
10
N content, C, %N (g[N]/100g[DM]
A
B
50 plant/m2
10
2
1
0.1
1.0
10.0
100.0
Plant mass, m , g[DM]/plant
1000.0
Fig. 2. Fit of the ‘general’ model (Eqs. (11) and (20)) to the data of Lemaire and Gastal, 1997. Axes, symbols and line formats are as in
Fig. 1. The N concentration of metabolic and support compartments are Cm = 6.0%N, Cs = 1.2%N, respectively. Note that the sloping
segment for the isolated plants is not a straight line (since Cs = 0).
∗ = 100 g[DM]m−2
until nmm becomes equal to Mm
∗
(Eq. (19) and fitted value of Mm ). Beyond this point,
∗ /n, and the crop
mm remains constant at mm = Mm
continues to grow linearly with time. Note that since
the value of k is arbitrary, so is the time scale. Harvest time for the density n = 10 plant/m2 was set at
the point where the simulated m reached 160 g[DM]
per plant (last couple of canopy-points in Fig. 1, but
Fig. 3. Dry matter yield and plant size at the end of a simulated growing season. Simulations were carried out for plant-densities ranging
from 10 to 160 plant m−2 . Solid lines are simulation results and dashed lines are based on the Shinozaki–Akira formula (Eq. (2)).
I. Seginer / Europ. J. Agronomy 21 (2004) 369–377
375
Fig. 4. Simulated ratio of metabolically active to total biomass as a function of time, for a range of planting densities. The break-point B
moves towards earlier time as the planting density increases. break-points A and B as in Figs. 1 and 2.
otherwise rather arbitrary). The same simulated harvest time was then used for all other planting densities. This procedure is compatible with the common
practice of relating plant development, and hence maturity and harvest time, to an integral of the environmental conditions (e.g. temperature integral, Adams
et al., 1997), rather than to the size of the plants.
Figs. 3 and 4 show the results of these ‘general’
model simulations for a wide range of plant densities (from 10 to 160 plant/m2 ). The simulation results for the harvest time are compared in
Fig. 3 with the corresponding values obtained with
the fitted yield formula of Shinozaki and Akira
(Eq. (2) with η = 1.55 plant kg−1 [DM] and ζ =
0.5 m2 kg−1 [DM]). The figure shows a fair qualitative
agreement between the simulation model and the yield
model.
Fig. 4 shows the simulated ratio mm /m as a function of time. This figure differs from Fig. 2 in certain details, but mainly in that the abscissa here is
time rather than plant size. The plot shows the two
break-points: (A) the onset of ‘self-shading’ (common to all plant-densities) and (B) the onset of
‘mutual-shading’ (moving towards earlier times with
increasing plat-density). The nitrogen concentration
(proportional to mm /m) is seen to depend, in this
model, on plant-density only in the intermediate stage
of growth. The growth rate up to A is exponential
and beyond B it is linear. The reduction in slope beyond B reflects the diminishing RGR. In reality, the
regime transitions at A and B are less abrupt than the
idealised model indicates.
5. Discussion
Two simple models, ‘specific’ and ‘general’, have
been examined as descriptors of the dependency of
nitrogen concentration on plant size and planting density. Both models follow the ideas of Caloin and Yu
(1984) and others, namely that the concentration of
nitrogen is different in the metabolically active and in
the support biomass of plants. The difference between
the models is only in the way the proportion of the two
kinds of biomass is formulated. The ‘specific’ model,
based on allometric relationships where Cs = 0, fits
a separate equation to data for each planting density,
while the ‘general’ model, where Cm > Cs > 0, attempts to fit data for all planting densities with a single
set of parameters.
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I. Seginer / Europ. J. Agronomy 21 (2004) 369–377
Both models predict the main feature of the Lemaire
and Gastal data, namely that the nitrogen concentration of plants of a given size is lower at higher plant
densities. In Figs. 1 and 2 the direct (vertical) comparison is between plants of different ages, those in
the dense arrangement being older than the isolated
plants. In contrast, Fig. 4 compares plants of the same
age and here, too, the denser canopies exhibit a lower
N-concentration during the intermediate growth stage.
The latter indication cannot, however, be tested by the
data of Lemaire and Gastal (1997).
Both approaches have their weaknesses, and a sound
choice can only be made on the basis of data for
more planting densities. With the available data, the
‘specific’ model appears to produce a slightly better
fit (Fig. 1 versus Fig. 2), but at the cost of two additional parameters for each planting density and a
less general point of view. The available data (straight
double-logarithmic lines) forces the ‘specific’ model
to assign no nitrogen to the support biomass, which
is not plausible. On the other hand, the general model
assumes that the canopy closes immediately after the
onset of mutual-shading, which is also not a credible
assumption.
A simulation utilizing the ‘general’ model formulation has been used to illustrate the relationships
between plants grown at different densities. The simulation results agree, qualitatively, with the empirical
Shinozaki–Akira relationship for the size and yield of
plants at harvest time. This agreement may be considered an indirect test of the ‘general’ model, which
cannot be performed, at this stage, with the ‘specific’
model.
Acknowledgements
This research has been supported by the Fund for
the Promotion of Research at the Technion.
Appendix A. Normalized allometric formula
Equation (1)
C = αM −β
(A.1)
is dimensionally awkward, because the units of α depend on the value of β. This can be rectified by normal-
izing the variables C and M. Convenient normalizing
factors which emerge from the ‘general’ model, are
Cm , the nitrogen concentration in the metabolic com∗ , the metabolically active biomass of
partment and Mm
a closed canopy. With these factors, the normalized
form of the allometric model becomes:
M −β
C
=α
,
(A.2)
∗
Cm
Mm
where α is dimensionless.
Appendix B. Notation
Symbols
a light extinction coefficient of canopy
(plant kg−1 [DM])
C nitrogen concentration on dry matter basis
(kg[N]kg−1 [DM])
k growth coefficient (s−1 )
M mass of dry matter per unit ground area
(kg[DM]m−2 )
m dry mass of a single plant (kg[DM] per plant)
n planting density (plant m−2 )
p coefficient in the allometric equation for mm
(Eq. (5)) (kg1−␥ [DM] plant−(1−␥) )
q coefficient in the asymptotic equation for mm
(Eq. (6)) (kg[DM] per plant)
t time (s)
α a coefficient in the canopy N-dilution formula
(Eq. (1)) (kg[N]kg−(1−␤) [DM]m−2␤ )
β an exponent in the N-dilution formula
(Eq. (1)) (–)
γ exponent in the allometric equation for mm
(Eq. (5)) (–)
ζ coefficient in the Shinozaki–Akira formula
(Eq. (2)) (m2 /kg[DM])
η coefficient in the Shinozaki–Akira formula
(Eq. (2)) (plant/kg[DM])
λ coefficient in the single-plant N-dilution formula
(Eq. (13)) (kg[N]kg−(1−␤) [DM] plant−␤ )
subscripts
c canopy
h harvest
i isolated plants
m metabolically active
s support
I. Seginer / Europ. J. Agronomy 21 (2004) 369–377
superscripts
* of closed canopy
Notes
{}
are used exclusively to enclose the arguments
of functions.
All areas (m2 ) refer to ground (field) area
(m2 [ground]).
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