Agriculture, Ecosystems and Environment 80 (2000) 71–85
On crop production and the balance of available resources
Ramun M. Kho∗
International Centre for Research in Agroforestry, P.O. Box 30677, Nairobi, Kenya
Received 11 March 1999; received in revised form 22 November 1999; accepted 23 January 2000
Abstract
One of the main insights achieved in the early days of agricultural science is that each environment has a specific balance of
resources, which is available to the crop. This balance determines crop production, the effect of resource addition and the effect
of agronomic operations. However, attempts to quantify this balance are scarce. It is normally taken into account indirectly
by a general description of soil, climate, topography, land use history, etc. This paper advocates quantifying this balance by
quantification of the degree of limitation of resources. A coefficient (between zero and one) is developed which implements
this idea. The paper shows that under a moderate assumption, the sum of the limitation coefficients of all resources equals
one. This makes the deduction possible of non-limiting resources. The original binary concept of limitation can be regarded
as a special case of this coefficient. The paper shows that crop response to addition of a resource can be viewed as the product
of: the limitation coefficient, the use efficiency, and the amount of the dose. General crop production principles as the law
of diminishing returns and the law of the optimum can be interpreted easily this way. Methods to estimate experimentally
the limitation coefficients are discussed. The methods are illustrated by estimating the degree of limitation of nitrogen and
phosphorus in southwest Niger. These two elements account for more than 70% of the total limitation (of carbon dioxide,
radiation, water, and all nutrients), which is in agreement with other scientists in this region who indicate these two elements
as the ‘principal’ limiting factors. © 2000 Elsevier Science B.V. All rights reserved.
Keywords: Agriculture; Concepts; Agroecosystem characterisation; Production ecology; Methods; Resource limitation
1. Introduction
Until the Second World War, much agronomic
research was directed towards the search for laws
governing the relation between input of resources and
crop yields (De Wit, 1994).
Almost one century ago, Blackman (1905) formulated the ‘law of limiting factors’ which states that
crop production shows only a response (in a propor∗ Present address: Agrotechnological Research Institute (ATO),
P.O. Box 17, 6700 AA Wageningen, The Netherlands. Tel.:
+31-317-475311; fax: +31-317-475347..
E-mail address: [email protected] (R.M. Kho)
tional relation) to modifications in the availability of
only one, the limiting, factor. If another factor becomes limiting, this imposes a plateau on the response
curve where a modification of the first factor does not
affect crop production any longer (Fig. 1). Half a century earlier Von Liebig (1855) had already found this
concept in slightly different terms as the ‘law of the
minimum’. The validity of the concept is confirmed by
many experiments in the sense that the response curve
shows diminishing returns and arrives at a plateau as
the availability of a resource increases (all other factors constant). Rabinowitch (1951) shows that the underlying kinetic view must be that plant growth is a
sequence of processes, whereby the process on which
0167-8809/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 7 - 8 8 0 9 ( 0 0 ) 0 0 1 3 5 - 3
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R.M. Kho / Agriculture, Ecosystems and Environment 80 (2000) 71–85
Fig. 1. Response curves according to Blackman (1905) with high
and low levels of a second resource.
the limiting factor acts determines the overall flow rate.
This slowest process creates so a ‘bottleneck’ for the
overall process. Originally, the concept is qualitative
(binary): a factor is limiting or not, and in each specific
environment there can be only one limiting factor.
However, in most environments the crop responds
to increased availability of several factors. Liebscher
(1895) formulated the ‘law of the optimum’, which
states that plants use more efficiently the production
factor which is in minimum supply, the closer other
production factors are to their optimum. In other
words, the initial slope of the response curve increases,
if the availabilities of the other limiting resources
increase (see Fig. 2). De Wit (1992) has shown that
the law of the optimum is confirmed by numerous ex-
Fig. 2. Response curves according to Liebscher (1895) with high
and low levels of a second resource.
periments in the past century. The underlying kinetic
view is that plant growth is not a sequence of processes whereby each process is determined by only
one factor. It is a sequence of processes whereby each
process is determined by two or more factors which
can also influence other processes and the plateau
(Rabinowitch, 1951). Crop production is still determined by the slowest process, but the crop does not
respond to a modification in the availability of only
one, but of several factors. In such circumstances, it
may be better to think and speak in terms of multiple
limiting factors, each with its own degree of limitation, instead of limitation as a binary variable. This is
widely recognised as appears from the use of terms as
‘major limitations’ (Sanchez, 1995) or ‘principal limiting factor’ (Shetty et al., 1995). Use of these terms
implies the existence of ‘minor limitations’ and ‘secondary limiting factors’. The original binary concept
of limiting factors has been evolved into a quantitative
concept in which the more a factor is in short supply,
the bigger its influence on crop production.
One of the main lessons learned from the early
days of agricultural science is that each environment
has a specific balance of resources that is available
to the crop. This balance determines crop production,
the effect of resource addition and the effect of agronomic operations. However, attempts to quantify this
balance are scarce. Jones and Lynn (1994) proposed a
‘relative resource limitation’ (see Section 2.2). Nijland
and Schouls (1997) discuss the concept of ’ecological
subspaces’ for interpretation of the Michaelis–Menten
growth model (see Section 2.4). The balance of
available resources is normally taken into account
indirectly by a general description of soil, climate, topography, land use history, etc. This makes it difficult
to extrapolate and to be aware of the (limited) scope
of experiments and the resulting recommendations.
After the Second World War, agricultural science
highlighted increasingly the physical, chemical and
biological processes that govern the growth of crops
(De Wit, 1994). The new paradigm studied the growth
rate (in, e.g. g dm m−2 per day) as a function of resource capture (e.g. MJ m−2 for light, mm for water,
and g m−2 for nutrients). Seasonal biomass production (g m−2 ) can than be found by integration. For
example, if the resource is light, the total biomass
accumulated over a growing season (W) can be found
by (Azam-Ali et al., 1994):
R.M. Kho / Agriculture, Ecosystems and Environment 80 (2000) 71–85
73
Z
W = εs
fS0 δt
where f is the fraction of incident radiation intercepted
by the crop canopy, S0 is the daily incident radiation
(MJ m−2 ), and εs is the conversion efficiency of solar
radiation (e.g. in g dm MJ−1 ).
If the resource is water, seasonal biomass production (W) can be expressed as (Ong et al., 1996):
X
Et
W = εw
P
where Et is the cumulative transpiration (mm H2 O)
and where ε w is the conversion efficiency of water
(g dm mm−1 H2 O transpired). The representation of
light capture by an integral and of water capture by
a sum is only a matter of convention. In both cases
the process runs in continuous time (integral) but is
usually calculated in discrete steps (sum). The conversion efficiencies are mostly considered species specific and conservative, which explains why they are
kept outside the integral. Concerning water, instead of
the conversion efficiency itself, its product with saturation vapour pressure deficit is also considered the
species specific constant (Cooper et al., 1987). Monteith (1994) describes the principles of resource capture by crop stands. The last decades, many research
efforts have been devoted to the measurement and
modelling of resource captures and to the estimation of
conversion efficiencies (see Hanks and Ritchie, 1991;
Monteith et al., 1994).
In line with De Wit (1992, 1994), this paper continues with the old paradigm of before the Second World
War. It aims to quantify the balance of available resources in the environment, make it measurable, and
explore some relationships with the old and the contemporary paradigm.
2. Quantifying the balance of available resources
2.1. Crop response, limitation, and the balance of
resources
Fig. 3 shows the response curve to availability of
one resource, given constant availabilities of other resources. Three states can be distinguished. In the first,
‘proportional’ state, the resource is the only limiting
factor and is used maximally. As soon as the resource
Fig. 3. The proportional (1), diminishing returns (2), and the
plateau (3) states of response curves.
is captured, it is used in the growth process contributing to more biomass. This results in the proportional
relation of production to availability, and in a minimum concentration of the resource in the crop. In the
second, ‘diminishing returns’ state, another factor influences the slowest process and becomes ‘limiting’
too. The plant cannot make maximum use of the first
resource and the slope of its curve decreases. When
the availability of the first resource continues to increase, its shortage relative to the availability of the
second continues to decrease. The curve of the first
shows diminishing returns (continuously decreasing
slope), and the concentration of the first resource in
the plant increases. In the third, ‘plateau’ state, the
second or a third factor limits another process that imposes a plateau on the response. The first resource has
been saturated and has reached its maximum concentration. A change in its availability does not affect its
capture nor the biomass production.
Note that the approach relates biomass production,
and not the yield of a particular plant organ, to availability. Biomass is closer related to the balance of
resources in the environment than a harvested plant
organ (e.g. grain). If resources are very out of balance, increased availability may reduce harvest index
and thus yield (e.g. by lodging). This possible fourth
state, showing decreasing production with increasing
availability, greatly complicates quantification of the
balance of resources with yield response data. This
problem is avoided by relating biomass production to
resource availability. Total (above-ground) biomass
production will have an asymptote, whereas yield of
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R.M. Kho / Agriculture, Ecosystems and Environment 80 (2000) 71–85
a plant organ may not. If harvest indices are constant,
yields of plant organs can be used too.
The balance of available resources in the environment influences the shape of the curve. In an idealised
environment where the resources influencing the process which imposes the plateau do not take part in
the slowest process, a Blackman-type of curve (only
states 1 and 3) will be the result (Rabinowitch, 1951).
In environments where resources are more in balance,
more factors are affecting simultaneously the slowest
process as well as the process imposing the plateau.
State 2 will occupy a significant part of the response
curve in these circumstances.
The time and spatial scale influence the shape of
the curve too. On a scale of hours or days and/or on a
scale of a single plant, one process may be the slowest process determining the overall growth rate. This
may lead to a Blackman-type of curve. On a time
scale of a season, the probability increases that different processes succeed one another as slowest process (because of addition and depletion of resources).
On a spatial scale of a crop, the probability increases
that different processes are simultaneously the slowest
process (because of increased spatial heterogeneity).
The overall response curve is then composed of several response curves, each with own initial slopes and
plateau’s. Abrupt transitions will be ‘averaged out’,
and the result will be a smooth curve with a diminishing returns state (see Fig. 4). So, the longer the time
scale and the larger the spatial scale, the smaller the
probability of a Blackman-type of curve. This implies
that the probability increases that in one specific en-
vironment several resources are limiting, each in their
own degree.
This discussion may make it clear that crop response
to availability of one resource depends on its degree
of limitation. This last can be viewed as a measure of
the shortage of a resource, relative to the availabilities
of other resources. If the degrees of limitation of all
resources can be quantified, the balance of available
resources can be quantified.
2.2. Defining limitation
A limiting resource is a resource of which a small
change in its availability affects biomass production
(i.e. resources in states 1 and 2 of the response curve).
The degree of limitation of a resource is related to
the slope (dW/dAi ) of the response curve; dW is the
change in production (W) responding to a small change
(dAi ) in availability of any resource i (other factors
equal). On the plateau, the slope equals zero and the
resource is non-limiting. Intuitively, it could be said
that the steeper the slope, the greater the response, and
the more the resource must be limiting. However, by
defining (the degree of) limitation as the slope of the
response curve, the limitation of different resources
with each other (e.g. the limitation of radiation versus
the limitation of water) cannot be compared. The slope
of the response curve depends on the arbitrary units
used for W and Ai . Jones and Lynn (1994) proposed
to normalise the slope to obtain a relative resource
limitation `i , defined by
`i =
dW/W
dAi /Ai
(1)
`i is dimensionless and independent of the units used
for W and Ai .
Crop production is a function of several resources
(carbon dioxide, radiation, water, nitrogen, phosphorus and other nutrients). Therefore, it is more appropriate to define limitation Li of any resource Ai with
partial derivatives:
Li =
∂W/W
∂Ai /Ai
Rearranging Eq. (2) gives
Fig. 4. The sum of several Blackman-type curves, each with own
slopes and plateau’s, will constitute a smooth curve.
Li =
∂W/∂Ai
W/Ai
(2)
R.M. Kho / Agriculture, Ecosystems and Environment 80 (2000) 71–85
Fig. 5. (A) Elements to determine the limitation coefficient L at
a specific point on the response curve (see text), and (B) the
resulting limitation coefficient as function of availability.
Accordingly, limitation Li at any point (Fig. 5A) is
equal to the ratio of the slope of the response curve
(other factors are taken constant) to the slope of the
straight line from origin to the response curve. The last
is defined to be the use efficiency or the productivity of
the resource (average production per unit available resource). If a resource is non-limiting (on the plateau),
the slope equals zero, so that the minimum value of Li
equals zero. If a resource is the only limiting resource,
production is proportional to availability. In this case,
the slope equals the use efficiency of the resource, so
that the maximum value of Li equals one. Between
the proportional state and the plateau, the slope decreases gradually to zero. The use efficiency will also
decrease, but will never reach the value zero. In the
diminishing returns state, the value of Li is thus between one and zero (Fig. 5B). Note that no assumption has been made about the mathematical form of
the response curve, except that it is smooth enough
to be differentiated with monotonic first derivative. A
Blackman-type of curve consists of only a proportional
state (the resource is limiting; Li =1) and a plateau (the
resource is non-limiting; Li =0). Therefore, regarding
one resource, the original binary concept of limitation
is a special case of coefficient Li (except in the point
of break which is not differentiable).
75
Fig. 6. (A) Crop response to availability A1 at high and low levels
of a second resource; and (B) the accompanying relations between
the limitation coefficient and availability A1.
2.3. The limitation of all resources
Li is only a good measure for the degree of limitation, if the ‘total limitation’ which is the sum of
the limitation coefficients of all (limiting) resources, is
constant. It can be argued that if one resource becomes
more limiting, other resources become relatively less
limiting (see Fig. 6). When there is only one limiting
resource, its limitation Li equals one and the limitation of all other resources equal zero (cf. Von Liebig’s
model of plant growth; or the binary concept of limitation of Blackman). In this case, total limitation equals
one. This suggests that (in order to be a generalisation of Blackman’s
concept) the sum of the limitation
P
coefficients, Li , should also be one when there are
several limiting resources. This section will demonstrate that this is indeed the case if the assumption of
constant returns to scale is made.
A crop transforms physical resources (inputs) to
biomass (output):
W = f (A1 , A2 , ......, An )
(3)
where the seasonal biomass production W (e.g. kg/ha)
is a function f of Ai (i=1,2,. . . ,n) which are the
availabilities of the resources radiation (e.g. MJ/m2 ),
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R.M. Kho / Agriculture, Ecosystems and Environment 80 (2000) 71–85
water (e.g. mm), nitrogen (e.g. kg/ha), phosphorus
(e.g. kg/ha), etc., n is the number of all resources.
It can be reasoned that if the availabilities of all resources (inputs) are multiplied with the same factor k,
the balance of resources and thus the degrees of limitation of the resources are not changed. Because concentration differences between the environment and
the plant are multiplied with this factor, resource captures will also change with this factor. Because their
mutual proportions do not change, the proportions of
the concentrations in the plant do not change. The rates
of all growth processes are probably multiplied with
this factor. On a field and seasonal scale, the biomass
production of the particular crop (the output) should
then also be multiplied with this factor k:
kW = f (kA1 , kA2 , ......, kAn )
(4)
De Wit (1992) shows data supporting this constant
returns to scale assumption (the proportional relation
of output to inputs). In the period 1945–1982, the use
of nitrogen fertiliser increased steadily in the USA.
Instead of showing diminishing returns, maize yields
increased proportionally with nitrogen use. This must
be explained by the technological change in this period: mechanisation, soil amelioration, better water
management, use of other inorganic fertilisers, use
of herbicides (decreased competition by weeds), etc.
This all resulted in increased availability of resources
to the crop, and led to a proportional increase of
biomass. New short-straw cultivars adapted to the
improved growing conditions prevented lodging, so
that the higher biomass production could also be converted into higher yields. Use of pesticides protected
the attained production against pests and diseases
better. Similar proportional relations were found for
rice yields versus nitrogen fertiliser in Indonesia from
1968 to 1988, and for nitrogen output in milk and
meat versus nitrogen input in highly intensive pastoral farming systems in the Netherlands from 1965
to 1985 (De Wit, 1992).
Note that radiation (solar irradiance) could not have
been increased in the here mentioned examples. The
proportional increase of output with inputs is then only
possible, if incident radiation was non-limiting. This
has been indeed confirmed by Monteith (1981). In
intensive systems, when supply of water and nutrients
is ample, incident radiation will become a limiting
resource. Eventually it determines the maximum
possible potential production.
According to the constant returns to scale assumption, the relative change of the output (biomass production) and the relative change of the inputs (resources)
are all equal (k−1):
dW
dAi
=
W
Ai
i = 1, 2, ..., n
(5)
Rearranging Eq. (5) gives for each resource i
dAi =
dW
Ai
W
(6)
According to the chain rule the derivative of W (Eq.
(3)) to x, when all resource availabilities Ai are some
function of x (the power changing all resource availabilities with the same factor), is
n
X ∂W
dAi
dW
×
=
dx
∂Ai
dx
(7)
i=1
Substituting Eq. (6) into Eq. (7) yields:
n
X ∂W
Ai
dW
dW
=
×
×
dx
∂Ai
W
dx
i=1
And dividing both sides by dW/dx gives
1=
n
X
∂W
i=1
∂Ai
×
Ai
W
Rearranging Eq. (2) and substitution yields
n
X
Li = 1
(8)
i=1
which shows that the sum of the limitations (as defined
by Eq. (2)) of all resources equals one, for any function
having constant returns to scale. If returns to scale
are approximately constant and the relative change of
output is q (≈1) times the relative change of all inputs
(dW/W = qdAi /Ai ), it is easily seen that the sum of
the limitations is constant and equals q.
This result seems to be in contrast with that of Jones
and Lynn (1994). Arguing that it is possible for growth
rate to be proportional simultaneously to changes in
several resources (so that each has a relative limitation
`i of one), they concluded that the sum of the relative limitations exceeds one. However, Jones and Lynn
R.M. Kho / Agriculture, Ecosystems and Environment 80 (2000) 71–85
(1994) defined the relative limitation with derivatives
(Eq. (1)). The sum of the relative limitations will indeed exceed one when several resources change simultaneously. If limitation is defined with partial derivatives (as in Eq. (2)) simultaneous proportionality of
several resources indicates constant returns to scale
and the sum of limitations will add to one.
Hence, the coefficients Li measure the degree of
limitation as fraction of the total limitation. Regarding
not only one resource, but regarding also all resources,
the original binary concept of Blackman (1905) limitation is a special case of coefficient Li .
The practical consequence of Eq. (8) is that if the
limitations of some resources are known and sum to
one, the inference can be made that all other resources
are non-limiting. The balance of available resources
has then been quantified completely.
2.4. Limitation and Michaelis–Menten ecological
subspaces
The Michaelis–Menten model gives a relation between crop production and resource availability. Nijland and Schouls (1997) have shown that this model
can be considered as one (of many) mathematical representation(s) of the theory of Liebscher. They have
re-analysed several published data and have shown
that the Michaelis–Menten model fits the data well.
Besides, Nijland and Schouls (1997) have shown that
the model has an elegant agronomic interpretation. For
one resource the model is
1
1
1
=
+
W
Wmax
αA
where
W is the dry matter production (kg dm/ha). The reciprocal (1/W) is the area (ha) needed for the production
of 1 kg dry matter,
Wmax is the maximum possible production (kg dm/ha)
when the resource is not limiting. The reciprocal
(1/Wmax ) is the minimum area that is needed for the
production of 1 kg dry matter,
α is a coefficient of response of production to availability A of the resource (e.g. kg dm/kg resource) and
A is the availability of the resource (e.g. kg/ha). The
reciprocal of αA (1/αA) is the area for deficiency of
the resource. It is the extra area (above the minimum
area) needed for the acquirement of the resource, in
77
order to produce 1 kg dry matter. If the resource is not
limiting, this area will approach zero.
The model can be easily generalised for more than
one resource. For two resources it is:
1
1
1
1
=
+
+
W
Wmax
αA1
βA2
(9)
1/αA1 is the area for deficiency of the first resource
and 1/βA2 is the area for deficiency of the second
resource. Eq. (9) can thus be read as
Area for production = minimum area
+area for deficiency of Resource 1
+area for deficiency of Resource 2
Note that the minimum area can be interpreted as the
sum of the areas of deficiency of all other resources
not explicitly taken in the model. This suggests that
the degree of limitation can be expressed with the area
for deficiency relative to the total area needed:
Limitation =
Area for deficiency of the resource
Total area needed for the production
of one unit dry matter
So, for the first resource it is
Limitation1 =
W
1/αA1
=
1/W
αA1
By rearranging Eq. (9) the Michealis–Menten model
is equal to:
W =
Wmax αA1 βA2
αA1 βA2 + Wmax αA1
The partial first derivative of this equation to A1 , the
availability of the first resource equals:
∂W
∂A1
(Wmax α βA2 )(αA1 βA2 +Wmax βA2 +Wmax αA1 )
−(Wmax αA1 βA2 )(α βA2 + Wmax α)
=
(αA1 βA2 + Wmax βA2 + Wmax αA1 )2
Wmax βA2
W
=
αA1 βA2 + Wmax βA2 + Wmax αA1 A1
W W
=
αA1 A1
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R.M. Kho / Agriculture, Ecosystems and Environment 80 (2000) 71–85
which shows that
Limitation1 =
W
∂W A1
= L1
=
αA1
∂A1 W
(10)
Hence, the area of deficiency of a resource relative
to the total area needed is nothing else than a special
case (if the Michaelis–Menten model applies) of the
limitation as defined in Eq. (2).
3. The production laws and the concept of
resource capture in retrospection
3.1. Limitation and the production laws
Since the discovery of systematic experimental design and the analysis of variance by R.A. Fisher in the
twenties, the ‘effect’ of resource addition in agronomic
experiments all over the world is tested for statistical
significance. How can this ‘effect’ be interpreted with
the limitation coefficients?
By rearranging Eq. (2), the ‘effect’ or the crop response (∂W) to a small addition of a resource can be
regarded as
∂W = Li
W
∂Ai
Ai
(11)
That is as the product of the limitation (Li ), the use
efficiency (W/Ai ) and the amount of the dose (∂Ai ).
The influence of these three components was already
known of course, but has now been made explicit in a
simple equation enabling quantification. Eq. (11) expresses a generalisation of the ‘law of limiting factors’
stating that the more a resource is limiting, the greater
its effect.
As discussed in Section 2.1, when the availability
of a resource increases (availabilities of all other resources constant) its shortage relative to the availabilities of other resources decreases. Consequently, its
limitation (Li ) decreases (see also Fig. 5B). This implies that, according to Eq. (11), the next dose (∂Ai )
will result in a decreasing response (∂W) and thus in a
decreasing use efficiency. The third and all following
doses will continuously have a lower limitation (Li )
and lower use efficiency (W/Ai ), and thus a lower response. This reflects the ‘law of diminishing returns’.
Addition of other limiting resources will decrease
their limitation and will thus, according to Eq. (8), in-
crease Li (see also Fig. 6B). Also, addition of other
limiting resources will increase production and thus
also the average production per unit available resource,
i.e. the use efficiency (W/Ai ). Addition of other limiting resources increases thus the ‘effect’ or the crop
response (∂W) to a certain dose (∂Ai ). This reflects
the ‘law of the optimum’.
If all resources are increased with an equal factor,
the use efficiencies of all resources will not change
(De Wit, 1992, 1994). Because the balance of available
resources does not change, limitations will not change.
Eq. (11) shows that in this case the crop response (∂W)
to a certain dose (∂Ai ) will not change. The decrease
in response because of the law of diminishing returns
has been compensated by the increase arising from the
law of the optimum.
3.2. The balance of available resources and resource
capture
Fig. 7 shows (on a time scale of one season) three
relations: the relation between (Quadrant I) resource
availability and capture; that between (Quadrant II)
capture and biomass production; and (Quadrant III)
that between resource availability and biomass production (cf. Van Keulen, 1982). Quadrant III is the
mirror image of the classical response curve (Figs. 2
and 3).
Fig. 7. Relations between (I) availability and capture; (II) capture and biomass production; and (III) availability and biomass
production (see text).
R.M. Kho / Agriculture, Ecosystems and Environment 80 (2000) 71–85
The relation in Quadrant I can be described by the
function C=f1 (A) (availabilities of other resources
constant), where C is the capture of the resource
(MJ m−2 for light, mm for water, and g m−2 for nutrients), and where A is its availability (same units).
Each point on the curve is associated with a certain
‘efficiency of resource capture’ ε cap : the amount of
resource captured per unit of available resource. This
efficiency depends on the crop’s demand and on the
crop’s ability to acquire the resource in the course of
the season (leaf area index, root density and rooted
depth). These depend on the attained biomass production and allocation to plant organs, which in turn
depend on the balance of available resources. For example, if nitrogen is limiting, vegetative growth will
be restricted and so the capture of light.
The relation in Quadrant II can be described by
the function W=f2 (C) (captures of other resources not
constant), where W is the biomass production (g m−2 ).
Each point on the curve is associated with a certain
‘conversion efficiency’ ε conv : dry matter production
per unit of captured resource. This depends on the capture of other resources. For example, if nitrogen has
its minimum concentration in the plant, additional uptake of phosphorus will not result in more production,
but in a higher phosphorus concentration.
Note that resource captures are confounded with
each other. Because capture is both a cause and a
consequence of growth that are difficult to separate,
capture can not be seen as an ‘independent’ variable
determining growth. The confounding may explain
why the relation between production and capture
often appears to be linear. Fig. 8 shows hypothetical relations (dashed lines) between production and
capture of one resource (e.g. light) at different fixed
captures of a second resource (e.g. nitrogen). The
empirically found relation is the solid line, which is
a correlation, not a causal relation. It is not possible to say which resource(s) increased production.
Increased capture of one resource will only lead
to a proportionally increased production if captures
of the other resources can increase proportionally.
This last is only possible if they are not limiting
which is determined by the balance of available resources. The use of an empirically found line between
production and capture for prediction in environments with another balance of resources is therefore
hazardous.
79
Fig. 8. A linear relation between production and the capture of
one resource (C1 ) may be found thanks to confounding with the
capture of other resources (e.g. C2 ).
The relation in Quadrant III can be described by
the function W=f3 (A) (availabilities of other resources
constant). Each point on the curve is associated with
a certain ‘use efficiency’ εuse : dry matter production per unit available resource. Note that resource
availabilities can be changed independently in a randomised experiment, whereas resource captures can
not. In contrast with the relation in quadrant II, it is
thus possible to find empirically causal relations for
quadrants I and III.
Table 1 shows relationships between the curves in
the three quadrants. It can be seen that (according to
the chain rule) the product of the slopes of the first
two curves is equal to the slope of the third curve.
Concerning this last, the law of diminishing returns
shows it decreases with increasing availability. Thus,
the first curve and/or the second show diminishing returns too. From the law of the optimum, the slope of
the third curve increases with addition of other limiting resources. Thus, the slope of the first curve and/or
the second will increase too. As the slope of a curve
changes, the efficiency will change in the same
Table 1
Relationships between properties of the curves in the three quadrants (see text)
Function
Slope
Efficiency
Limitation
I
II
III
C=f1 (A)
f10 =dC/dA
ε cap =f1 (A)/A
Lcap =f10 /ε cap
W=f2 (C)
f20 =dW/dC
ε conv =f2 (C)/C
Lconv =f20 /ε conv
W=f3 (A)=f2 (f1 (A))
f30 =dW/dA=f20 f10
ε use =f3 (A)/A=ε conv ε cap
Luse =f30 /ε use =Lconv Lcap
80
R.M. Kho / Agriculture, Ecosystems and Environment 80 (2000) 71–85
direction. So, on theoretical grounds it can be expected that the efficiency of resource capture εcap
and/or the conversion efficiency εconv will decrease
with increasing availability of the resource, and will
increase with addition of other limiting resources.
This has also been found empirically. Azam-Ali et al.
(1994, Table 8.2) show reported radiation conversion
efficiencies of three C4 crops (maize, sorghum and
millet) and nine C3 crops (wheat, rice, barley, potato,
cassava, sweet potato, soyabean, groundnut and sugar
beet) when water and nutrients are ample and when
there is a shortage of one or both of them. In the first
case (i.e. if radiation is the only limiting resource), the
conversion efficiencies were significantly (p<0.001)
larger (on average more than 2.1 times) than in the
second case (i.e. if other resources are limiting).
Analysis of the data of Azam-Ali et al. (1994) by
means of variance components (e.g. Longford, 1993)
shows that the variance component between species
is 0.0659. That between environments is more than
seven times larger (0.4886). The residual variance
is 0.1233. In other words, 72% of the total variance
of the conversion efficiency in this data set can be
attributed to the environment, whereas only 10% to
species. This suggests that conversion efficiencies are
more determined by (the balance of resources in) the
environment, than by species. Efficiencies are most
likely only conservative within the set of environments with the same balance of available resources.
In analogy with the limitation of quadrant III (Luse
which is defined in Section 2.2), ‘limitations’ for the
first two quadrants can be defined (Lcap and Lconv ).
The sum over all resources of each will most likely exceed one. Generalisations of the function in quadrant
I (like C1 =f1 (A1 ,...,An )) do not have meaningful partial derivatives. That in quadrant II (W=f2 (C1 ,...,Cn ))
does not have real existing partial derivatives (because
of the confounding). Therefore, Lcap and Lconv lack
important properties that Luse has.
4. Methods to estimate limitations
4.1. Approximating limitations from published
experiments
Resource use efficiencies are sometimes reported
in published experiments. An approximation of the
limitations from those publications can be found by
L≈
1W/W
1W
=
1A/A
(W/A)1A
(12)
where 1W is the change in production, responding
to the change 1A in the availability of the resource,
and where W/A is the use efficiency of the resource.
This approach uses the average slope of the response
curve instead of the slope in the control environment
(see Section 2.2). In addition experiments (fertilisation and irrigation) this average slope is because of
the law of diminishing returns lower than, and therefore an under-estimation of, the slope in the control environment. The approach may thus lead to an
under-estimation (of the limitation of nutrients and
water) in the control environment. In case of a shade
cloth experiment the reverse (over-estimation) may be
the case. In general, three conditions can be formulated for the validity of this approach: (1) the dose 1A
must have been small (leading to a small bias); (2) the
use efficiency must have been determined with respect
to the total availability of the resource; and (3) the
use efficiency must have been determined with total
biomass production, or (if yields were used) harvest
indices in the experiment must have been constant.
4.2. Estimation from response data
The limitations can be estimated from experiments
in which the availabilities of resources have been varied systematically (e.g. in a 3k factorial design; see
also Cochran and Cox, 1957, Chapter 8A). The availability of nutrients can be varied by addition of fertiliser, that of water by irrigation, and that of radiation
by the use of shade cloths.
The biomass production (W) can then be fitted as
function of the resource availabilities (Ai ). Table 2
shows some empirical response functions and the appropriate limitation. Wmax and the Greek letters are
environment specific parameters that must be determined empirically. The resource availability is the sum
of two availabilities: that in the control environment
at zero application (Ai,0 ) and the application (Ai,appl ),
i.e. Ai =Ai,0 +Ai,appl . (In a shade cloth experiment the
‘application’ has a negative value.)
The availabilities at zero application are often not
known and can also be regarded as parameters. A disadvantage of this approach is that a curved line/surface
R.M. Kho / Agriculture, Ecosystems and Environment 80 (2000) 71–85
81
Table 2
Some empirical response functions and the appropriate limitation
Model
Limitation
MitscherlichQ(exponential)
W = Wmax ni=1 (1 − e−αi Ai )
Li = αi Ai /(eαi Ai − 1)
Michaelis–Menten (hyperbolic)
P
(1/W ) = (1/Wmax ) + ni=1 (1/αi Ai )
Li = W/αi Ai
Polynomial (quadratic)
P
P
W = α0 + ni=1 (αi Ai + βi A2i + nj>i γij Ai Aj )
Li = (αi Ai + 2βi A2i +
is extrapolated (see Fig. 9). This may result in unstable estimations of Ai,0 with large standard errors. A
better approach may be one in which additional information is used for the estimation of availability at
zero application.
4.3. Estimation from response data with additional
information
Resource captures (intercepted radiation, transpired
water and nutrients taken up) are closer related to the
availability of the resource than is biomass. Especially
concerning nutrients, captures are linearly related to
availability over a larger range than is biomass, because the diminishing returns are compensated by increasing concentrations.
Dean (1954) related nutrient uptake (capture) to application of the nutrient. He estimated the availability to the control crop (the parameter Ai,0 ) by linear
extrapolation of this relation until intersection with
the horizontal at zero uptake. He called this the ‘a’
value. Dean (1954) showed that the ‘a’ value was much
smaller using the readily soluble superphosphate, than
Pn
j =1 γij Aj )/W
using the poorly soluble fused tricalcium phosphate.
Apparently, the ‘a’ value measures availability of the
nutrient in the soil in a form that is as available as the
nutrient in the used fertiliser. Therefore, it should not
be viewed as an absolute, real existing quantity, but
as a concept: a measurement of availability relative to
the standard of measurement (measured here as differences in application: ∂A). Because the interest is not in
absolute values of availability, but in relative changes
in availability (∂A/A; see Eq. (2)), the method of Dean
(1954) is appropriate for the present purpose.
The method can be generalised easily for other resources (radiation and water). Intercepted radiation
can be related to different levels of incident radiation
(using shade cloths), and transpired water can be related to different levels of applied water (by irrigation). By linear extrapolation until intersection with
the horizontal at zero capture, the ‘a’ values (Ai,0 ) for
radiation, water and nutrients can be found.
After the estimation of the ‘a’ values, the approach
in Section 4.2 can be followed, where the Ai,0 are now
taken as known by replacing them with the estimated
‘a’ values.
5. The degree of limitation of nitrogen and
phosphorus in sandy millet fields in Niger
This section illustrates the experimental quantification of the limitation coefficients as developed in this
paper.
5.1. Material and methods
Fig. 9. Response curve to resource application. Parameter A0 has
to be found by extrapolation of the curve (dashed line).
In the 1996 season, pearl millet (Pennisetum glaucum (L.) R.Br.) was grown on farmers fields near
N’Dounga, south–west Niger (13◦ 230 N and 2◦ 160 E).
82
R.M. Kho / Agriculture, Ecosystems and Environment 80 (2000) 71–85
Table 3
Soil properties at N’Dounga for three depths at the onset of the
experiment
Depth (m)
Org. C (%)
Total N (ppm)
Total P (ppm)
Bray1 P (ppm)
pH-H2 O 1:2.5
pH-KCl 1:2.5
H+ (meq/100 g soil)
Al3+ (meq/100 g soil)
Na+ (meq/100 g soil)
K+ (meq/100 g soil)
Ca2+ (meq/100 g soil)
Mg2+ (meq/100 g soil)
Sand (%)
Silt (%)
Clay (%)
0–0.15
0.15–0.40
0.40–0.90
0.259
164
310
4.3
6.0
4.8
0.044
0.02
0.037
0.146
1.37
0.67
88.9
3.8
7.3
0.173
131
314
1.7
5.6
4.3
0.100
0.19
0.040
0.095
1.43
0.91
84.2
3.5
12.3
0.138
118
294
1.3
5.8
4.4
0.071
0.09
0.046
0.053
2.17
1.03
82.5
3.7
13.8
Soils consist of (loamy) sand, are moderately acidic,
and have low to very low fertility (Table 3). Payne et al.
(1991) gives hydrological characteristics of a nearby
similar soil. The surface is flat with an average slope of
less than 1%. The climate is characterised by one rainy
season from May/June until September/October. Total annual rainfall in 1996 was 428 mm, slightly lower
than the long-term average (Sivakumar et al., 1993).
The years before the experiment, the soils were cultivated by intercrops millet/cowpea. The local cultivar
of millet, being the staple crop, was grown with and
without nitrogen (urea), and with and without phosphorus (Single Super Phosphate) fertiliser. The experimental design was a 22 factorial with addition of one
centre point (Table 4; see also Fig. 10). Each treatment
was replicated five times. Plots were 10 m×10 m gross
and 7 m×7 m without borders. Nitrogen was broadcast, half of the dose shortly before sowing and the
Table 4
Treatments of the experiment
Fig. 10. Biomass response surface (Eq. (13)) to nitrogen and
phosphorus availability in south–west Niger (all units are in kg/ha).
The capital letters A–E denote the place of the treatments used to
fit the surface (see text).
other half in the fifth week after sowing. Phosphorus
was broadcast together with the first portion of nitrogen. According to farmer practice, the millet was
sown in hills with a density of 10 000 hills/ha. Three
weeks after sowing, all plots were weeded and all hills
were thinned, leaving the three or four best established
plants in each hill. Three days later, when the crop was
recovered from the thinning, the height of the highest leaf tip (when all leaves were held vertically) of
each individual hill was measured. The residuals of
the height after fitting of the full model seemed to be
associated with plots with patches with a hard crust.
These residuals were used as covariable in the analysis (Buerkert et al., 1995). The second weeding was
done in the eighth week after sowing.
At harvest, samples of leaves, tillers, rachis, and
grains were taken in each plot of which nitrogen
and phosphorus contents were determined. Leaves,
tillers, rachis, and grains were harvested separately,
oven-dried and weighed.
5.2. Results
Treatment
N application (kg/ha)
P application (kg/ha)
A
B
C
D
E
0
180
0
180
90
0
0
60
60
30
Fig. 11 shows the relation of nitrogen (N) uptake
to nitrogen application (appl). Regression led to the
equation
(N uptake) = 26.6 + 0.285(N appl),
S.E.
4.4
0.036,
R 2 = 0.75
R.M. Kho / Agriculture, Ecosystems and Environment 80 (2000) 71–85
83
P uptake = 5.62+0.264(P appl)−0.0034(P appl)2 ,
S.E.
0.70
0.065
0.0010,
R 2 = 0.63
where the phosphorus uptake and application are in kg
P/ha. Extrapolation of the slope of the regression curve
at zero application until intersection with the horizontal at zero uptake estimates the phosphorus availability
at zero application as 5.62/0.264=21.3 kg P/ha. The
standard error is approximated as 6.6 kg P/ha.
Fitting of Eq. (9) with N and P availability at zero
application taken as 93 and 21.3, respectively gave
Fig. 11. Relation between nitrogen uptake and application, and
estimation of the nitrogen availability at zero application (all units
are in kg/ha).
where the nitrogen uptake and application are in kg
N/ha. Extrapolation of the regression line until intersection with the horizontal at zero uptake estimates
the nitrogen availability (‘a’ value of Dean, 1954,) at
zero application as 26.6/0.285=93 kg N/ha. The standard error is approximated as 25 kg N/ha.
Fig. 12 shows the relation of phosphorus (P) uptake to phosphorus application. Regression led to the
equation:
1
1
= 0.000089 + 0.0109
W
93 + Nappl
1
+0.00214
,
21.3 + Pappl
S.E.’s are 0.000021, 0.0022 and 0.00046
respectively
(13)
Fig. 10 gives the surface (of biomass W) described by
this Eq. (13). The application of Eq. (10) estimates
the limitation of nitrogen in the control environment
(zero application) as 0.38 (approximated standard error 0.10), and the limitation of phosphorus as 0.33 (approximated standard error 0.098). These two elements
account thus for (0.38+0.33)100%=71% of the total
limitation (of carbon dioxide, radiation, water, and all
nutrients). The result is in agreement with Penning de
Vries and Djitéye (1982) who also found that these
two nutrients are the major limiting factors in the Sahel (see also Van Keulen and Breman, 1990; Shetty
et al., 1995).
Table 5 gives for each treatment the limitations and
the average efficiencies (ε cap , εconv and εuse ; see Section 3.2) of nitrogen and phosphorus. The table shows
that the efficiencies vary greatly and that they are
strongly positively correlated with the degree of limitation of the resource.
6. Discussion and conclusions
Fig. 12. Relation between phosphorus uptake and application, and
estimation of the phosphorus availability at zero application (all
units are in kg/ha).
A coefficient (Eq. (2)) has been derived which quantifies the degree of limitation of growth resources in
a specific environment. It generalises the binary concept of Blackman (1905), making it applicable for circumstances that are more realistic, by fractionating his
total limitation for one resource as a sum of degrees
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R.M. Kho / Agriculture, Ecosystems and Environment 80 (2000) 71–85
Table 5
Limitations and average efficiencies per treatment
Treatment
A
B
C
D
E
Nitrogen
Phosphorus
LN
εcap
ε conv (kg/kg)
ε use (kg/kg)
LP
εcap
ε conv (kg/kg)
ε use (kg/kg)
0.38
0.17
0.50
0.26
0.31
0.27
0.25
0.27
0.32
0.31
137
72
194
72
97
36
17
49
22
29
0.33
0.44
0.11
0.17
0.22
0.22
0.30
0.10
0.12
0.20
758
764
600
587
519
159
220
56
75
103
of limitations of several resources. The coefficient enables quantification of the insight that an effect of a
resource increases with its (degree of) limitation. It
co-operates well with general accepted crop production principles as the law of diminishing returns and
the law of the optimum.
With these limitation coefficients, the balance of
available resources in any agro-ecosystem can be
quantified. Hence, agro-ecosystems can be characterised with a few parameters and more accurately
than with qualifications as ‘major limitations’, ‘principal limiting factors’ or ‘(highly) deficient’.
The validity of the constant returns to scale assumption (Eq. (4)) is plausible, but should be investigated
further. However, even if returns to scale are not exactly constant, the sum of the limitations will be a
constant close to one.
The balance of available resources and thus the
limitation coefficients are mainly determined by soil
and climate. However, the limitation coefficients may
vary with crop, time and management. Firstly, the
balance of available resources in one specific environment may not be equal for all crops. For example,
water and/or nutrients may be less limiting for a
deep-rooted crop, than for a shallow rooted crop.
Therefore, the limitation coefficients for plants with
different phenology and morphology (e.g. trees and
annual crops) may be different on the same site.
The nitrogen limitation (LN ) for leguminous crops
is likely to be lower than that for non-leguminous
crops. The radiation limitation (LR ) for C4 crops is
likely to be higher than that for C3 crops. Secondly,
the limitation coefficients of one site may also vary
with time, because of mineralisation/depletion, variation in rainfall, and variation in cloudiness. Thirdly,
if above- and below-ground interspecies competition
starts at different times and densities, it is possible to
change the balance between above- and below-ground
limitations with the stand density. Other farm operations as (time of) weeding and method of tillage
may favour the availability of one resource above that
of others, and may influence the limitation coefficients too. Determining the limitation coefficients of
agro-ecosystems and exploring their variation (with
crops, time and management) are subjects for further
research.
Resource captures are confounded with each other
and cannot be changed independently in a randomised
experiment. This implies that an empirically found relation between production and capture of a resource
is a correlation, not a causal relation. Conversion efficiencies and/or efficiencies of resource capture are,
like use efficiencies, most likely only conservative
within the set of environments with the same balance
of available resources. Efficiencies are strongly positively correlated with the degree of limitation of the
resource. The use of efficiencies for the prediction of
crop production in environments with another balance
of available resources is thus hazardous.
Quantification of the limitation of growth resources
will probably facilitate extrapolation of research results because it takes the balance of resources in each
environment explicitly into account. This will be especially the case for inherent multidisciplinary sciences
as ecology, weed science, intercropping, and agroforestry (Kho, 2000). These ‘holistic’ sciences, studying interactions between plants of different species
in different environments, have to deal with a complex of all resources, their mutual proportions and
interrelations.
R.M. Kho / Agriculture, Ecosystems and Environment 80 (2000) 71–85
Acknowledgements
The Directorate General International Co-operation
(DGIS) of the Netherlands’ Ministry of Foreign Affairs supported this research financially. Thanks are
due to R. Coe, C.K. Ong, M.R. Rao, M. van Noordwijk and J. Goudriaan for critical and constructive
comments on the manuscript.
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