R.M. Kho Agriculture, Ecosystems and Environment 80 2000 71–85 73
W = ε
s
Z fS
δt where f is the fraction of incident radiation intercepted
by the crop canopy, S is the daily incident radiation
MJ m
− 2
, and ε
s
is the conversion efficiency of solar radiation e.g. in g dm MJ
− 1
. If the resource is water, seasonal biomass produc-
tion W can be expressed as Ong et al., 1996: W = ε
w
X E
t
where P
E
t
is the cumulative transpiration mm H
2
O and where ε
w
is the conversion efficiency of water g dm mm
− 1
H
2
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 con- version efficiencies are mostly considered species spe-
cific and conservative, which explains why they are kept outside the integral. Concerning water, instead of
the conversion efficiency itself, its product with sat- uration vapour pressure deficit is also considered the
species specific constant Cooper et al., 1987. Mon- teith 1994 describes the principles of resource cap-
ture 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 contin- ues with the old paradigm of before the Second World
War. It aims to quantify the balance of available re- sources in the environment, make it measurable, and
explore some relationships with the old and the con- temporary 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 re-
sources. 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 contribut- ing to more biomass. This results in the proportional
relation of production to availability, and in a mini- mum concentration of the resource in the crop. In the
second, ‘diminishing returns’ state, another factor in- fluences 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 in- crease, 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 im- poses a plateau on the response. The first resource has
been saturated and has reached its maximum concen- tration. 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 avail-
ability. 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 bal- ance, 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
74 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 environ- ment influences the shape of the curve. In an idealised
environment where the resources influencing the pro- cess 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 andor on a
scale of a single plant, one process may be the slow- est 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 dif-
ferent processes succeed one another as slowest pro- cess 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 sev-
eral 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 diminish- ing 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-
Fig. 4. The sum of several Blackman-type curves, each with own slopes and plateau’s, will constitute a smooth curve.
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 dWdA
i
of the response curve; dW is the change in production W responding to a small change
dA
i
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 A
i
. Jones and Lynn 1994 proposed to normalise the slope to obtain a relative resource
limitation ℓ
i
, defined by ℓ
i
= dWW
dA
i
A
i
1 ℓ
i
is dimensionless and independent of the units used for W and A
i
. Crop production is a function of several resources
carbon dioxide, radiation, water, nitrogen, phospho- rus and other nutrients. Therefore, it is more appro-
priate to define limitation L
i
of any resource A
i
with partial derivatives:
L
i
= ∂WW
∂A
i
A
i
2 Rearranging Eq. 2 gives
L
i
= ∂W∂A
i
WA
i
R.M. Kho Agriculture, Ecosystems and Environment 80 2000 71–85 75
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 L
i
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 re-
source. If a resource is non-limiting on the plateau, the slope equals zero, so that the minimum value of L
i
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 L
i
equals one. Between the proportional state and the plateau, the slope de-
creases gradually to zero. The use efficiency will also decrease, but will never reach the value zero. In the
diminishing returns state, the value of L
i
is thus be- tween one and zero Fig. 5B. Note that no assump-
tion 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; L
i
= 1 and a plateau the
resource is non-limiting; L
i
= 0. Therefore, regarding
one resource, the original binary concept of limitation is a special case of coefficient L
i
except in the point of break which is not differentiable.
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 L
i
is only a good measure for the degree of lim- itation, 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 L
i
equals one and the limita- tion of all other resources equal zero cf. Von Liebig’s
model of plant growth; or the binary concept of limita- tion of Blackman. In this case, total limitation equals
one. This suggests that in order to be a generalisa- tion of Blackman’s concept the sum of the limitation
coefficients,
P L
i
, should also be one when there are several limiting resources. This section will demon-
strate 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 A
1
, A
2
, ......, A
n
3 where the seasonal biomass production W e.g. kgha
is a function f of A
i
i=1,2,. . . ,n which are the availabilities of the resources radiation e.g. MJm
2
,
76 R.M. Kho Agriculture, Ecosystems and Environment 80 2000 71–85
water e.g. mm, nitrogen e.g. kgha, phosphorus e.g. kgha, etc., n is the number of all resources.
It can be reasoned that if the availabilities of all re- sources inputs are multiplied with the same factor k,
the balance of resources and thus the degrees of limi- tation of the resources are not changed. Because con-
centration differences between the environment and the plant are multiplied with this factor, resource cap-
tures 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 kA
1
, kA
2
, ......, kA
n
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 pe-
riod: 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 con-
verted 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 pas-
toral 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 assump-
tion, the relative change of the output biomass produc- tion and the relative change of the inputs resources
are all equal k−1:
dW W
= dA
i
A
i
i = 1, 2, ..., n
5 Rearranging Eq. 5 gives for each resource i
dA
i
= dW
W A
i
6 According to the chain rule the derivative of W Eq.
3 to x, when all resource availabilities A
i
are some function of x the power changing all resource avail-
abilities with the same factor, is dW
dx =
n
X
i= 1
∂W ∂A
i
× dA
i
dx 7
Substituting Eq. 6 into Eq. 7 yields: dW
dx =
n
X
i= 1
∂W ∂A
i
× dW
W ×
A
i
dx And dividing both sides by dWdx gives
1 =
n
X
i= 1
∂W ∂A
i
× A
i
W Rearranging Eq. 2 and substitution yields
n
X
i= 1
L
i
= 1
8 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
dWW = qdA
i
A
i
, 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
ℓ
i
of one, they concluded that the sum of the rela- tive limitations exceeds one. However, Jones and Lynn
R.M. Kho Agriculture, Ecosystems and Environment 80 2000 71–85 77
1994 defined the relative limitation with derivatives Eq. 1. The sum of the relative limitations will in-
deed exceed one when several resources change simul- taneously. If limitation is defined with partial deriva-
tives 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 L
i
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 limi-
tation is a special case of coefficient L
i
. 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 be- tween crop production and resource availability. Nij-
land and Schouls 1997 have shown that this model can be considered as one of many mathematical rep-
resentations 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 W
= 1
W
max
+ 1
αA where
W is the dry matter production kg dmha. The recip- rocal 1W is the area ha needed for the production
of 1 kg dry matter, W
max
is the maximum possible production kg dmha when the resource is not limiting. The reciprocal
1W
max
is the minimum area that is needed for the production of 1 kg dry matter,
α is a coefficient of response of production to avail-
ability A of the resource e.g. kg dmkg resource and A is the availability of the resource e.g. kgha. The
reciprocal of αA 1αA 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 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
W =
1 W
max
+ 1
αA
1
+ 1
βA
2
9 1αA
1
is the area for deficiency of the first resource and 1βA
2
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 Limitation
1
= 1αA
1
1W =
W αA
1
By rearranging Eq. 9 the Michealis–Menten model is equal to:
W = W
max
αA
1
βA
2
αA
1
βA
2
+ W
max
αA
1
The partial first derivative of this equation to A
1
, the availability of the first resource equals:
∂W ∂A
1
= W
max
α βA
2
αA
1
βA
2
+ W
max
βA
2
+ W
max
αA
1
− W
max
αA
1
βA
2
α βA
2
+ W
max
α αA
1
βA
2
+ W
max
βA
2
+ W
max
αA
1 2
= W
max
βA
2
αA
1
βA
2
+ W
max
βA
2
+ W
max
αA
1
W A
1
= W
αA
1
W A
1
78 R.M. Kho Agriculture, Ecosystems and Environment 80 2000 71–85
which shows that Limitation
1
= W
αA
1
= ∂W
∂A
1
A
1
W =
L
1
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
