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
3.1. Limitation and the production laws Since the discovery of systematic experimental de-
sign 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 re- sponse ∂W to a small addition of a resource can be
regarded as ∂W = L
i
W A
i
∂A
i
11 That is as the product of the limitation L
i
, the use efficiency WA
i
and the amount of the dose ∂A
i
. 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 ex-
presses 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 re-
sources constant its shortage relative to the availabil- ities of other resources decreases. Consequently, its
limitation L
i
decreases see also Fig. 5B. This im- plies that, according to Eq. 11, the next dose ∂A
i
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 L
i
and lower use efficiency WA
i
, and thus a lower re- sponse. 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 L
i
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 WA
i
. Addition of other lim- iting resources increases thus the ‘effect’ or the crop
response ∂W to a certain dose ∂A
i
. 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 ∂A
i
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 pro- duction 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 cap- ture and biomass production; and III availability and biomass
production see text.
R.M. Kho Agriculture, Ecosystems and Environment 80 2000 71–85 79
The relation in Quadrant I can be described by the function C=f
1
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 nu- trients, 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 pro-
duction and allocation to plant organs, which in turn depend on the balance of available resources. For ex-
ample, 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=f
2
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 cap-
ture of other resources. For example, if nitrogen has its minimum concentration in the plant, additional up-
take 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 hypotheti- cal 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 possi-
ble to say which resources 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 re-
sources. The use of an empirically found line between production and capture for prediction in environ-
ments with another balance of resources is therefore hazardous.
Fig. 8. A linear relation between production and the capture of one resource C
1
may be found thanks to confounding with the capture of other resources e.g. C
2
.
The relation in Quadrant III can be described by the function W=f
3
A availabilities of other resources constant. Each point on the curve is associated with
a certain ‘use efficiency’ ε
use
: dry matter produc- tion per unit available resource. Note that resource
availabilities can be changed independently in a ran- domised 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 andor the second show diminishing re- turns too. From the law of the optimum, the slope of
the third curve increases with addition of other limit- ing resources. Thus, the slope of the first curve andor
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 quad-
rants see text I
II III
Function C=f
1
A W=f
2
C W=f
3
A=f
2
f
1
A Slope
f
′ 1
= dCdA
f
′ 2
= dWdC
f
′ 3
= dWdA=f
′ 2
f
′ 1
Efficiency ε
cap
= f
1
AA ε
conv
= f
2
CC ε
use
= f
3
AA=ε
conv
ε
cap
Limitation L
cap
= f
′ 1
ε
cap
L
conv
= f
′ 2
ε
conv
L
use
= f
′ 3
ε
use
= L
conv
L
cap
80 R.M. Kho Agriculture, Ecosystems and Environment 80 2000 71–85
direction. So, on theoretical grounds it can be ex- pected that the efficiency of resource capture ε
cap
andor the conversion efficiency ε
conv
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 p0.001
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 environ-
ments with the same balance of available resources.
In analogy with the limitation of quadrant III L
use
which is defined in Section 2.2, ‘limitations’ for the first two quadrants can be defined L
cap
and L
conv
. The sum over all resources of each will most likely ex-
ceed one. Generalisations of the function in quadrant I like C
1
= f
1
A
1
,...,A
n
do not have meaningful par- tial derivatives. That in quadrant II W=f
2
C
1
,...,C
n
does not have real existing partial derivatives because of the confounding. Therefore, L
cap
and L
conv
lack important properties that L
use
has.
4. Methods to estimate limitations
