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
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 ≈
1WW 1AA
= 1W
WA1A 12
where 1W is the change in production, responding to the change 1A in the availability of the resource,
and where WA 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 fertilisa-
tion and irrigation this average slope is because of the law of diminishing returns lower than, and there-
fore an under-estimation of, the slope in the con- trol 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 formu-
lated 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 var- ied systematically e.g. in a 3
k
factorial design; see also Cochran and Cox, 1957, Chapter 8A. The avail-
ability of nutrients can be varied by addition of fer- tiliser, 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 A
i
. Table 2 shows some empirical response functions and the ap-
propriate limitation. W
max
and the Greek letters are environment specific parameters that must be deter-
mined empirically. The resource availability is the sum of two availabilities: that in the control environment
at zero application A
i,
and the application A
i,appl
, i.e. A
i
= A
i,
+ A
i,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 dis-
advantage of this approach is that a curved linesurface
R.M. Kho Agriculture, Ecosystems and Environment 80 2000 71–85 81
Table 2 Some empirical response functions and the appropriate limitation
Model Limitation
Mitscherlich exponential W = W
max
Q
n i=
1
1 − e
− α
i
A
i
L
i
= α
i
A
i
e
α
i
A
i
− 1
Michaelis–Menten hyperbolic 1W = 1W
max
+ P
n i=
1
1α
i
A
i
L
i
= Wα
i
A
i
Polynomial quadratic W = α
+ P
n i=
1
α
i
A
i
+ β
i
A
2 i
+ P
n j i
γ
ij
A
i
A
j
L
i
= α
i
A
i
+ 2β
i
A
2 i
+ P
n j =
1
γ
ij
A
j
W
is extrapolated see Fig. 9. This may result in unsta- ble estimations of A
i,
with large standard errors. A better approach may be one in which additional in-
formation 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, be- cause the diminishing returns are compensated by in-
creasing concentrations.
Dean 1954 related nutrient uptake capture to ap- plication of the nutrient. He estimated the availabil-
ity to the control crop the parameter A
i,
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
Fig. 9. Response curve to resource application. Parameter A has
to be found by extrapolation of the curve dashed line.
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 differ- ences in application: ∂A. Because the interest is not in
absolute values of availability, but in relative changes in availability ∂AA; see Eq. 2, the method of Dean
1954 is appropriate for the present purpose.
The method can be generalised easily for other re- sources radiation and water. Intercepted radiation
can be related to different levels of incident radiation using shade cloths, and transpired water can be re-
lated to different levels of applied water by irriga- tion. By linear extrapolation until intersection with
the horizontal at zero capture, the ‘a’ values A
i,
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 A
i,
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
