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
This section illustrates the experimental quantifica- tion of the limitation coefficients as developed in this
paper. 5.1. Material and methods
In the 1996 season, pearl millet Pennisetum glau- cum L. R.Br. was grown on farmers fields near
N’Dounga, south–west Niger 13
◦
23
′
N and 2
◦
16
′
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
0–0.15 0.15–0.40
0.40–0.90 Org. C
0.259 0.173
0.138 Total N ppm
164 131
118 Total P ppm
310 314
294 Bray1 P ppm
4.3 1.7
1.3 pH-H
2
O 1:2.5 6.0
5.6 5.8
pH-KCl 1:2.5 4.8
4.3 4.4
H
+
meq100 g soil 0.044
0.100 0.071
Al
3+
meq100 g soil 0.02
0.19 0.09
Na
+
meq100 g soil 0.037
0.040 0.046
K
+
meq100 g soil 0.146
0.095 0.053
Ca
2+
meq100 g soil 1.37
1.43 2.17
Mg
2+
meq100 g soil 0.67
0.91 1.03
Sand 88.9
84.2 82.5
Silt 3.8
3.5 3.7
Clay 7.3
12.3 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 MayJune until SeptemberOctober. To-
tal 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 culti- vated by intercrops milletcowpea. The local cultivar
of millet, being the staple crop, was grown with and without nitrogen urea, and with and without phos-
phorus Single Super Phosphate fertiliser. The exper- imental design was a 2
2
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 broad-
cast, half of the dose shortly before sowing and the
Table 4 Treatments of the experiment
Treatment N application kgha
P application kgha A
B 180
C 60
D 180
60 E
90 30
Fig. 10. Biomass response surface Eq. 13 to nitrogen and phosphorus availability in south–west Niger all units are in kgha.
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 ni-
trogen. According to farmer practice, the millet was sown in hills with a density of 10 000 hillsha. 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 high-
est 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 analy- sis 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 Fig. 11 shows the relation of nitrogen N uptake
to nitrogen application appl. Regression led to the equation
N uptake = 26.6 + 0.285N appl, S.E.
4.4 0.036,
R
2
= 0.75
R.M. Kho Agriculture, Ecosystems and Environment 80 2000 71–85 83
Fig. 11. Relation between nitrogen uptake and application, and estimation of the nitrogen availability at zero application all units
are in kgha.
where the nitrogen uptake and application are in kg Nha. Extrapolation of the regression line until inter-
section with the horizontal at zero uptake estimates the nitrogen availability ‘a’ value of Dean, 1954, at
zero application as 26.60.285=93 kg Nha. The stan- dard error is approximated as 25 kg Nha.
Fig. 12 shows the relation of phosphorus P up- take to phosphorus application. Regression led to the
equation:
Fig. 12. Relation between phosphorus uptake and application, and estimation of the phosphorus availability at zero application all
units are in kgha.
P uptake = 5.62+0.264P appl−0.0034P appl
2
, S.E.
0.70 0.065
0.0010, R
2
= 0.63
where the phosphorus uptake and application are in kg Pha. Extrapolation of the slope of the regression curve
at zero application until intersection with the horizon- tal at zero uptake estimates the phosphorus availability
at zero application as 5.620.264=21.3 kg Pha. The standard error is approximated as 6.6 kg Pha.
Fitting of Eq. 9 with N and P availability at zero application taken as 93 and 21.3, respectively gave
1 W
= 0.000089 + 0.0109
1 93 + N
appl
+ 0.00214
1 21.3 + P
appl
, 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 er-
ror 0.10, and the limitation of phosphorus as 0.33 ap- proximated standard error 0.098. These two elements
account thus for 0.38+0.33100=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 Sa- hel 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
, ε
conv
and ε
use
; see Sec- tion 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 limi-
tation of the resource.
6. Discussion and conclusions
