Directory UMM :Data Elmu:jurnal:A:Agriculture, Ecosystems and Environment:Vol80.Issue1-2.Aug2000:

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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) formu-lated 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: r.m.kho@ato.wag-ur.nl (R.M. Kho)

tional relation) to modifications in the availability of only one, the limiting, factor. If another factor be-comes 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 cen-tury 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 fac-tors constant). Rabinowitch (1951) shows that the un-derlying kinetic view must be that plant growth is a sequence of processes, whereby the process on which

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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 pro-cesses 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 deter-mined 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 limita-tion, 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 lim-iting factor’ (Shetty et al., 1995). Use of these terms implies the existence of ‘minor limitations’ and ‘sec-ondary 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 agro-nomic 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, to-pography, 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}

re-source capture (e.g. MJ m−2_{for light, mm for water,}

and g m−2 _{for nutrients). Seasonal biomass}

produc-tion (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):

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W =εs

fS0δt

where f is the fraction of incident radiation intercepted by the crop canopy, S0is 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 =εw

wherePEtis the cumulative transpiration (mm H2O)

and where εw is the conversion efficiency of water

(g dm mm−1 _H

2O 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

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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 and/or 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 (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 limitationℓi, defined by

ℓi =

dW/W dAi/Ai

(1) ℓi 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, phospho-rus and other nutrients). Therefore, it is more appro-priate to define limitation Li of any resource Ai with

partial derivatives:

Li =

∂W/W ∂Ai/Ai

(2) Rearranging Eq. (2) gives

Li =

∂W/∂Ai

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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 re-source). 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 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 Li 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; 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).

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

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 Li 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 limitalimita-tion equals one. This suggests that (in order to be a generalisa-tion of Blackman’s concept) the sum of the limitageneralisa-tion coefficients,PLi, 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 (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

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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 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 (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 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):

W =

dAi

i=1,2, ..., n (5)

Rearranging Eq. (5) gives for each resource i dAi =

W Ai (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 avail-abilities with the same factor), is

dx =

i=1

∂W ∂Ai

×dAi

dx (7)

Substituting Eq. (6) into Eq. (7) yields: dW

dx =

i=1

∂W ∂Ai

×dW

W ×

And dividing both sides by dW/dx gives

i=1

∂W ∂Ai

×Ai W

Rearranging Eq. (2) and substitution yields

i=1

Li =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

(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 ℓi of one), they concluded that the sum of the

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(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 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) limi-tation 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 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-resentation(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

W =

Wmax

+ 1

αA where

W is the dry matter production (kg dm/ha). The

recip-rocal (1/W) is the area (ha) needed for the production of 1 kg dry matter,

Wmaxis 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,

αis a coefficient of response of production to avail-ability A of the resource (e.g. kg dm/kg resource) and

A is the availability of the resource (e.g. kg/ha). The

reciprocal ofαA (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

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:

W =

Wmax

+ 1

αA1

+ 1

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=

1/αA1

1/W =

αA1

By rearranging Eq. (9) the Michealis–Menten model is equal to:

W = WmaxαA1βA2

αA1βA2+WmaxαA1

The partial first derivative of this equation to A1, the

availability of the first resource equals: ∂W

∂A1

(Wmaxα βA2)(αA1βA2+WmaxβA2+WmaxαA1)

−(WmaxαA1βA2)(α βA2+Wmaxα)

(αA1βA2+WmaxβA2+WmaxαA1)2

= WmaxβA2

αA1βA2+WmaxβA2+WmaxαA1

W A1

= W

αA1

W A1

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which shows that Limitation1=

αA1

= ∂W

∂A1

W =L1 (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 =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) 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 (Li) decreases (see also Fig. 5B). This

im-plies 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

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 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

lim-iting 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 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).

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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

nu-trients), and where A is its availability (same units). Each point on the curve is associated with a certain ‘efficiency of resource capture’ εcap: 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=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’ εconv: 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 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 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 (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

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 and/or 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 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 quad-rants (see text)

I II III

Function C=f1(A) W=f2(C) W=f3(A)=f2(f1(A)) Slope f′

1=dC/dA f ′

2=dW/dC f ′

3=dW/dA=f ′ 2f

′ 1 Efficiency εcap=f1(A)/Aεconv=f2(C)/C εuse=f3(A)/A=εconvεcap Limitation Lcap=f′

1/εcap Lconv=f′

2/εconvLuse=f′

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direction. So, on theoretical grounds it can be ex-pected 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 environ-ments 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 ex-ceed one. Generalisations of the function in quadrant I (like C1=f1(A1,...,An)) do not have meaningful

par-tial 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

1A/A =

(W/A)1A (12)

where 1W is the change in production, responding

to the change1A 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 (fertilisa-tion and irriga(fertilisa-tion) 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 dose1A

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 3k 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 (Ai). Table 2

shows some empirical response functions and the ap-propriate limitation. Wmax 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 (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 dis-advantage of this approach is that a curved line/surface

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Table 2

Some empirical response functions and the appropriate limitation

Model Limitation

Mitscherlich (exponential) W=WmaxQni=1(1−e

−αiAi₎ _L

i=αiAi/(eαiAi−1) Michaelis–Menten (hyperbolic)

(1/W )=(1/Wmax)+Pni=1(1/αiAi) Li=W/αiAi Polynomial (quadratic)

W=α0+Pn_i₌₁(αiAi+βiA2i+

j >iγijAiAj) Li=(αiAi+2βiA2i+

j=1γijAj)/W

is extrapolated (see Fig. 9). This may result in unsta-ble estimations of Ai,0 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 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

Fig. 9. Response curve to resource application. Parameter A0 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 (∂A/A; 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 (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,0are 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 limitaquantifica-tion 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

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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-H2O 1:2.5 6.0 5.6 5.8

pH-KCl 1:2.5 4.8 4.3 4.4

H+_{(meq/100 g soil)} _0.044 _0.100 _0.071 Al3+ _{(meq/100 g soil)} _0.02 _0.19 _0.09 Na+_{(meq/100 g soil)} _0.037 _0.040 _0.046 K+_{(meq/100 g soil)} _0.146 _0.095 _0.053 Ca2+_{(meq/100 g soil)} _1.37 _1.43 _2.17 Mg2+_{(meq/100 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 May/June until September/October. 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 millet/cowpea. 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 22factorial 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 (kg/ha) P application (kg/ha)

A 0 0

B 180 0

C 0 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 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 ni-trogen. 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 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+0.285(N appl),

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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 inter-section 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 stan-dard error is approximated as 25 kg N/ha.

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 kg/ha).

P uptake=5.62+0.264(P appl)−0.0034(P appl)2,

S.E. 0.70 0.065 0.0010, R2=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 horizon-tal 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

W =0.000089+0.0109

1 93+Nappl

+0.00214 1

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 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+0.33)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 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 (εcap,εconvandεuse; 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

A coefficient (Eq. (2)) has been derived which quan-tifies the degree of limitation of growth resources in a specific environment. It generalises the binary con-cept of Blackman (1905), making it applicable for cir-cumstances that are more realistic, by fractionating his total limitation for one resource as a sum of degrees

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Table 5

Limitations and average efficiencies per treatment

Treatment Nitrogen Phosphorus

LN εcap εconv (kg/kg) εuse(kg/kg) LP εcap εconv(kg/kg) εuse (kg/kg)

A 0.38 0.27 137 36 0.33 0.22 758 159

B 0.17 0.25 72 17 0.44 0.30 764 220

C 0.50 0.27 194 49 0.11 0.10 600 56

D 0.26 0.32 72 22 0.17 0.12 587 75

E 0.31 0.31 97 29 0.22 0.20 519 103

of limitations of several resources. The coefficient en-ables quantification of the insight that an effect of a resource increases with its (degree of) limitation. It co-operates well with general accepted crop produc-tion 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 charac-terised with a few parameters and more accurately than with qualifications as ‘major limitations’, ‘prin-cipal limiting factors’ or ‘(highly) deficient’.

The validity of the constant returns to scale assump-tion (Eq. (4)) is plausible, but should be investigated further. However, even if returns to scale are not ex-actly 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 envi-ronment 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, vari-ation in rainfall, and varivari-ation 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 oper-ations as (time of) weeding and method of tillage may favour the availability of one resource above that of others, and may influence the limitation coeffi-cients too. Determining the limitation coefficoeffi-cients 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 re-lation between production and capture of a resource is a correlation, not a causal relation. Conversion ef-ficiencies and/or efef-ficiencies 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 posi-tively 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 re-sults because it takes the balance of resources in each environment explicitly into account. This will be espe-cially the case for inherent multidisciplinary sciences as ecology, weed science, intercropping, and agro-forestry (Kho, 2000). These ‘holistic’ sciences, study-ing interactions between plants of different species in different environments, have to deal with a com-plex of all resources, their mutual proportions and interrelations.

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Acknowledgements

The Directorate General International Co-operation (DGIS) of the Netherlands’ Ministry of Foreign Af-fairs supported this research financially. Thanks are due to R. Coe, C.K. Ong, M.R. Rao, M. van Noord-wijk and J. Goudriaan for critical and constructive comments on the manuscript.

References

Azam-Ali, S.N., Crout, N.M.J., Bradley, R.G., 1994. Perspectives in modelling resource capture by crops. In: Monteith, J.L., Scott, R.K., Unsworth, M.H. (Eds.), Resource Capture by Crops. Proceedings of Easter Schools in Agricultural Science, Nottingham University Press, Nottingham, pp. 125–148. Blackman, F.F., 1905. Optima and limiting factors. Ann. Bot. 19,

281–295.

Buerkert, A., Stern, R.D., Marschner, H., 1995. Post stratification clarifies treatment effects on pearl millet growth in the Sahel. Agron. J. 87, 752–761.

Cochran, W.G., Cox, G.M., 1957. Experimental Designs, 2nd Edition. Wiley, New York, 611 pp.

Cooper, P.J.M., Gregory, P.J., Tully, D., Harris, H.C., 1987. Improving water use efficiency of annual crops in the rainfed farming systems of west Asia and north Africa. Exp. Agric. 23, 113–158.

Dean, L.A., 1954. Yield-of-phosphorus curves. Proc. Soil Sci. Soc. Am. 18, 462–466.

De Wit, C.T., 1992. Resource use efficiency in agriculture. Agric. Syst. 40, 125–151.

De Wit, C.T., 1994. Resource use analysis in agriculture: a struggle for interdisciplinarity. In: Fresco, L.O., Stroosnijder, L., Bouma, J., van Keulen, H. (Eds.), The Future of the Land: Mobilising and Integrating Knowledge of Land Use Options. Wiley, New York, pp. 41–55.

Hanks, J., Ritchie, J.T. (Eds.), 1991. Modelling Plant and Soil Systems. Agronomy Monograph No. 31, American Society of Agronomy, Madison, 545 pp.

Jones, H.G., Lynn, J.R., 1994. Optimal allocation of assimilate in relation to resource limitation. In: Monteith, J.L., Scott, R.K., Unsworth, M.H. (Eds.), Resource Capture by Crops. Proceedings of Easter schools in Agricultural Science, Nottingham University Press, Nottingham, pp. 111–123. Kho, R.M., 2000. A general tree–environment–crop interaction

equation for predictive understanding of agroforestry systems. Agric. Ecosyst. Environ. 80, 87–100.

Liebscher, G., 1895. Untersuchungen über die Bestimmung des Düngerbedürfnisses der Ackerböden und Kulturpflanzen. J. Landwirtsch. 43, 49–216.

Longford, N.T., 1993. Random Coefficient Models. Oxford Statistical Science Series 11, Clarendon Press, Oxford, 270 pp.

Monteith, J.L., 1981. Does light limit crop production? In: Johnson, C.B. (Ed.), Physiological Processes Limiting Plant Productivity. Butterworths, London, pp. 23–38.

Monteith, J.L., 1994. Principles of resource capture by crop stands. In: Monteith, J.L., Scott, R.K., Unsworth, M.H. (Eds.), Resource Capture by Crops. Proceedings of Easter Schools in Agricultural Science, Nottingham University Press, Nottingham, pp. 1–15.

Monteith, J.L., Scott, R.K., Unsworth, M.H. (Eds.), 1994. Resource Capture by Crops. Proceedings of Easter Schools in Agricultural Science, Nottingham University Press, Nottingham, 469 pp. Nijland, G.O., Schouls, J., 1997. The relation between crop

yield, nutrient uptake, nutrient surplus and nutrient application. Wageningen Agricultural University Papers 97–3, Agricultural University, Wageningen, 151 pp.

Ong, C.K., Black, C.R., Marshall, F.M., Corlett, J.E., 1996. Principles of resource capture and utilisation of light and water. In: Ong, C.K., Huxley, P. (Eds.), Tree–Crop Interactions: A Physiological Approach. CAB International, Wallingford, UK, pp.73–158.

Payne, W.A., Lascano, R.J., Wendt, C.W., 1991. Physical and hydrological characterisation of three sandy millet fields in Niger. In: Sivakumar, M.V.K., Wallace, J.S., Renard, C., Giroux, C. (Eds.), Proceedings of the International Workshop on Soil Water Balance in the Sudano–Sahelian Zone, Niamey, Niger, February 1991. IAHS Publ. No. 199, IAHS Press, Institute of Hydrology, Wallingford, UK, pp. 199–207.

Penning de Vries, F.W.T., Djitèye, M.A. (Eds.), 1982. La productivité des pâturages sahéliens. Une etude des sols, des végetations et de l’exploitation de cette ressource naturelle. Agric. Res. Rep. 918, Pudoc, Wageningen, 525 pp.

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direction. So, on theoretical grounds it can be ex-pected 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 environ-ments 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 ex-ceed one. Generalisations of the function in quadrant I (like C1=f1(A1,...,An)) do not have meaningful

par-tial 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

1A/A =

(W/A)1A (12)

where 1W is the change in production, responding

to the change1A in the availability of the resource,

4.2. Estimation from response data

The biomass production (W) can then be fitted as function of the resource availabilities (Ai). Table 2

shows some empirical response functions and the ap-propriate limitation. Wmax and the Greek letters are

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 dis-advantage of this approach is that a curved line/surface

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Table 2

Some empirical response functions and the appropriate limitation

Model Limitation

Mitscherlich (exponential)

W=WmaxQni=1(1−e

−αiAi₎ _L

i=αiAi/(eαiAi−1) Michaelis–Menten (hyperbolic)

(1/W )=(1/Wmax)+Pni=1(1/αiAi) Li=W/αiAi Polynomial (quadratic)

W=α0+Pn_i₌₁(αiAi+βiA2i+

j >iγijAiAj) Li=(αiAi+2βiA2i+

j=1γijAj)/W

is extrapolated (see Fig. 9). This may result in unsta-ble estimations of Ai,0 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

Dean (1954) related nutrient uptake (capture) to ap-plication of the nutrient. He estimated the availabil-ity to the control crop (the parameter Ai,0) by linear

Fig. 9. Response curve to resource application. Parameter A0 has

to be found by extrapolation of the curve (dashed line).

absolute values of availability, but in relative changes in availability (∂A/A; see Eq. (2)), the method of Dean

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,0are 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 limitaquantifica-tion 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

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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-H2O 1:2.5 6.0 5.6 5.8

pH-KCl 1:2.5 4.8 4.3 4.4

H+_{(meq/100 g soil)} _0.044 _0.100 _0.071

Al3+ _{(meq/100 g soil)} _0.02 _0.19 _0.09

Na+_{(meq/100 g soil)} _0.037 _0.040 _0.046

K+_{(meq/100 g soil)} _0.146 _0.095 _0.053

Ca2+_{(meq/100 g soil)} _1.37 _1.43 _2.17

Mg2+_{(meq/100 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

Table 4

Treatments of the experiment

Treatment N application (kg/ha) P application (kg/ha)

A 0 0

B 180 0

C 0 60

D 180 60

E 90 30

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+0.285(N appl),

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Fig. 11. Relation between nitrogen uptake and application, and estimation of the nitrogen availability at zero application (all units are in kg/ha).

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 kg/ha).

P uptake=5.62+0.264(P appl)−0.0034(P appl)2,

S.E. 0.70 0.065 0.0010, R2=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 horizon-tal 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

W =0.000089+0.0109

1 93+Nappl

+0.00214 1

21.3+Pappl ,