The trajectory followed from 1961 to 2009 by a number of countries in terms of crop yield and total N inputs into cropland is shown in figure 1. The results for all countries of the FAO data base are provided in supplementary material (S2). The Y versus F trajectory drawn by most countries shows, at least for periods of several decades, a distinct curvilinear relationship. Linear trajectories, like those described by de Wit (1992) for individual crops were seldom observed. Several mathematical formulations of the yield-fertilization relationship in a given pedo-climatic and technical-agronomical context have been proposed in the agronomical literature, most of them involving negative exponential functions (Llewely and Featherstone 1997, Harmsen 2000). Nijland et al (2008) proposed to integrate the production functions of Liebig, Mitscherlich and Liebscher (de Wit 1992) into one system model based on Michaelis−Menten hyperbolic relationships. Because we are expressing both output and input in exactly the same unit (kgN ha–1 yr–1) and because we are looking for a simple long-term integrative theoretical relationship, we decided to make use of the simplest possible function obeying the three following properties: (i) the function intercept should be zero; (ii) the slope of the function should be 1 at low fertilization; (iii) the function should reach a plateau at high fertilization. The first two properties reflect the fact that, in the long run, harvest cannot exceed N restitutions to the soil, and that the effect of low fertilization in strongly N-limited systems is characterized by a NUE close to 1. The third property expresses the classical law of diminishing return and the fact that, in constant technical-agronomical context, some other limiting factor will always impose a ceiling to production at saturating N availability. Two mathematical functions with only one parameter obey both conditions: a hyperbolic function of the form Y = Ymax*F/(F + Ymax) [1] and a negative exponential function such as Y = Ymax [1 − exp(−F/Ymax)] [2]. We observed that the former generally provides the best fit to the data. In both cases the parameter Ymax represents the yield value reached at saturating N fertilization, as well as the value of fertilization at which a definite fraction of this maximum yield is reached (this fraction being 0.5 in the case of relation [1] or 0.63 for relation [2]). Over the 1961–2009 period, certain countries that we will call 'type I', such as China, Egypt and India, present a simple trajectory with regularly increasing fertilization and gradual reduction in the crop yield response, following a consistent and unique Y versus F relationship (figure 1(a)).

Other countries (called 'type II'), such as the USA, Brazil and Bangladesh, display a historical trajectory with first a regularly increasing fertilization and yield, fitting the Y versus F relationship with a definite Ymax, then a turning point with a shift of the trajectory to another relationship with a significantly higher Ymax. This likely reflects improved agronomical practices in terms of production factors other than nitrogen, together with the pursuit of increasing fertilization. The turning point seems to have occurred in the 1980s or later depending on the country (figure 1(b)). The case of the USA, for example, is consistent with a slowdown in the increase of synthetic fertilizers inputs from the 1980s parallel to a moderate increase in the yields of the most important crops (Howarth et al 2002, Alston et al 2010).

Figure 1. Examples of trajectories followed by countries in the Y versus F diagram. (a) Examples of type I trajectories. (b) Examples of type II trajectories. (c) Examples of type III trajectories. (d) Examples of type IV trajectories. R2 is the coefficient of determination, defined as: where obs i are the observed values of yield, calc i the yield value calculated with the relationship and meanobs is the average value of the observed yields over the period considered. Negative values of R2 indicate poor fit of the relationship on the observed values. This is often the case for the most recent period of type III trajectories because of still evolving agronomical conditions. Download figure: Standard image High-resolution image Export PowerPoint slide

In most European countries (see the example of France, the Netherlands and Greece in figure 1(c)), the trajectory also shows a bi-phasic pattern, describing a regular increase in both fertilization and yield during the 1960–1975 period, followed by a shift towards improved yields without further increasing fertilization and even decreasing fertilization from the 1980s on ('type III'). The case of the Netherlands is the most spectacular, as in this country, which has always used very high rates of fertilization, the level applied in recent years has been reduced to the same as in the 1960s with, however, doubled yields. This trend is related to the reduction of N inputs prescribed by European environmental policies and regulations (van Grinsven et al 2012), which interestingly did not prevent significant yield increases thanks to a better N management. Note, however, that despite the increase of NUE and decrease in N surpluses, the nitrogen surplus emitted to the environment in many cases remains much higher than that of other countries belonging to types I and II.

Finally, there is a small group of countries, such as Morocco, Benin and Nigeria, whose trajectory does not display any consistent Y versus F relationship (type IV). These countries have always very low inputs and yields. Very often, their trajectory in the Y versus F diagram crosses the 1:1 line, indicating higher yield than fertilization. High inter-annual variation in the agricultural performance observed in some of these countries could be explained by weather phenomena, such as persistent water droughts, socio-political questions, or sometimes could be even an artefact due to the poor quality of our estimates of total nitrogen inputs to agricultural soil: in particular, in those countries where shifting agriculture is practiced, the fertilization of agricultural soil by the nitrogen stock accumulated in forested soil during the fallow period is not taken into account in our input estimations. However the 'negative' nitrogen balance displayed in the Y versus F diagram can also represent the signature of an unsustainable nitrogen mining of agricultural soils.

For type I to III countries, we were able to define the Ymax values providing the best fit of the hyperbolic relationship [1] to the points corresponding to the 1961–1980 period or later, and another Ymax for the most recent 10–15 years. The two Ymax values obtained characterize the past and current agricultural potential respectively, defined as the protein yield that could be obtained from cropland at a maximum N fertilization rate, with the corresponding cropping practices (figure 2). Comparison of the two periods shows a significant increase of Ymax in 45 countries (type II and III trajectories). For a large number (55) of countries, however, nearly the same parameter value or Ymax holds over the 50-year period (type I trajectory), as is the case for China, Egypt, Turkey, Chile, India and a many others. Possible N mining is indicated by higher crop yield than fertilization for 18 countries such as Canada, Morocco, Algeria, Iraq and Mozambique in the 1960–1980 period (see S2 for the complete list). In recent years, N mining continues in 10 African countries, as well as in Former Soviet Union countries, Afghanistan and Paraguay. N mining has been observed in Argentina for the entire studied period. This result is coherent with that recently reported by Álvarez et al (2014) that indicate a budget of the copping system in the pampean agroecosystems, which only becomes positive when including pasture lands. The severe problem of nutrient mining and loss of soil fertility in African countries has been frequently highlighted (Vitousek et al 2009, Liu et al 2010). In these countries yields are among the lowest in the world but have apparently wide margins for improvement through better fertilization practices, including an increasing use of legumes in crop rotations (Vanlauwe et al 2014). However, imbalances with other nutrients such as P could limit yield responses to N addition (van der Velde et al 2014). In the Former Soviet Union, after the abrupt changes which occurred from 1989, crops may have benefitted from nutrient legacies. The results of our calculations for this country, however, might also be affected by recent and poorly documented changes like massive land abandonment not fully documented by the FAO (Schierhorn et al 2013).