**1. Introduction**

The increase in human population and the demand for life quality have induced the growing production of food and alternative vegetal energy sources in replacement to petrol. Soybean responds to more than 80% of biodiesel production, and will reach 5% inclusion in the fossil diesel in the next years in Brazil. This trend will increase pressure to new areas for soybean production on actually human food production areas, as well on pasture and untouched forests areas.

The progress of agriculture has been based on increase in animals and plants productivity per unit of area, which only has application when land availability is the sole limiting factor. However, the efficiency of use of limiting resources (including water, fertilizers and petrol) has to be considered. This mistaken vision is leading to excessive use of non renewable natural resources and environmental pollution. The reserves of phosphate in the world that can be explored at low cost are enough for 40 to 100 years and the world reserves of potassium are enough for 50 to 200 years. The situation is worse for micronutrients, in which the reserves of copper and zinc are enough for 60 years, manganese for 35 years and selenium for 55 years (Herring & Fantel, 1993; Roberts & Stewart, 2002; Aaron, 2005).

In addition to the depletion of natural reserves, the excessive use of fertilizers can contribute to soil and water courses contamination with nitrate (Angus, 1995; Bumb, 1995), soil acidification (Helyar & Poter, 1989), and emissions of carbon dioxide (CO2), nitrous oxide (N2O) and ammonia to the atmosphere. The pollution with nitrate has being an actual preoccupation in Europe and North America. The fertilization with phosphorus and nitrogen cause decrease in water oxygenation by excessive increase in the population of toxic algae in the oceans (Kebreab et al., 2002).

The agriculture participates in 20% of annual increase in the anthropogenic emission of greenhouse gases, mainly CH4 and N2O. Approximately 70% of all anthropogenic emission of N2O is attributed to agriculture. The current methodology used in Canada to estimate the flow of N2O is based in the direct relation between the emission of N2O and the application of nitrogen fertilizers (Lemke et al., 1998).

The possible deleterious effects of emissions of N2O are global warming and catalytic destruction of the ozone chain in the stratosphere, in which the N2O retains 13 times more heat than methane (CH4) and 270 times more than CO2 (Granli & Bockman, 1994). The atmospheric level of N2O has increased in growing fashion since 1960, associated with increase in utilization of nitrogen fertilizers (Bumb, 1995; Strong, 1995).

Rationality in the Use of Non Renewable Natural Resources in Agriculture 269

Fig. 2. Theoretical growth curve of life beings as a function of saturation by limitation of nutrients (under nutrition) or by products of metabolism (environmental pollution).

are being explored actually.

changes, among others.

2009).

fermentation end products – the volatile fatty acids.

As an example, a bacterium with volume of 1 m3 and duplication time of 20 minutes has potential to reach a biomass much greater than the earth in only 48 hours or 144 generations (Russell, 2002; p.57-58). Therefore, the archaea and bacteria are the first life beings of the planet, are hungry most of time, and can be the survivors of a biological collapse, such as those that occurred 65,000,000 years ago with the dinosaurs and 250,000,000 years ago, when more than 90% of life beings were extinct, leading to the formation of petrol reservoirs that

Another example of cessation of growth and death of population by environmental pollution is in the silage production, where the bacteria die and the nutrients are conserved to be used by ruminants, as a consequence of the acidity caused by accumulation of

Speaking of food production crisis lead we back to the Malthus theory, which although there are some conceptual errors, it will threaten the humanity and all life beings forever. According to Malthus, the population growth curve follow geometric progression and the food production arithmetic progression, leading to the crisis of food supply in some situations or in some periods of our existence (Thomas Malthus, accessed in March 02,

However, both population growth and food production follow a sigmoid curve up to the plateau or in form of a bell or double sigmoid over time (the second goes down hill, similar to the first in a mirror). The population growth curve is cumulative, as a consequence of the sum of the annual growth rates (Figure 3), which depend on the annual rate of food production (productivity), that by its time is consequence of the annual rate of soil utilization and exploration of non renewable natural resources (fertilizers and petrol), that follow the Hubbert curve. Changes in these curves can be caused by men or naturally, with discoveries of new food production technologies, population death (caused by diseases, wars, predations, among others), proliferation of plagues and diseases in plants, climatic

Fig. 1. Hubbert curve of exploration of petrol (non renewable natural resource), and altered curve by artificial maintenance of peak of production.

An worried phenomena about the use of non renewable natural resources can be visualized in the Hubbert curve (by Dr. Marion King Hubbert), which was a Shell Geologist, who predicted in 1956 that the global production of petrol would present a peak in the beginning of the XXI century (Hubbert Peak theory, accessed in March 02, 2009), and the curve of exploration follows the bell shape (Figure 1).

The phenomena observed by Hubbert related to petrol exploration is applicable to any other limiting natural resource, such as fertilizers, soil, and water and, consequently, food production. As more persistent is the maintenance of the maximum exploration of a resource, more drastic is the fall in the exploration of the final reserves in a short space of time, occurring the called blackout or sharply decay in the rate of production (Figure 1). Therefore, after the peak of exploration, if there is no new reserves to be discovered, no alternatives to produce more food without dependence on the available resources, or control of excessive exploration based on efficiency of use of these resources, catastrophic consequences can occur with mankind in some time of this century, as predicted by the Club of Rome in 1972, in the known publication "The limits of growth" (Meadows et al., 1972). The alert of the Club of Rome was based in the model associating accelerated industrialization, rapid population growth, depletion of non renewable natural resources, widespread malnutrition, and environmental pollution.

The objective of this work is to demonstrate the application of saturation kinetic models to improve efficiency of use of non renewable natural resources in agriculture, avoid the complete depletion as predicted by the Hubbert curve, and minimize the problems related to environmental pollution.

#### **2. Population growth curve**

The growth curve of populations of life beings in the absence of factors that affects the physical integrity, such as sickness and predation, has sigmoid curve (Gompertz or hyperbolic curve), including latency, exponential growth, plateau and senescence or death. The plateau occurs due to the saturation phenomena associated with depletion of nutrients or in some cases by environmental pollution, which acts in feedback against the uncontrolled growth (Figure 2).

Fig. 1. Hubbert curve of exploration of petrol (non renewable natural resource), and altered

An worried phenomena about the use of non renewable natural resources can be visualized in the Hubbert curve (by Dr. Marion King Hubbert), which was a Shell Geologist, who predicted in 1956 that the global production of petrol would present a peak in the beginning of the XXI century (Hubbert Peak theory, accessed in March 02, 2009), and the curve of

The phenomena observed by Hubbert related to petrol exploration is applicable to any other limiting natural resource, such as fertilizers, soil, and water and, consequently, food production. As more persistent is the maintenance of the maximum exploration of a resource, more drastic is the fall in the exploration of the final reserves in a short space of time, occurring the called blackout or sharply decay in the rate of production (Figure 1). Therefore, after the peak of exploration, if there is no new reserves to be discovered, no alternatives to produce more food without dependence on the available resources, or control of excessive exploration based on efficiency of use of these resources, catastrophic consequences can occur with mankind in some time of this century, as predicted by the Club of Rome in 1972, in the known publication "The limits of growth" (Meadows et al., 1972). The alert of the Club of Rome was based in the model associating accelerated industrialization, rapid population growth, depletion of non renewable natural resources,

The objective of this work is to demonstrate the application of saturation kinetic models to improve efficiency of use of non renewable natural resources in agriculture, avoid the complete depletion as predicted by the Hubbert curve, and minimize the problems related

The growth curve of populations of life beings in the absence of factors that affects the physical integrity, such as sickness and predation, has sigmoid curve (Gompertz or hyperbolic curve), including latency, exponential growth, plateau and senescence or death. The plateau occurs due to the saturation phenomena associated with depletion of nutrients or in some cases by environmental pollution, which acts in feedback against the

curve by artificial maintenance of peak of production.

widespread malnutrition, and environmental pollution.

to environmental pollution.

**2. Population growth curve** 

uncontrolled growth (Figure 2).

exploration follows the bell shape (Figure 1).

Fig. 2. Theoretical growth curve of life beings as a function of saturation by limitation of nutrients (under nutrition) or by products of metabolism (environmental pollution).

As an example, a bacterium with volume of 1 m3 and duplication time of 20 minutes has potential to reach a biomass much greater than the earth in only 48 hours or 144 generations (Russell, 2002; p.57-58). Therefore, the archaea and bacteria are the first life beings of the planet, are hungry most of time, and can be the survivors of a biological collapse, such as those that occurred 65,000,000 years ago with the dinosaurs and 250,000,000 years ago, when more than 90% of life beings were extinct, leading to the formation of petrol reservoirs that are being explored actually.

Another example of cessation of growth and death of population by environmental pollution is in the silage production, where the bacteria die and the nutrients are conserved to be used by ruminants, as a consequence of the acidity caused by accumulation of fermentation end products – the volatile fatty acids.

Speaking of food production crisis lead we back to the Malthus theory, which although there are some conceptual errors, it will threaten the humanity and all life beings forever. According to Malthus, the population growth curve follow geometric progression and the food production arithmetic progression, leading to the crisis of food supply in some situations or in some periods of our existence (Thomas Malthus, accessed in March 02, 2009).

However, both population growth and food production follow a sigmoid curve up to the plateau or in form of a bell or double sigmoid over time (the second goes down hill, similar to the first in a mirror). The population growth curve is cumulative, as a consequence of the sum of the annual growth rates (Figure 3), which depend on the annual rate of food production (productivity), that by its time is consequence of the annual rate of soil utilization and exploration of non renewable natural resources (fertilizers and petrol), that follow the Hubbert curve. Changes in these curves can be caused by men or naturally, with discoveries of new food production technologies, population death (caused by diseases, wars, predations, among others), proliferation of plagues and diseases in plants, climatic changes, among others.

Rationality in the Use of Non Renewable Natural Resources in Agriculture 271

Although the use of saturation kinetic model to explain the nutrients responses by the superior forms of life is not being adopted (Morgan et al., 1975), the Michaelis-Menten model allows to explain the curvilinear relationship of plants and animals to the nutrients and the model of Lineweaver-Burk allows to obtain the kinetic constants, ks (the amount of substrate needed to reach half theoretical maximum response in rate of growth or production of milk, wool, eggs, among others) and kmax (theoretical maximum response in

The responses of plants and animals to nutrients as saturation phenomena have important implications in addition to calculation of the rate of decreasing economical return and estimates of nutrients recommendations, such as the consciousness about the excessive use of non renewable natural resources; soil, water, and air pollution; and global warming. The knowledge about the efficiency of utilization of fertilizers in agriculture will play an important role in the political decisions about the rational use of non renewable natural resources in the future. The natural fertilizer sources have to be used with maximum

Recommendations of fertilization are mostly based in the method of calculation of nutrients requirements of a culture and the mineral contribution of the soil. The fertilizers are then calculated to supply the deficiencies. This method allows recommendation of the lower level that maximize the production. However, the method does not indicate changes in the recommendation based on changes in the costs of nutrients and grains. Also, it does not give direct information of the effect of application of other level than the recommended one

It has being utilized a variety of empirical models to predict the responses to nutrients and to calculate the optimum levels of nutrients. Among then, it is included the model of Mitscherlich, square root (Mombiela et al., 1981; Sain & Jauregui, 1993), exponential, linearplus-plateau, linear-plus-hyperbola, quadratic and quadratic-plus-plateau (Cerrato &

The use of saturation kinetics to explain the nutritional responses to nutrients by superior life beings are rarely employed (Morgan et al., 1975). The model of Michaelis-Menten has not being evaluated to make recommendations of fertilization. This model has a great potential in recommendation of use of nutrients in agriculture, by considering the efficiency of use of nutrients and the Law of diminishing return, as observed by Mitscherlich (1909). This model can aggregate important concepts such as responses to different levels of nutrients, benefit-cost ratio, efficiency of use of nutrients, rationality of use of non renewable

Linear regressions of reciprocal of plants responses as a function of reciprocal of nutrients supply, methodology known as data transformation of Lineweaver-Burk (Lineweaver &

1/Y = a + b \* (1/X)

Burk, 1934; Champe & Harvey, 1994), were proposed by Lana et al. (2005) as follow:

rate of growth or production), according to Lana et al. (2005).

efficiency and with minimum negative effects in the environment.

(Makowski et al., 1999).

where:

a = intercept,

**4. Marginal response or Law of diminishing return in plants** 

Blackmer, 1990; Bullock & Bullock, 1994; Makowski et al., 1999, 2001).

natural resources and consciousness about environmental pollution.

Y = responses of plants (grain yield, x 1,000 kg/ha),

Fig. 3. Theoretical population growth (cumulative and growth rate) as function of time.

Studies of these phenomena lead us to understand the need for rational use of non renewable natural resources. The kinetic saturation models are important tools generated by science in order to evaluate the efficiency and allow the rational use of non renewable natural resources (Lana, 2005; Lana et al., 2005; Lana, 2007a,b; Lana et al., 2007a,b; Lana, 2008). As a result, they can avoid the complete depletion of the resources and the collapse in food and energy supply, with dramatic consequences for our civilization, as predicted by Malthus, Hubbert and Club of Roma.
