Methods for calculating fertilizer application rates in agriculture

Despite the high expenditure of time and resources, methods for determining fertilizer application rates based on field experiments are imprecise. Firstly, when determining optimal rates, experiments are set up with a large number of variants; in this case, the variability of soil fertility inevitably has an effect. Secondly, the correction factors used are general in nature, as they are not fully substantiated experimentally. Thirdly, due to the constant cultivar rotation in agriculture, the introduction of new technologies, and new forms of fertilizers, obtaining field experiment results that account for changed conditions is slower than required by production, which is why recommendations from scientific institutions become outdated and serve only as a guide. Certain shortcomings also arise when planning the results of fertilizer application, as the field experiment method does not allow for targeting a specific amount of yield increase.

All this necessitates the search for more accurate and efficient methods for establishing fertilizer application rates, which led to the emergence of calculation-based methods. Among their great variety, the normative, balance, and mathematical methods should be highlighted.

Normative method. Its application allows for the monitoring and regulation of soil fertility (with full fertilizer availability). However, there are also drawbacks associated with the fact that data on nutrient uptake by the harvest are inaccurate (according to reference books), and correction factors, being derivatives of nutrient use efficiency coefficients from fertilizers, can change significantly.

The calculation of fertilizer application rates based on cost norms per unit of yield is performed using the formula:

D = Yp · N1 · K, where: D – nutrient application rate for the planned yield, kg/ha;

Yp – planned yield, t/ha;

N1 – nutrient cost norms for producing a unit of yield at the optimal rate;

K – correction for soil agrochemical properties.

To calculate fertilizer application rates based on cost norms per unit of yield increase, the following formula is used:

D = ∆Yp · N2 · K, where D – nutrient application rate for the planned yield, kg/ha;

∆Yp – possible yield increase due to fertilizers, t/ha;

N2 – nutrient cost norms per unit of yield increase at the optimal rate;

K – correction for soil agrochemical properties.

The yield increase due to fertilization is determined by the formula:

∆Y = Yp – B · Cb, where B — arable land rating;

Cb – price of the arable land rating, t/ha.

The arable land rating is determined by land appraisal data. The price of the rating is calculated by the formula:

Cb = Yk / B · K,

where: Yk – yield without fertilizer application (t/ha), based on field experiment data;

B – arable land rating based on land appraisal data;

K – correction factor for production conditions (Table 175; Eysert E.K., Achkanov A.Ya., Durgaryan N.G. et al., 1987).

Table 177 – Correction factors for fertilizer application rates to account for agrochemical properties of Precaucasian chernozems Nutrient content Level in soil Spring cereals, corn, sunflower Winter wheat, sugar beet Vegetables, pome fruits, stone fruits, grapes

 Phosphate fertilizers Very low 1.2 1.4 1.5 Low 1.1 1.3 1.2 Medium 1.0 1.0 1.0 Elevated 0.5-0.7 0.7 0.7 High 0.2-0.3 0.3 0.5 Very high 0.2 0.2 0.3 Potash fertilizers Very low 1.0 1.3 1.5 Low 1.0 1.1 1.3 Medium 1.0 1.0 1.0 Elevated 0.3-0.5 0.5-0.7 0.7 High - 0.3 0.5 Very high - 0.2 0.3

Balance method. There are several modifications of balance methods. Of these, only two have gained the most widespread use and recognition: a) calculation of fertilizer rates based on soil nutrient reserves; b) calculation of fertilizer rates based on yield increase.

Calculation of fertilizer rates based on nutrient element reserves in the soil. The essence of this method lies in determining the fertilizer rate based on the difference between the expected nutrient uptake by the planned yield and their existing reserves in the topsoil layer. This accounts for the nutrient use efficiency coefficients from both the soil and fertilizers. Schematically, this calculation can be represented as follows:

N = (V – Z · Kp) / Ku, where: N – required fertilizer rate, kg/ha;

V – estimated nutrient uptake by the planned yield, kg/ha;

Z – nutrient reserve in the topsoil layer, kg/ha;

Kp – coefficient of nutrient use by the plant from the soil;

Ku – coefficient of nutrient use by the plant from fertilizers.

The method is very simple, and if one does not delve into its essence, it seems that it correctly reflects the calculation of the fertilizer rate to cover the needs of the crop being grown.

The drawbacks of the method include the linearity of the elementary balance equation, which results in its ability to approximate only a relatively narrow section of the "Mitscherlich curve". Practically, this means that the elementary balance equation can be used when calculating the nutrient regime for a yield not exceeding 35–40% of the potential maximum.

Calculating mineral fertilizer application rates using formulas often diverges from the actual situation in the field. Due to the variability of soil fertility, weather conditions, and biological traits of hybrids, calculated doses can deviate from the actual plant requirements by more than 50%. Errors also arise when planning yield, which depends on the entire range of agricultural practices, not just nutrition. Attempts to apply maximum rates for the sake of records lead to the overconsumption of active ingredient and a decline in yield.

• Deviation of calculated rates from requirements — more than 50%
• Estimation of actual yield — for 3–5 years
• Average potato yield — 15 t/ha
• Planned potato yield increase — 5 t/ha

Calculation of fertilizer rates for yield increase

This method is based on the fact that the basic portion of the harvest is formed by the plant due to natural soil fertility. Fertilizers are used only to ensure the planned increase. The main difference between this method and calculating for total planned yield lies in the assessment of effective fertility. Instead of soil analysis results for mobile elements, this approach uses the final product — the field's average yield over recent years.

1. Estimate the average actual crop yield and the actual fertilizer rate for the last 3–5 years.
2. Determine the planned yield increase.
3. Calculate the nutrient uptake for the planned increase.
4. Adjust the obtained rate taking into account the nutrient utilization coefficients from fertilizers.

To calculate the rate of a specific nutrient, the basic formula is used: R = R_a + ((Y_p - Y_a) * U) / C_u. In this equation, the target rate (R) is calculated based on the actual average rate (R_a), the difference between planned and actual yield (Y_p and Y_a), nutrient uptake (U), and its utilization coefficient (C_u). Each indicator requires adaptation to a specific field.

The following variables are used in the formula:

• R — target fertilizer rate, kg/ha;
• R_a — actual average fertilizer rate applied previously, kg/ha;
• Y_p — planned yield, centners/ha;
• Y_a — actual average yield for the last 3–5 years, centners/ha;
• U — nutrient uptake per 1 centner of yield, kg;
• C_u — nutrient utilization coefficient from fertilizer.

The method attracts practitioners due to its simplicity, but it has serious limitations. Its advantages include the fact that nutrient availability is assessed based on actual yields under current agricultural practices. However, the method proposes a uniform approach for all elements without accounting for the natural influx of nitrogen, phosphorus, and potassium into the soil. Because of this, it is difficult to maintain a non-deficit balance of phosphorus and potassium.

The main problem with the method is the instability of the nutrient utilization coefficient (C_u). It constantly changes under the influence of soil moisture and absorption capacity, fertilizer doses, crop biology, and its predecessors in crop rotation. It is practically impossible to accurately determine C_u for each field in the current season.

Mathematical modeling of nutritional requirements

More accurate application rates can be calculated using production functions. Mathematical models describe the quantitative relationship between doses of mineral nutrition and yield in a specific soil-climatic zone. They allow an agronomist to calculate the optimal combination of elements to obtain the maximum economic effect at any level of expenditure.

This dependency is most accurately described by a regression equation with powers of 0.5 and 1 for active ingredients and 0.5 for their pair interactions. When applying complete mineral fertilizer, the formula is as follows:
Y = a_o + a₁N^0,5 + a₂N + a₃P^0,5 + a₄P + a₅K^0,5 + a₆K + a₇(NP)^0,5 + a₈(NK)^0,5 + a₉(PK)^0,5

In this model, the indicators are decoded as follows:

• Y — expected yield, centners/ha;
• a_o — free term representing yield in the control (without fertilizers);
• a₁...a₉ — regression coefficients showing the strength of the influence of fertilizer doses and combinations on the yield.

Limitations of calculation methods and production functions

Mathematical modeling using a production function allows for a more accurate description of the Mitscherlich curve over a wide range of doses. However, in practice, this method has serious limitations. The formulas are strictly tied to the soil and weather conditions of the specific field where the initial experiment was conducted. To create a nutritional scheme even for a single crop rotation, one would have to calculate a whole set of such functions. Furthermore, this method allows for ambiguity: the formula can show the same yield for different combinations of nitrogen, phosphorus, and potassium, which contradicts the law of indispensability of growth factors.

Balance-based calculation methods are attractive due to their simplicity, but it is difficult to draw up an accurate balance under the conditions of a specific farm. Calculations usually use averaged reference indicators for nutrient uptake and their utilization coefficients from soil and fertilizers. These standards, developed for large soil-climatic zones, often do not match the real situation on a specific field. Furthermore, the balance method does not take into account the physiological yield limit of a crop: to increase grain harvest, it is not enough to simply proportionally increase NPK doses.

Below is a practical example of calculating doses of organic and mineral fertilizers for a planned yield increase. The calculation accounts for nutrient uptake, plant utilization coefficients of elements, and the initial soil nutrient availability. All indicators are summarized in a unified technological chain, from nutrient uptake to the physical weight of fertilizers.

• Nitrogen (N) uptake per 1 t of harvest — 5.0 kg
• Phosphorus (P₂O₅) uptake per 1 t of harvest — 2.2 kg
• Potassium (K₂O) uptake per 1 t of harvest — 8.0 kg

How to design an effective fertilization system on a farm

A rational fertilization system is not just a one-time application of fertilizers for a specific crop. It is a scientifically grounded plan for the distribution of organic and mineral fertilizers across crop rotation fields, which is compiled for one full rotation. Such a system solves three tasks simultaneously: increasing yield, improving product quality, and preserving soil fertility. Proper nutrient distribution becomes especially important when transitioning to intensive growing technologies.

The development of a nutrition system on a farm is built as a sequential process. An agronomist needs to consider a set of factors to minimize nutrient losses and increase cost-effectiveness.

1. Conduct an agrochemical analysis of the farm's soils, determining their type and level of element availability.
2. Study the agrobiological characteristics of crops in the crop rotation and their nutritional requirements by growth stages.
3. Assess the properties of available mineral and organic fertilizers (solubility, availability, physiological acidity).
4. Compare the needs of the fields with the actual fertilizer resources available on the farm.
5. Determine optimal methods and timing of application (basal, at-sowing, top dressing) for each crop.

A fertilization system works effectively only as an element of the overall farming system, alongside tillage and crop rotation. It should cover not only arable land but also perennial plantations, as well as the farm's hayfields and pastures. The developed nutrition scheme is adapted to the specific type of technology intensity:

• extensive;
• normal;
• intensive;
• highly intensive.