Derivation and Application of a Method for Determining Minimum Recommended Rates of Fertilization1
- John Pesek and
- Earl O. Heady2
Where the cost of fertilizer applied in the field can be expressed as the cost of application plus the cost of fertilizer, it is important to determine the rate which gives the highest net return per dollar invested. This rate describes both the economic minimum rate of fertilizer application and the lower limit that should be included in making agronomic fertilizer recommendations.
If yield increase, Y1, is expressed in the form of the quadratic equation, Y1 = sx + tx2, where x is the rate of fertilizer and s and t are constants; and the cost of fertilizer applied, Y2, is given by Y2 = m + rx, where m is the fixed application cost per unit area, and r is the price ratio of a unit of fertilizer to a unit of yield increase; then it is necessary to maximize the relationship by setting d/dx (Y1/Y2 − 1) = 0 in order to derive the rate of fertilization that will produce the highest profit per dollar invested in fertilization.
A direct solution of the quadratic functional relationship above can be made and leads to while a method of successive approximations may be used to solve relationships based on exponential functions of the Mitscherlich or Spillman form.
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