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Inefficiency of Fertilizer Use Resulting from Nonuniform Spatial Distribution: I. Theory¹

ABSTRACT

Nonuniform physical distribution of a fertilizer nutrient was expressed as a function of space coordinates and substituted into an equation which characterizes yield response to that nutrient. The resulting differential equation was then integrated over the area fertilized to give a new yield potential function which is valid under any specified nonuniformity restriction. In particular, a periodic cosine function with amplitude "a" was selected to represent the spatial distribution function for a single fertilizer nutrient, while a quadratic equation of the form Y = b₀ + b₁X + b₁₁X²was used to express the relationship between yield and the nutrient. The new yield potential equation
was derived from these particular functional forms. Yield loss associated with the cosine distribution pattern then was obtained as the difference
between the ideal and the restricted production equations.

A corresponding derivation was made for the case of n variable factors of production under the assumption of a quadratic response equation and distinct cosine laws for physical distribution of each variable. Yield losses were expressed in the form
where b_ij is the coefficient for interaction of the i'th and j'th factors, and a_i and a_j are respective amplitudes of the cosine distribution funcations of the i'th and j'th nutrients.

¹ Journal Paper No. J-4023 of the Iowa Agr. and Home Econ. Exp. Sta., Ames. Project No. 1189. Research jointly sponsored by the National Science Foundation and the Department of Agronomy.

² National Science Foundation Graduate Fellow, and Professor of Soils, respectively.

Received for publication February 6, 1961. Accepted for publication April 5, 1961.