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

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 = b 0 + b 1 X + b 11 X 2 was used to express the relationship between yield and the nutrient. The new yield potential equation urn:x-wiley:03615995:saj2sssaj196203615995002600020022x:equation:saj2sssaj196203615995002600020022x-math-0001 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 urn:x-wiley:03615995:saj2sssaj196203615995002600020022x:equation:saj2sssaj196203615995002600020022x-math-0003 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.