A Differential Equation to Estimate Fertilizer Response Curves
1988
Cochrane, T. T. (Thomas T.)
Second order polynomial equations are generally used to estimate fertilizer response curves. However when new crops are introduced, and especially in the lesser developed tropics, experimental data are often inadequate for their application; consequently alternative functions might be considered. This study was conducted to develop a simple differential equation to predict fertilizer response. The equation was formulated on the hypothesis that the slope at any point along the response curve, but beyond the inflection point if it is sigmoidal in shape, is proportional to the difference between the crop yield at that point and maximum yield; dY/dF = k(Y - Yₘ), in which Y = yield, F = fertilizer increment from a chosen initial condition point at or beyond the inflection point, k = a constant and Yₘ = maximum yield. The equation was explained with the help of a theoretical corn (Zea mays L.) response to N curve, and was tested successfully against the response data of fertilizer trials recorded for sugar-cane (Saccharum cv. L.), bananas (Musa cv. L.), potatoes (Solanum tuberosum L.), rice (Oryza sativa L.), bean (Phaseolus vulgaris L.) and Panicum maximum (L.) grass. Predictions of yields along the response curves, compared with equivalent yields of the recorded curves, exceeded r² = 0.995. No advanced mathematical skills are needed to use the equation, which is easily derived and applied with only the help of a hand-held calculator with an Inx key. Further, it is readily incorporated into profit equations for estimating economically optimal fertilizer dressings. As the equation may be derived using data that is too tenuous for other functions, it may warrant wider testing.
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