Modified‐Quadratic/Plateau Model for Describing Plant Responses to Fertilizer

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Abstract
Accurate estimation of optimal fertilizer rates is becoming more critical because of growing economic and environmental concerns associated with fertilizer use. Curve‐fitting techniques are often used to estimate optimal fertilizer rates, but significant problems exist in selecting a proper model for a particular cropping situation. One solution is to develop a model sufficiently general to preclude the need for selecting a proper model for each cropping situation. We developed a modified‐quadratic/plateau (MQ/P) segmented model (four or five parameters) and demonstrated its general applicability compared with the quadratic/plateau (MQ/P) segmented model (three parameters) for describing fertilizer responses. The MQ/P model contains an efficiency index (Ex) that is a quadratic function of the applied rate of nutrient X. In the Q/P model, Ex remains constant relative to the rate of nutrient X. The MQ/P and Q/P segmented models set yield equal to the maximum yield (Ym) when the rate of nutrient X is greater than that required to achieve Ym. The general applicability of the MQ/P model in comparison with the Q/P model was evaluated by determining how well these models fit hypothetical input data calculated with linear/plateau (L/P), Mitscherlich (M), and square root/plateau (SR/P) functions. The MQ/P model gave R2 values >0.99 and predicted economically optimum fertilizer rates (i.e., rates at which the first derivative of yield response functions equaled the fertilizer/crop price ratio) close to those calculated with the input functions. The Q/P model gave R2 values ranging from 0.95 to 0.99, but gave poor predictions of economically optimum fertilizer rates. The ability of the MQ/P model to fit functions as diverse as L/P, M, and SR/P, and to predict economically optimum N rates associated with these functions, suggests that the MQ/P model has utility as a general curve‐fitting technique for responses in which the right‐hand segment is a plateau. An advantage of the MQ/P model is that the regression parameters can be interpreted as physically meaningful constants.

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