Spline Response Functions for Direct and Carry-over Effects Involving a Single Nutrient

Two continuous models which interpret the principles underlying the law of the minimum in crop response to an essential nutrient are presented. Such single-nutrient response functions are the linear response and plateau model (LRP), and the linear, quadratic and plateau model (LQP). A spline algorithm was developed to estimate these functions. Both functions were applied to data from a field experiment carried out in Salta, Argentina, to study the response of three cuts of Rhodes grass, Chloris gayana Kunth, to N fertilization with urea. Furthermore, two associated models for carry over of available N in the soil, as well as carry-over effects on yield, either within a growth season or between consecutive seasons, were developed. These are the platform and linear response model (PLR), and the platform, quadratic and linear response model (PQL). The LQP model, and its counterpart for carry over (PQL), imply the existence of an apparent, although restricted substitution between the limiting nutrient and the next potential limiting factor, whereas the LRP and PLR models are based on the non-substitution assumption. It is emphasized that the restricted substitution of the LQP and PQL models is strongly situation specific, and that they can collapse into LRP and PLR fits without a quadratic domain, as they often did in this study. The quadratic polynomial equation was used throughout as a reference model with extended substitution. At worst, the spline functions performed equally well as the quadratic polynomial, and in some situations they provided a substantial increase in the goodness of fit. Hence, it is reconfirmed in this study that, while polynomial responses tend to overestimate the optimum fertilizer rate, the LRP and LQP models are better suited to provide more realistic optimum levels of fertilization. The carry-over effect on yield was found to be more relevant within a growth season (R² = 0.58) than between seasons (R² = 0.38). Both the PLR and PQL models were able to explain 90 % of the variability in residual N in the soil as a function of the level of urea applied at the beginning of the growth period. The estimated minimum levels of fertilization required to attain the plateau yield showed a tendency to decrease as a result of the carryover effect. These doses were 430 and 250 kg N ha⁻¹ for the two cuts of the first season, and 223 kg N ha⁻¹ for the first cut of the second season. The methodological contributions of this paper are that: (i) it provides continuity for the LRP model and (ii) the knot(s) of the spline functions are endogenously obtained through an iterative procedure which maximizes the goodness of fit.