Fitting the Law of the Minimum to Fertilizer Applications and Crop Yields1
- P. E. Waggoner and
- W. A. Norvell2
Liebig's Law of the Minimum is a simple, logical, and successful predictor of crop response to fertilization. A statistical method is reported for fitting the Law by least squares regression to yield responses to two essential factors, with optional inclusion of a limit to maximum yield.
The Law maybe written: y = MIN(a+bq, c+dz2 m); where y is yield, z1 and z2 are two essential nutrients, and m is a maximum imposed by another factor. The expression MIN( ) defies y as the minimum of the specified values; and a,b,c and d are the intercepts and slopes relating y to z1 and z2. The parameters were estimated from crop responses to fertilizer by searching for the minimum residual sum of squares: where for each treatment with yield y1, X1 = z2 if c+dz2 ≤ a+bz1 and m, X1 = (a−c+bz1/d if a+bz1 < c+dz2 and m, X1 = (m−c)/d If m < a+bz1 and c+dz2.
This logical law fit yields of corn that received N and P fertilizer more closely than three empirical functions, and the standard error of estimate for the law was within 14% of that of the most closely fitting function for two legumes that received K and P. Where the Law of the Minimum is logical, it can now be fitted statistically to observations, identifying nutrients limiting each yield and estimating the return from added nutrient.Please view the pdf by using the Full Text (PDF) link under 'View' to the left.
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