Formulae and graphics for overfertilization with several fertilizers are given. It is shown, that a result, similar to Liebig's Law is valid: If by overfertilization with one fertilizer the crop-yield becomes zero, it remains zero, independent of the other fertilizers.
Schneeberger (2009b, paper 3) presented a theory of overfertilization for one fertilizer x. The ascending part I of the fertilizer-yield curve is a Mitscherlich curve
or according to Baule (1a)
and the descending part (1b)
is an inverse Mitscherlich curve. See the symmetry of (1a) and (1b) in Baule's form! x_{A} and x_{B} are characterized by_{1}(x_{A})=0 and _{2}(x_{B})=0.
The parameters of formula (1a) and (1b) are calculated in paper 3 separately with the data of the ascending part I and with the data of the descending part II. The result led to the hypothesis: The parameters a_{1} and a_{2} are identical (=a), which is already realized in formulae (1a) and (1b). So one in future will make use of this hypothesis and estimate the five parameters c, a, b_{1}, b_{2}, x_{B} of formulae (1a) and (1b) together by minimizing according to Gauss the combined function
(2)
with the nonlinear Simplex-Method of Nelder and Mead (1965). z_{1i} resp. z_{2i} are the experimental values of the crop-yield with fertilizer x_{1i} resp. x_{2i} in I resp.II, _{1i} resp. _{2i} the corresponding hypothetical values. The result is in table 1, line 1.
c | a | b_{1} | b_{2} | x_{B} | |
---|---|---|---|---|---|
line 1 | 62.75 | 112.6 | 0.0187 | 0.0122 | 418.9 |
line 2 | 62.61 | 111.54 | 0.0198 | 0.01197 | 413.7 |
Line 2 repeats the results of paper 3. The good coincidence is a good sign for the hypothesis a_{1}=a_{2}=a. Figure 3 shows this result (cf. also figure 1 in paper 3).
Figure 3: Crop-yield (x) with parameters of table 1, line 1
For illustration I confine to two fertilizers x and y; the further generalization is obvious. Schneeberger (2010 b, paper 4) gave for the case of no overfertilization the generalization of Mitscherlich's formula for two fertilizers x and y in the form
(3)
see there the illustration of (x,y)=const. as contour-lines in figure 3a.
For the generalized case - two variables and overfertilization - formulae (2) and (3) are combined. For this we divide the (x,y)-plane in four sections (see figure 4):
For clearness I must modify the symbolic of formulae (1)
Then for estimation of the nine parameters c, a_{1}, b_{x1}, b_{x2}, x_{B}, a_{2}, b_{y1}, b_{y2}, y_{B} we minimize the function
(4)
with the method of Nelder and Mead (1965). Summation is taken over the points of fertilizing (x_{i},y_{i}) of the respective section (11), (21), (12), (22). With the resulting parameters we get the crop-yields (x,y)in the four sections:
(5)
I regret it very much, that I have only the data of the example in paper 3, for overfertilization with one fertilizer, no data for two fertilizers. But I think that I can demonstrate the essential characteristics by choosing the parameters of variable x (those of table 1, line 1) also as parameters of variable y (a_{2}=a_{1}, b_{y1}=b_{x1}, b_{y2}=b_{x2}, y_{B}=x_{B} ) . As result we get the contour-lines of formula (5) in figure 4.
Figure 4: Contour-lines (x,y)=const.
The symmetry to the bisector of the first quadrant of course comes from the symmetry of the parameters in x and y.
A conclusion, which can be drawn from figure 4: If overfertilization for one variable (=fertilizer) yields =0, then remains zero, independent of the other fertilizers. This is an analogous statement to Liebig's Law, but now at the end of the fertilizing process.
It was assumed, that overfertilization ends with process (1b). The author tends to this hypothesis. In paper 3 another hypothesis with a part III of exponential dying was discussed. Then overfertilization would be generalized in analogous way. But the decision on the "right" hypothesis must be found with experiments.
Nelder, J.R. and Mead, R. (1965). A Simplex Method for function minimization. The Computer Journal 7, 303-313
Schneeberger, H. (2009b). Overfertilization: An Inverse Mitscherlich Process. Internet: http://www.soil-statistic.de, paper 3
Schneeberger, H. (2010b). Mitscherlich's Law: Generalization with several fertilizers. Internet: http://www.soil-statistic.de, paper 4
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