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Abstract

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.

Introduction to Generalzed Overfertilization

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 by₁(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 a₁ and a₂ 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, b₁, b₂, 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.

Line 2 repeats the results of paper 3. The good coincidence is a good sign for the hypothesis a₁=a₂=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

Overfertilization with two fertilizers

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):

• in section (1,1) there is overfertilization not for x, not for y
• in section (2,1) there is overfertilization for x, not for y
• in section (1,2) there is overfertilization not for x, for y
• in section (2,2) there is overfertilization for x, for y

For clearness I must modify the symbolic of formulae (1)

• for variable x before overfertilization
• for variable x with overfertilization
• for variable y before overfertilization
• for variable y with overfertilization

Then for estimation of the nine parameters c, a₁, b_x1, b_x2, x_B, a₂, 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 (a₂=a₁, 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.

Remark

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.

References