Institute of Statistics, University of Erlangen-Nürnberg, Germany

2010, 1

It was shown in Paper 1 (Mitscherlich's Law: Sum of two exponential Processes. Conclusions), that Mitscherlich's curve: crop ŷ in dependence on fertilizer x, can be partitioned into two exponential processes ŷ_{01} and ŷ_{02}. It will be shown that this partition is indeed of special importance, but it is not the only possible one. There is one further partition of special importance, called here ŷ_{11} and ŷ_{12} , and with these two special partitions an infinite number of others is given.

The "experiment", seed and crop, with certain soil, seed, fertilizer etc., as described in Paper 1, in general doesn't result in partition ŷ_{01} and ŷ_{02} (this must be corrected), but in one of these infinite partitions. If this one would be known, the loss of soil-immanent fertilizer by the crop and the degree of exploitation of the external fertilizer could be calculated.

In Paper 1 already one particular partition of ŷ = ŷ_{01} + ŷ_{02} with

(9)

(10)

is given, see figure 1 and figure 3. ŷ_{01} is the crop from the external fertilizer, ŷ_{02} from the soil-immanent fertilizer. ŷ_{01} is an exponential growing process with asymptote ŷ = a, ŷ_{02} an exponential declining process with asymptote ŷ =0. So ŷ_{01} makes most (crop) of the external fertilizer. ŷ_{02} makes least use of the soil-immanent fertilizer; (9) and (10) is the optimal partition.

The contrary is the case with the partition ŷ = ŷ_{11} + ŷ_{12} with

(12a)

(12b)

see figure 3. Now the crop from the external fertilizer ŷ_{11} has the asymptote ŷ = a - c, much worse than with ŷ_{01}; ŷ_{12} = c means, that a maximum of the soil-immanent fertilizer is spent, independent of the quantity of the external fertilizer x. (12a/b) is the poorest partition.

Figure 3: Partition of Mitscherlich's curve ŷ into two components ŷ_{i1} and ŷ_{i2} (i=0,1,2,...)

But with partitions (9), (10) and (12a), (12b) we have an infinite number of further partitions ŷ = ŷ_{•1 + }ŷ_{•2} ( • for 2, 3, ...):

(13a)

(13b)

(0 ≤ α ≤ 1)

α = 0 gives the poorest, α = 1 the optimal partition. I will call α parameter of affinity.
Because of ŷ_{•1} = ŷ_{11} + α(ŷ_{01} -ŷ_{11} ) for formula (13a), ŷ_{•1}(x) for fixed x grows from ŷ_{11}(x) to ŷ_{01}(x) with parameter α in a linear way. The same is right with ŷ_{•2}. See the α-scala in figure 3. In addition the curves ŷ_{•1} = ŷ_{21} and ŷ_{•2} = ŷ_{22} for α = 0.7 are plotted there.

For a special "experiment" a certain value of α will exist. The knowledge of this α would be of great importance for the knowledge of the loss of soil-immanent fertilizer and the effectiveness of the external fertilizer x. The greater α, the better for both results.

If α would be known (e.g. α=0.7, we would have for given x (e.g. 200 kg/ha of N):ŷ_{22}(x) = c_{2}= 24.13(100kg/ha) of winter-wheat, and herewith d_{2} according to formula (11a) of paper 1: -d_{2} = 31.65 (kg/ha) of N is the soil-immanent fertilizer, needed for the total crop ŷ(x=200)= 100(100kg/ha) of winter-wheat. In table 2 results for some further values of α are given.

α | c_{2} |
d_{2} |
---|---|---|

0 | 53.20 | -83.80 |

0.60 | 28.28 | -37.95 |

0.65 | 26.20 | -34.76 |

0.70 | 24.13 | -31.65 |

0.75 | 22.05 | -28.60 |

0.80 | 19.98 | -25.63 |

1 | 11.67 | -14.36 |

In reversed direction we get α from d_{2}. So the problem of finding α is that of determining
the value of d_{2}.

**continued 09.01.2012**

To demonstrate the dependence of ŷ_{1} and ŷ_{2} (in short for ŷ_{•1} and ŷ_{•2}) on the parameter α, figures 4a, 4b ... 4e give the curves ŷ_{1}, ŷ_{2} and ŷ = ŷ_{1} + ŷ_{2} for α = 0, 0.25, 0.50, 0.75, 1.

Fig. 4a: α = 0; c_{2} = 53.2, d_{2} = -83.8

Fig. 4b: α = 0,25; c_{2} = 42.8, d_{2} = -62.7

Fig. 4c: α = 0.5; c_{2} = 32.44, d_{2} = -44.6

Fig. 4d: α = 0.75; c_{2} = 22.1, d_{2} = -28.6

Fig. 4e: α = 1; c_{2} = 11.7, d_{2} = -14.4

Fig. 5: α = 0.75; c_{2} = 32.0, d_{2} = -43.8

The aim of external fertilization and soil-care must be maximizing ŷ_{1} and minimising ŷ_{2}, or in short, maximizing α.

How can the value of a fertilizer-soil combination be computed?

With the original data, given in Paper 1, this cannot be done. For that the registration of (at least) one pair of data (x_{0}, d_{2}(x_{0})), for example for x_{0}=200, is necessary; -d_{2}(x_{0}) > 0 is the quantity of soil-immanent fertilizer, which gives the part of crop ŷ_{2}(x_{0}) = c_{2}(x_{0}).
Δ = - d - (-d_{2}(x_{0})) > 0 is the soil-immanent fertilizer after crop, which can be measured for example with a chemical analysis - as I assume. Herewith we get d_{2}(x_{0}) = d + Δ. Then with d_{2}(x_{0}) the value of c_{2}(x_{0}) - signed in the figures as c_{2} - is found as solution of

c_{2} + (a - c_{2})(1-e^{-bd2}) = 0 or c_{2} = a(1-e^{bd2}) = ŷ_{2}(x_{0}) (14)

and herewith (see figure 3)

(15)

If for example, an experiment with the above soil-fertilizer combination for external fertilizing with x_{0} = 200 gives the value d_{2} = -62.7, then the affinity is α = 0.25 (see Fig. 4b). d_{2} = -28.6 gives α = 0.75 (Fig. 4d). One can see from figures 4b and 4d, that

- The soil-fertilizer combination with higher value of α yields by far the better exploitation of the external fertilizer (see curves ŷ
_{1}): for x_{0}= 200 there is ŷ_{1}(α=0.75)/ŷ_{1}(0.25) = 77.89/57.13 = 1.36; that means, that with the same quantity of external fertilizer x_{0}= 200 the exploitation is 36 % higher. - To yield the same total crop ŷ, the part of crop ŷ
_{2}from the soil-immanent fertilizer reduces to about one half in our example: for x_{0}= 200 we get ŷ_{2}(α = 0.75)/ŷ_{2}(α = 0.25) = c_{2}(α = 0.75)/c_{2}(α = 0.25) = 22.1/42.2 = 0.52. This means: Only 45.6 % of the soil-immanent fertilizer (d_{2}(α = 0.75)/d_{2}(α = 0.25) = 28.6/62.7 = 0.456) is spent in a soil-fertilizer combination with α = 0.75 against one with α = 0.25. The quantity of external fertilizer is the same in both cases: x_{0}= 200.

I think, this is active soil-conservation.

With n test points (i = 1,...,n) instead of the one x_{0} we get values α, which are all estimates of the same "true" &alpha (cf. figures 4d and 5 (with x_{0} = 100)). So our final estimate of α then is Σ α _{i} / n. For gaining the n experimental values α_{i} of course the assumptions of physical experiments must be fulfilled: All experimental parameters are constant, only the value of x is varied. This will be hard to realise in agronomy.

Further interesting questions would arise by changing another parameter, for example the soil, etc.

I praise the internet! It gives the possibility to publish works, which are out of the ordinary. I choose this way, after my first paper was returned with the comment "is in its contents fully out of the scope of the journal" (translated from German).

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