Mitscherlich s Law: Sum of two exponential Processes; Conclusions 2009, 1 st July

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1 Mitschelich s Law: Sum of two exponential Pocesses; Conclusions 29, st July Hans Schneebege Institute of Statistics, Univesity of Elangen-Nünbeg, Gemany Summay It will be shown, that Mitschelich s fomula, the cop yield as function of the fetilize, can be patitioned into two pats. One comes fom the fetilize in the soil, the othe fom the fetilize, supplied fom outside. As a consequence, the quantity of the fetilize, which is extacted fom the soil by the cop, can be computed. Coespondingly new fetilizing can be planned. Intoduction One poblem in fetilizing, on which many authos have woked, ecently was epoted on by Giebel et al. (26): Until now one may be able to deive a pognosis upon the spatial aveage of soil mineal-n content of a field. Howeve, no appoaches ae available that would allow the estimation of the spatial distibution of plant-available N ove an entie field within the same o a following yea. Nevetheless, adequate epesentation of spatial N distibution within a field is impotant to site-specific and envionmentally sustainable N fetilization. This esult of spatial aveages of soil mineal-n content of a field was obtained by Giebel et al. with samples of gound-dillings and a following analysis. Anothe poblem is the question, what pat of the cop-yield comes fom the plant-available fetilize in the soil, and what pat is caused by the given fetilize. This poblem is solved hee in quite a diffeent way without technical devices, only with analytical methods and the fact, that the Mitschelich- pocess can be patitioned into two exponential pocesses. One of these pocesses is the so-called standadized Mitschelich pocess, the othe is an exponential dying pocess. Fom the quantity of lost fetilize in the soil consequences fo futue fetilizing can be dawn. The Law of Mitschelich (99) is an exponential gowing pocess with hoizontal asymptote. This sot of pocesses is widely known in natual sciences: In biology we find it as Mendel s law of genetics, in mechanics it is the genealization of Hooke s law (Schneebege 25): the fist (linea) tem of the Taylo-seies of this genealization is Hooke s law; in electical engineeing chaging of a battey follows this law, etc. Mitschelich fist fomulated the law in fom of a diffeential equation: d / dx = ' = b(a ) () whee x is the quantity of fetilize, y the cop-yield and (as usual in statistics) the hypothetical value of y; a is the maximum value of the yield (we exclude ove-fetilization!) and b> is the most impotant exponential facto. Fomula () means, that the gowing of the cop is popotional to (a- ). With bounday condition = ) = c we have as solution of equation () bx = c + (a c)( e ) (2)

2 The bounday condition = ) = c means, that c is the cop-yield with only the plantavailable fetilize in the soil, with no supplementay fetilize, i.e. x=. See the cuve in figue. MATERIALS AND METHODS Fo ou analysis we use the esults of an expeiment with winte-wheat at the Univesity of Technology München/Weihenstephan in 22, given in table : Fo evey of the =5 values x of fetilize, s=4 values of cop-yield y s ae given. Fo claity in the figues not the expeimental points ( x, y s ) with the geat vaiance of the y-values, but ( x, y ) ae given as stas; see also the values y in table. Table Yield y (winte-wheat in kg/ha) fo given fetilize x (N in kg/ha) s x y s y continuation s x y s y The thee paametes a,b,c of the Mitschelich-cuve (2) ae estimated with the method of Least Squaes of Gauss, i.e. a, b and c ae solutions of the condition 2 2 ( y s (c + (a c)( e ))) = (y s ) s s Min (3) This poblem is solved iteatively with the non-linea Simplex-Method of Nelde and Mead (965). The esult is a=3., b=.7583, c=53.2 (a c=59.9) (4).7583 x = ( e ) (5) It can easily be shown, that computation with the values of y (instead of y) yields the same esult. The cuve, plotted in figue, is extapolated to negative values of x (dotted line). We get x = d = 83.8 as solution of ) =. d(>) is the plant-available fetilize in the soil, which

3 gives the coesponding cop-yield =c=53.2, i.e. without fetilizing fom outside. With this soil-immanent fetilize d we can wite fomula (2) also in the fom of the Mitschelich- Baule function (Baule 98). See also Fank, Beattie and Embleton (99): with d = b d) = a( e ) (6) ln b a a c (<) (7) Figue. Patition of the Mitschelich-cuve into the two pats and 2 The following theoem immediately esults fom fomula (6): Theoem: The geneal Mitschelich-cuve ) of fomula (6) with d >, o fomula (2) with c>, oiginates fom the standadized Mitschelich cuve ) = a( e ) with d= in fomula (6) o c= in fomula (2) by simply shifting the cuve ) to the left fo d > units (of vaiable x). See figue. Fo the stategy of fetilizing in the futue, to avoid too much o too less fetilizing with the known consequences, it is impotant to state, that accoding to Liebig s Law we have fo the total quantity q of plant-available fetilize in the soil: q d. Patition of Mitschelich s Fomula We can wite fomula (2) in the fom = a( e ) + ce = + 2 (8)

4 The pats and 2 ae of specific impotance: a) =a(-e ) (9) is the standadized Mitschelich pocess, beginning at the oigin, )=(,), i.e. c=. So this pocess is geneated alone by the fetilize, supplied fom outside. b) 2 ce = () is a diminishing exponential pocess, stating fom 2 =)=c. So 2 ) is the cop-yield, caused by the plant-available fetilize in the soil alone. Note the emak to fomula (2)! Remak: Pocesses of the fom (b) ae the most impotant dying pocesses in natual sciences (e.g. adioactive decay), biology, medicine, economics, etc. Hee one bette would speak of an exhaustion pocess (of the fetilize in the soil). In figue the cuves and 2 ae plotted as dashed lines. RESULTS AND DISCUSSION If the quantity of fetilize fom outside is zeo (i.e. x=), we have = and the cop is = 2 = c = 53.2 ( kg/ha of winte-wheat), which comes alone fom the fetilize in the soil d=83.8 (kg/ha of N). Now we assume, that fo example a quantity of fetilize x = x = kg/ha of N is given fom outside. Then we get fom fomula (9) the quantity of cop o ), which comes fom the given x and with fomula () we get the quantity of cop 2 ), which comes fom the fetilize in the soil. The total cop is ) = ) ). We get + 2 ) = 6.; 2 ) = 24.9, ) = 85. The poblem now is: What quantity of fetilize in the soil gives the cop 2 )? This is found with the auxiliay Mitschelich pocess = c + (a c )( e ); = ) () c 2 = which yields x= d = as solution of ) =. d=32.8 (kg/ha) is the soil-immanent plant-available pat of fetilize, which, togethe with the fetilize x = (kg/ha) fom outside, yields the total cop ) = 85. ( kg/ha). See figue 2. Note: Equation () can also be witten in the equivalent fom: with b d ) = a( e ) (a) a d = ln < b a c

5 Figue 2: Example of fetilizing with x = kg/ha of N. Calculation of the quantity of fetilize d = kg/ha of N, supplied by the soil A necessay supplement follows with Pape 2: Mitchelich's Law: A Supplement, ACKNOWLEDGEMENT I have to thank D. Maidl of the Univesity of Technology München (Weihenstephan)/Gemany fo poviding me with the data, used in this aticle. REFERENCES Baule, B. (98). Zu Mitschelichs Gesetz de physiologischen Beziehungen, Landwitschaftliche Jahbüche 5, Fank M.D.,Beattie B.R. and Embleton M.E. (99). A Compaison of Altenative Cop Response Models, Ameican Jounal of Agicultual Economics, 72, Giebel, A. et al.(26). How epesentatively can we sample soil mineal nitogen? Jounal of Plant Nutition and Soil Sciences, 69, Mitschelich, E.A.(99). Das Gesetz des Minimums und das Gesetz des abnehmenden Bodenetags, Landwitschaftliche Jahbüche, 38, Nelde, J.R. and Mead, R. (965). A Simplex Method fo function minimization The Compute Jounal, 7, Schneebege, H. (25). Theoy of the stess-stain elationship of concete and steel. In: R.K.Dhi, M.D.Newlands and L.J.Csetenyi (eds.): Applications of Nanotechnology in Concete Design. Poceedings of the Intenational Confeence at the Univesity of Dundee, Scotland, UK, 5-2.

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