If only one fertilizer x is used, the dependence of yield z(x) on x first was given by Mitscherlich (1909) in form of the differential equation

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1 Mitsherlih s Lw: Generliztion with severl Fertilizers Hns Shneeerger Institute of Sttistis, University of Erlngen-Nürnerg, Germny 00, 5 th August Astrt: It is shown, tht the rop-yield z in dependene on two fertilizers x nd y is the produt of two omponents: z in dependene on x lone nd z in dependene on y lone, divided y, the yield without extern fertilizers, i.e. with xy0. For n fertilizers, we hve the n produt of n omponents, divided y. Introdution If only one fertilizer x is used, the dependene of yield z(x) on x first ws given y Mitsherlih (909) in form of the differentil eqution dẑ dx ( ẑ) () where is the symptoti vlue of z, the ftor of proportionlity. As usul in sttistis, ẑ is the hypothetil vlue, z the experimentl vlue of the rop-yield. For eqution () it is ssumed: No over-fertiliztion. For the se of overfertiliztion see Shneeerger (009). Solution of formul () with oundry ondition ẑ (x 0) is Mitsherlih s urve in the espeil instrutive form x ẑ(x) + ( )( e ) () demonstrted in figure with 0.5,., 0,75. For estimtion of the prmeters in prtie see Shneeerger (009). Figure : Crop-yield ẑ(x) s funtion of one fertilizer x

2 An other form of formul () is: used in the following. ẑ(x) x ( )e () Bule (98) gve the solution of eqution () in the form (x d) ẑ ( e ) (3) with d ln (<0) (3) in figure we hve d Generliztion Now we ssume two vriles (fertilizers) x nd y (see figure ) Figure : Crop-yield ẑ (x, y) s funtion of two fertilizers x nd y

3 Then we hve for the rop-yield ẑ (x, y) with formul () ẑ(0, y) nd ẑ(x,0) y ( )e (4) x (x) ( )e (4) Note: For short we write (y 0), (x 0), (y 0) (x 0), (y 0), (x 0), d (y 0) d, d (x 0) d. Generlizing formule (4) we hve ẑ(x, y) ẑ(x, y) (x)y (x) ( (x) (x))e (5) x ( )e (5) nd herewith (x) x (6) (x) (x) (x)y e (x) e Now I mke use of result of Mitsherlih (947): given different fertilizers, the prmeter (in Mitsherlih s nottion) is onstnt for fixed fertilizer, s I ould prove in tens of yers of work. Mitsherlih s Wirkungsgrd is our prmeter exept for the onstnt ln0. This mens: Herewith we get from formul (6): nd (x) independent of y resp. x (7) is independent of y, i.e. (y 0) (8) (y 0) nd (x) independent of x (x) (8) nd with this d independent of y (9) ln ln d (x) d nd d independent of x. (9) Finlly we hve from formul (5) (or (5)): y y ẑ(x, y) (x) (x)e (x)( ( )e ) (x) (0) or in the most instrutive form x y ẑ(x, y) ( + ( )( e ))( + ( )( e )) ()

4 RESULT: The generlized Mitsherlih formul in two vriles is the produt of the onedimensionl formule, multiplied y /. With formule (9) one n show, tht formul () is identil with the formul of Bule (98). ẑ(x, y) (x d ) (y d ) A( e )( e ) with () A Equtions (0) nd () n esily e generlized for n fertilizers: Applition ẑ(x,...x ) ẑ(x,0,...0)ẑ(0, x,...0)...ẑ(0,0,...x n ) (3) n n The following dt re from n exmple of Steinhuser, Lngehn nd Peters (99) with x (in 00 kg/h of P O 5 ), y (in 00 kg/h of O), z (in 000kg/h of rye). K Tle : Crop-yield ẑ (x, y) in 000 kg/h of rye, x in 00 kg/h of P O 5, y in 00 kg/h of K O x 0.5 x x x x 5. 5 x y y y y y y The 5 prmeters,,,, were determined with the method of Lest Squres of Guss f (,,,, ) (z(x, y) ẑ(x, y)) Min (4) x y summing over ll 36 dt-points. The minimum ws gined itertively with the non-liner Simplex-Method of Nelder nd Med (965). The result is 0.75, 0.947, 0.899,.9438,. 07 nd ẑ 3.683( ( e 0.899x ))( y.673(- e )) In figure 3 the ontour-lines ẑ (x, y) 0., 0.4, 4.0 re drwn, in figure 3 interseting urves ẑ (x, y onst.) for y d,0, 0.5,.0,.5 nd re plotted. It is ovious tht they re Mitsherlih-urves. We hve d 0.376, d The symptotes of the urves ẑ (x, y ) re horizontl dotted stright lines in figure 3. Espeilly for y we get

5 ẑ(, ) 6.78, Bule s prmeter A in formul (). ẑ(x, y ) is urve (x) of figure. In nlogy interseting urves for xonst. ould e plotted. Figure 3: Contour-lines ẑ (x, y) onst. - in 000 kg/h of rye, x in 00 kg/h of P O 5, y in 00 kg/h of K O

6 Figure 3: Mitsherlih-urves ẑ (x, y onst.) Exmple: With fertilizer 00 kg/h of P O 5 (x) nd 50 kg/h of K O (y0.5) we get the rop-yield 335 kg/h of rye ( ẑ.335).. With xy0 (i.e. without externl fertilizers) we would get 7 kg/h of rye ( ẑ 0.7). Aknowledgement I hve to thnk Dr. Emher, Munih, who helped me to pulish these ppers in the internet. A generliztion with overfertiliztion is given in pper 5 (Pper 5: Mitsherlih's Lw: Generliztion with severl Fertilizers nd Overfertiliztion) Referenes Bule B. (98). Zu Mitsherlihs Gesetz der physiologishen Beziehungen, Lndwirtshftlihe Jhrüher 5, Mitsherlih E.A. (909). Ds Gesetz des Minimums und ds Gesetz des nehmenden Bodenertrgs, Lndwirtshftlihe Jhrüher 38, Mitsherlih E.A. (947). Ds Ergenis von üer 7000 Feld-Düngungsversuhen, Zeitshrift für Pflnzenernährung, Düngung, Bodenkunde 38. Bnd, 947, Verlg Chemie Nelder J.R. nd Med R. (965). A Simplex Method for funtion minimiztion. The Computer Journl 7, Shneeerger H. (009). Mitsherlih s Lw: Sum of two Exponentil Proesses. Conlusions. Internet: pper nd pper Shneeerger H. (009). Over-Fertiliztion: An Inverse Mitsherlih Proess. Internet: pper 3 Steinhuser H., Lngehn C. nd Peters U. (99). Einführung in die lndwirtshftlihe Betrieslehre, Bnd, 5.Auflge