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
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-0.79. 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
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 )) ()
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 0. 50 x 3 0. 75 x 4. 00 x 5. 5 x 6. 50 y 0.5.00..4.58.73.87 y 0.50.4.79.09.34.55.73 y 3 0.75.7..59.90 3.5 3.35 y 4.00.00.55.98 3.3 3.59 3.8 y 5.5..8 3.9 3.65 3.94 4.8 y 6.50.4 3.05 3.55 3.93 4.4 4.50 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(0.75 + 0.6757( e 0.899x ))(0.75 +.07y.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 --0.46. The symptotes of the urves ẑ (x, y ) re horizontl dotted stright lines in figure 3. Espeilly for y we get
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
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, 363-385 Mitsherlih E.A. (909). Ds Gesetz des Minimums und ds Gesetz des nehmenden Bodenertrgs, Lndwirtshftlihe Jhrüher 38, 537-55 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, 303-33 Shneeerger H. (009). Mitsherlih s Lw: Sum of two Exponentil Proesses. Conlusions. Internet: www.soil-sttisti.de, pper nd pper Shneeerger H. (009). Over-Fertiliztion: An Inverse Mitsherlih Proess. Internet: www.soil-sttisti.de, pper 3 Steinhuser H., Lngehn C. nd Peters U. (99). Einführung in die lndwirtshftlihe Betrieslehre, Bnd, 5.Auflge