Vol. 6 (1997): 151-160. Long-term fertilizer field trials: comparison of three mathematical response models Stefan T. Bäckman Department ofEconomics and Management, P.O. Box 27, FIN-00014 University ofHelsinki, Finland, e-mail: Stefan, backman @helsinki.fi S. Vermeulen, V.-M. Taavitsainen Kemira Agro, Luoteisrinne 2, FIN-02771 Espoo, Finland Accession to the European Union caused a drop of nearly 60 per cent from 1994 to 1995 in prices of wheat, barley and oats in Finland. The economic use of fertilizer therefore decreased accordingly. To calculate the effect of the price changes on the economic optima, the physical production function must be known. Three physical production functions, the quadratic, the linear response and plateau (LRP) and the exponential function were estimated for this purpose. The models differed little in respect of the R >dj value (0.82-0.90) but the calculated optimum varied, depending on the production function. Data on a long-term field trial (21 years) were analysed. The field trial was established in 1973 to demonstrate the effect of mineral fertilizer in crop production. The crops grown in the trial were barley, wheat and oats. Different varieties were included in the models. Key words: economic optimum, nitrogen, wheat, barley, oats, Finland ntroduction Fertilizer field trials yield a vast amount of data that cannotbe analysed without the use of a prop- er mathematical model. Fertilizer recommenda- tions and political decisions on fertilizer use de- pend directly on the parameters estimated from the data. However, in agronomy there is no stand- ard system for choosing between models to rep- resent the relationship between fertilizer input and yield formation (Cerrato and Blackmer 1990). All the models chosen, the quadratic model, the exponential model usually known as the Mitscherlich model, and a modified form of the plateau model, are commonly used in crop re- sponse analyses (Bock and Sikora 1990, Cerra- to and Blackmer 1990, Frank et al. 1990, Paris 1992, Sumelius 1993). Even though the models give comparable R 2 values, they may give dif- ferent optimal fertilizer rates. Optima based on the quadratic function or on the exponential func- tion have been criticised for giving excessively high optimal fertilizer rates. A model based on a linear response and plateau (LRP) function gives © Agricultural and Food Science in Finland Manuscript received June 1996 151 AGRICULTURAL AND FOOD SCIENCE IN FINLAND https://www.c-info.fi/en/info/?token=mmED3UU_Y5ijMtEe.hpLoJM4gRugoBumCqo-1dg.YuV-UvVqwNhLKsYYjCZY_pUglGt5040--l0ILvsggqc_jYex_dtAz-krzDdAyiSJE2zwPohzz4g-0VfbAaW30ljK2xo04ni3R4XYquouyb5c6Rs4WE0_qk6QX6f6OjBbK2hWjEzhwGBFsuXhdqacErNXSFGIXivwJsMSsi8nNOEhdnJdIYbBhYj755r9JDlXY7GhQt6WzQ9rEQGTaqqD19H-EuF-qKDW8PJeAFhss7KTk5HLoMurWROTUerPzibxBkBvtcEuekZ3YzEw9UHKnmdhIyS854bBqtXyilsAPsiaMQFxoxCAAz39eHZ5ZQw Bäckman, S. T. et al. Fertilizer trials: mathematical response models Table I. Potential benefits and drawbacks of the LRP, quadratic and exponential models. LRP Quadratic Exponential Fit with theory von Liebig law of diminishing return Mitscherlich Fit with existing recommendations lower or equal higher higher Computing feasibility very complex easy complex Diminishing phase no yes no Symmetric no yes no Used by extension service seldom often seldom lower optima (Paris 1992). Frank et al. (1990) found that the costs of wrong specification of optimal input amounts are substantial. They claimed that neither polynomial nor plateau growth should be assumed a priori on agronom- ic grounds. Paris (1992) pointed out that most people interpret the von Liebig hypothesis as a linear relation to limiting nutrient, and that this interpretation is not consistent with von Liebig’s concept. According to Paris, the von Liebig hy- pothesis claims that there is a direct, but not nec- essarily linear, relation to the limiting nutrient. Incorrect specification of optimal input has not only economic but also environmental im- pacts. Excessive application rates lead to nutri- ent losses to the environmentby leaching or vol- atilization. Miettinen (1993) studied the effect of fertilizer policy and leaching and found that imposition of input quotas caused farmers the lowest additional costs in barley production. The loss of profits was greater with other measures and the decrease in nitrogen leaching was small- er. Evaluation of these input quotas (or nutrient quotas) again requires the use of mathematical models. We here compare three models (quadratic, LRP, exponential) in an effort to establish their benefits and drawbacks and to calculate optima according to them. Some of the potential bene- fits and drawbacks are assumed in advance (Ta- ble 1). The use of these models in practice and the theory behind them are also discussed. To test the models we used data on a long-term field trial (21 years) in which compound NPK ferti- lizers were used. We found that nitrogen ex- plained most of the yield response; fertilization with phosphorus did not increase the yield, though at the lowest levels the yield was deter- mined by both nitrogen and phosphorus (Yli- Halla 1991). Material and methods The fertilizer field trial was located on the same plot, with five fertilizer levels being applied each year. The trial was established in 1973 and the data used are from the period 1973-1993. New varieties replaced old ones but the cultivation techniques remained the same throughout the period. The field trial was conducted at Kemira Agro's research farm at Kotkaniemi, 40 km northeast of Helsinki. The change in varieties was gradual. Barley (29 varieties), wheat (21 varieties) and oats (16 varieties) were cultivat- ed. The five fertilizer levels are referred to as A, B, C, D and E in increasing order of fertilizer application, with level A as zero. The primary purpose of the trial was demonstrative, and the treatments were not randomized. The amounts of fertilizer applied in the trial were reduced from 1986 onwards (Table 2), ex- cept at level A, at which it was zero throughout. The NPK fertilizer applied was the compound fertilizer most commonly used in Finland. The composition and amounts of this fertilizer are listed in Table 2. The nutrients N, P and K are considered as linear combinations and N is assumed to be the main limiting nutrient. Partial correlation anal- 152 AGRICULTURAL AND FOOD SCIENCE IN FINLAND Vol. 6(1997): 151-160. Table 2, Applied NPKcompound fertilizer (kg/ha) in different periods. Year 1973-1979 1980-1985 1986-1988 1989 1990-1993 Fertilizer 15-9-12 16-7-13 16-7-13 17-6-12 20-4-8 Level A, kg/ha 0 0 0 0 0 Level B, kg/ha 300 300 300 300 200 Level 6OO 600 500 500 400 Level D, kg/ha 900 900 700 700 600 Level E, kg/ha 1200 1200 900 900 800 yses also show a higher value for the N fertilizer than for the compound fertilizer as a whole (Ta- ble 3). The N fertilizer applied was chosen as the independent variable in the model, but note that P and K were also applied as macronutri- ents in the field trial. Differences in the compo- sition of the compound fertilizer had little effect on output (Fig. 1). Annual effects were analysed with F tests. Inclusion of annual effects allowed for annual physical production functions. The effect of residual N on levels B, C, D and E was assumed to be nil from year to year. The effect of residual P was assumed to be close to nil at level C. At level A, yield decreased from 1973 onwards. The functions are quadratic, LRP and expo- nential. The LRP and exponential functions were chosen because they reflect a level at which no further increase in yield response is achieved. The LRP function showed a “breakpoint" accord- ing to von Liebig’s Law of the Minimum. The exponential function increases asymptotically to a maximal yield level. Polynomial (quadratic) function y.. = a + b x.. + c..x 2. + e JU U U >J u u u (1) The indices i and j denote variety and year variationsrespectively, y stands for the yield re- sponse in kg/ha and x for the N fertilizer applied in kg/ha. a, b and c are parameters and e are er- ror terms that are assumed to be normally dis- tributed and with zero mean. This model, in which all coefficients vary both among varieties and in time, is easy to calculate and is the one most commonly used in fertilizer trial analyses. Table 3. Partial correlations between fertilizer components and wheat yield. Controlling forYield P N fertilizer 0.3525 0.000 P fertilizer P fertilizer 0.0272 0.356 N fertilizer NPK fertilizer -0.0386 0,190 N fertilizer and P fertilizer The parameter c is expected to give a negative estimate in order to reflect diminishing returns. There are separate calculations for every crop. All indices i.,.n and j...k on every coefficient are tested separately with F tests and refer to effects in this text. Only effects significant according to the F test are included in the final results. Linear response and plateau (LRP) function {a. + b., x.., O < x. < x u .u u r u b'J I + e« (2) y , x > X... jJ max’ ij bij J The indices i and j denote variety and year variations respectively, y stands for the yield re- Fig. 1.Average wheat yields at five fertilizer levels. 153 AGRICULTURAL AND FOOD SCIENCE IN FINLAND Bäckman, S. T. et al. Fertilizer trials: mathematical response models sponse in kg/ha and x for N fertilizer applied in kg/ha. The index b on x stands for the level of fertilizer thatachieves the breakpoint in the yield response to additional fertilizer nitrogen, y is the yield response level achieved in a given year for a specific variety. The function imposes a maximal yield level at which further application of a specific input will not give any further in- crease in the yield. Exponential function (Mitscherlich) y . = a (1-b ec »x "001 ) + e (3)u u u u The indices i and j denote variety and year variations, respectively, y stands for the yield response in kg/ha and x for the N fertilizer ap- plied in kg/ha. A factor, 0.01, was included in the model because of a problem with decimals when printing the results, and this factor was thereafter considered in the results. The func- tion gives an asymptotic maximal yield level. The growth to the maximal level diminishes expo- nentially, which means that the parameter c. is negative. The quadratic function was computed with IMSL subroutines; RGLM for fitting the multi- variate linear regression model and RSTAT for computing and printing statistics. These subrou- tines were retrieved from the MATLAB program. The quadratic function could thereafter be esti- mated simply by changing the dummy variable effects. For estimation of the LRP and the expo- nential functions we used the subroutine DUNLSF. This IMSL command solves a non- linear least squares problem using a modified Levenberg-Marquardt algorithm and a finite-dif- ference Jacobian. The Levenberg-Marquardt method is a modification of the Gauss-Newton algorithm for solving non-linear least squares problems (Appendix 1). From one current point another point is calculated by the trust region approach (Dennis and Schnabel 1983). This pro- cedure is repeated until stopping criteria are sat- isfied. Each separate estimation was made by a different FORTRAN program. All effects were analysed with F tests. For estimation of the LRP model, a program loop was used to restrict the movement of val- ues of the parameter xb| , with values outside the domain of the field trial (0-192) giving a penal- ty to the minimization of the non-linear least squares. The first restricting penalty loop (algo- rithm x b = (200-x b ) 2 ) we tried did not perform effectively enough on all data. The second re- stricting penalty loop, x h = xb (Fig. 2), was more effective and x. finally varied between -4bij J and 4, which meant optimal solutions between 0 and 200 kg of N per hectare. These penalty loops served to bring the routines to a clear stop. The first algorithm probably did not work well because the sample range was too small to re- flect the decreasing response phase in all cases. The barley data were nevertheless calculated with the first algorithm, and so barley reflects a higher plateau response than wheat or oats (Fig. 3). We assumed that all species and varieties in the trial would respond differently to fertilizer, and that all species and varieties in the trial would give differentyields without fertilizer. We also assumed that every year would give differ- ent yield responses. F tests were performed to test our assumptions. The R a 2 d value will give the estimated func- tion response to the distribution pattern, is the R a 2 d value corrected for the number of degrees of freedom and is therefore better suitedfor com- paring different models with different numbers Fig. 2. Restricting penalty algorithm for the LRP model. 154 AGRICULTURAL AND FOOD SCIENCE IN FINLAND Vol. 6 (1997): 151-160. ofparameters. The number of parameters is also a criterion in which the number of parameters as well as the flexibility of the function should be related to the information they give. Preference should be given to function estimates that are easy both to explain and to calculate. Table 4. Optimal input for the maximal economic return. Quadratic model LRP model Exponential model f -b X = X " i„ (_E*E_) X=X= * 2c c Economic analysis Making marginal cost equal to marginal reve- nue gives us the maximal economic return. This occurs, according to the production theory, when the marginal physical product (MPP) is equal to the price ratio, w/p, where w is the unit price of fertilizer and p the unit selling product price. Due to the short growing season in Finland, annual variations in the corresponding yield responses are high. Knowledge of the physical production function is therefore valuable. Another factor adding to the interest of the physical production function interesting is the change in price rela- tions in Finland caused by accession to the Eu- ropean Union in 1995. The ensuing switch from an agricultural policy favouring price support to one favoring more direct support meant a drop of about 60 per cent in the price of cereals. The maximal yield is the point at which the slope of the production function equals zero. The fertilizer input for maximal economic output, when costs are included, is lower than that for maximal yield. The maximal economic return calculated with theLRP function is approximat- ed to equal zero, or breakpoint. The first-order derivatives of the exponential function show four solutions, depending on the signs selected for the parameters. We chose the solution that is most likely to occur on the basis of diminishing returns (Table 4). The optimal economic output is calculated from the mean LS estimate, which gives a lower result than the average of the sum of the optima calculated with the included ef- fects. The prices used for calculating maximal eco- nomic output are w = FIM 9.43/kg, oats FIM 1.54/kg, barley FIM 1.63/kg and wheat FIM 2.19/kg in 1993 and w = FIM 6.41 /kg, oats FIM 0.70/kg barley FIM 0.73/kg and wheatFIM 0.87/ kg in 1995. Results The values estimated for the quadratic function were checked with F tests. All the effects were significant except for the variations in the re- sponse of the oats variety to N. The second de- gree variety variations were not significant ei- ther. Estimation of the quadratic function suc- ceeded without major drawbacks but resulted in many optima that were explicit (outside the sam- ple range). The parameters estimated (Table 5) are the values that can be fitted to the models present- ed. Note that the intercepts (parameter a, LRP and quadratic model) for oats and barley are higher than those for wheat. Wheat also responds less to N fertilizer than does oats or barley (Fig. 3). The restricting penalty algorithm used for barley did not allow us to include as many ef- fects as did the algorithm used for wheat and oats. This is probably why the breakpoint for barley is higher than that for oats and wheat. The parameters a for the exponential model show (Table 5) the asymptotic yield level. The stand- ard error in the parameter of the asymptotic lev- el for wheat is remarkably high, possibly imply- ing that the annual and variety variations do not explain the entire yield variation for wheat at high fertilizer levels. A comparison of the distribution pattern and the LRP model estimated for oats shows that a 155 AGRICULTURAL AND FOOD SCIENCE IN FINLAND AGRICULTURAL AND FOOD SCIENCE IN FINLAND Bäckman, S. T. et al. Fertilizer trials: mathematical response models reduction in the yield response for oats at high fertilizer levels cannot be reflected by the LRP model. Oats shows a clear decrease in yield re- sponse at high fertilizer rates, but the LRP mod- el does not have a decreasing phase at which MPPcO. The residuals now cross the zero line twice and the optimum should therefore be found between the crossings. The yearly variation in the breakpoint at the x-axis was not significant for the LRP model, but the resulting yield pla- teau or N response had significant variations. There were too few replications for each variety in the data on the barley experiments. The LRP and the exponential results for barley were there- fore calculated without considering variations due to variety. Estimation of the exponential function suc- ceeded well. The R 2 values were higher than those for the quadratic function. The exponen- tial function does not have a phase for a decreas- ing yield response and so is it less suitable for yield response analysis than the quadratic func- tion. Only a few of the optima obtained were implicit. Moreover, the sample range was too Table 5. Estimated parameters and standard errors for quadratic, LRP and exponential models. a SK b SE c SE Quadratic Oats 1414 (64.5) 51.5 (1.6) -0.204 (0.01) Barley 1010 (73.3) 52.9 (1.3) -0.173 (0.01) Wheat 1274 (72.0) 35.8 (1.2) -0.094 (0.06) LRP Oats 1379 (114) 43.17 (3.1) 0.27 (0.08) Barley 1051 (42.5) 43.13 (1.2) 9,16 (0.09) Wheat 1336 (81.3) 28.56 (1.5) 0.27 (0.07) Exponential Oats 5339 (341) -0.71 (0.02) -1.94 (0.17) Barley 6044 (332) -0.81 (0.01) -2.13 (0.66) Wheat 7493 (1953) -0.79 (0.02) -1.23 (0.15) Fig. 3. Oats, barley and wheat yield responses to applied nitrogen (21 years) with the LRP model. 156 Vol. 6(1997): 151-160. Table 6. Statistical results of estimation of the three models. R 2 Hst. SSres SStot Number of Calculation of number error parameters* of parameters Quadratic function Oats 85.2637.5 3.910E+08 1.637E+10 82 =3+3*21+16 87.8664.0 6.088E+08 2.501E+10 124 =3+3*21+2*29 88.3526.6 2.922E+08 1.393E+10 106 =3+3*21+2*20 Barley Wheat LRP function Oats 82.7688.8 4.516E+08 1.637E+10 88 =3+2*2l-2+3*16-3 86.2705.4 7.229E+08 2.501E+10 63 =3+3*2l-3 87.1552.3 2.914E+08 1.393E+10 100 =3+2*2l-2+3*20-3 Barley Wheat Exponential function Oats 83.3675.6 4.322E+08 1.637E+10 93 =3+3*2l-3+2*16-2 86.0711.0 7.264E+08 2.501E+10 63 =3+3*2l-3 90.2479.8 2.382E+08 1.393E+10 120 =3+3*2l-3+3*20-3 Barley Wheat *The numbers of parameters with included effects. Effect included on the basis of F-tests. There are 16 varietes foroats, 29 for barley and 20 for wheat. The number of years are 21. All year variation effects included except those for the LRP function on oats and wheat. All the year effects in the exp. model could only be included for wheat. In the LRP model all wheat and oats year effects could be included. small when the exponential function was used on the data for wheat. The number of parameters varies according to the number of effects included(Table 6). When only R * was compared, the quadratic function performed better than the exponential or LRP function on oats and barley. The exponential function performed best on wheat, but there was a problem with the sample range, which may ex- plain the high error term for the coefficient a for wheat.The LRP function performed only slightly worse than the others. R 2 , , however, is not aadj very good criterion for selecting a model, as shown by, among others, Kvålseth (1985), McGuirk and Driscoll (1995). It would be more appropriate to test the models against each oth- er. This would be an interesting topic for a fol- low-up study. We did not do so here because our main interest was in finding out which effects to include, and also because we lacked reliable methods for testing the models against each other. It is striking that, according to the quadratic and exponential models, a decrease in optimal fertilizer use was a result oflower prices in 1995 (Table 7). According to the LRP model, howev- er, changes in price do not affect economic opti- mal fertilizer use. This is because the MPP is zero after the breakpoint, which tells us that any additional fertilizer input gives no further in- crease in yield. Optimal fertilizer use according to the LRP model is, however, still lower than according to the quadratic or exponential mod- el. Newer varieties show an increase in optima. Discussion Recommendations based on an “incorrect" mod- el have an impact on both profitability and on the environment. Preference has been given to functions that enable us to relate the parameters achieved to biological and physical processes. Biological theories and economic theories are not, however, always comparable. In agronomy, there is no satisfactory relation between growth 157 AGRICULTURAL AND FOOD SCIENCE IN FINLAND Bäckman , S. T. et al. Fertilizer trials: mathematical response models Table 7. Economic fertilizer use and corresponding yield with three price relations and three different regression models, kg/ha. Crop Quadratic LRP Exponential Nitrogen Yield Nitrogen Yield Nitrogen Yield 1993Prices Oats 111 4619 77 4720 128 5024 Barley 136 5005 116** 6054 136 5772 Wheat 168 4633 113 4575 230* 7143 1995 Prices Oats 104 4563 77 4720 108 4870 Barley 127 4942 116** 6054 116 5630 Wheat 151 4540 113 4575 187 6897 * Value beyond sample range and therefore not reliable. ** Calculated with a different restricting loop from oats and wheat. inputs and yield formation. Attempts have been made to construct deterministic models, but as van Keulen and Stol (1991) remarked: "These models cannot be used in practice. They are an aid to structure thinking about the system." Sto- chastic models are more often used, but they may lack any idea with which the results can be ex- plained. The optimization of fertilizer use depends on the aim ofa particular optimum. Possible optima are: a) biological (includes quality and quantity) b) economic (includes micro- and macroeco- nomic optima) c) environmental. The biological optimum depends on the quan- tity measures and will give the biologically max- imal yield. The biologically optimal input of fertilizers may be different if quality is taken into account. Economic optima are calculated to give the maximal economic return for the individual farmer at the microlevel. The optimal macroin- put of fertilizer is the optimal input for a certain number of farmers with different production options (functions). The results of these experi- mental data cannot be used as such at the mac- rolevel. Here we have concentrated on deriving the economic and biological quantity optima for this experiment, and considered only the varia- ble fertilizer costs. The quadratic function has been criticized for giving excessively high estimates (Ackello-Ogu- tu et al. 1985, Paris 1992). We, however, found that it was easy to both calculate and use for optima. The quadratic function has a diminish- ing phase that was accurate if comparisons are made with the distribution pattern of the data on oats. The high estimates are partly due to the inflexibility of the quadratic function but partly to the size of the domain. The lack of the diminishing phase in the ex- ponential function is a disadvantage. A dimin- ishing phase would, however, require inclusion of a fourth parameter in the model. The advan- tage of a four-parameter exponential function over the quadratic function would be that the former could include a steeper increasing phase and a flatter diminishing phase. The disadvan- tage of a fourth parameter is the decrease in the degree of freedom. The quadratic function is symmetric in the increasing and diminishing phases. The distribution pattern appears not to be symmetric. The diminishing phase is inter- esting only ifit affects the earlier (increasing and rational) phases; otherwise it has no significance for the optima. A non-linearfunction with three parameters that could include a steeper increas- ing phase and a flatter diminishing phase would probably perform well. These non-linear func- tions are available but they do not necessarily have any relevance to biological theories. These 158 AGRICULTURAL AND FOOD SCIENCE IN FINLAND Vol. 6(1997): 151-160. functions might be an interesting topic for a fol- low-up study. The LRP model gives optimal input levels that are closer to the recommendations used to- day than the optimal input levels given by the exponential or quadratic models. According to the exponential function, the optimal use of fer- tilizer is highly dependent on the unit price rela- tion between input and output. This is not so according to the LRP function. The LRP func- tion is, however, quite cumbersome to calculate and therefore probably not that useful. If the error term were brought further into the calculation of the optimal solutions in the form of an economic risk, the optimizations ob- tained would be more accurate. This was done for the quadratic function by Carmer et al. (1991). It is, however, a very time-consuming task and the results should be considered in the perspective of variability in actual growing con- ditions. Optimising under varying conditions can give us higher optima because of the price rela- tion and attitudes to risk. The average curve is still, however, the expectation curve. In conclusion, we stress that none of these three models, quadratic, LRP or exponential, performed satisfactorily in all respects. The quadratic model performed best for oats because of the diminishing phase. The LRP performed best in view of the recommendations, and the exponential function performed best in respect of the steep increasing phase. As a topic for fur- ther research new models should be tried out or new methods introduced that include other fac- tors affecting economic output. References Ackello-Ogutu, C, Paris, Q. & Williams, W.A. 1985. Test- ing a von Liebig crop response function against pol- ynomial specifications. American Journal ofAgricul- tural Economics 67: 873-880. Bock, B.R. & Sikora, F.J. 1990. Modified-quadratic/pla- teau model for describing plant responses to fertiliz- er. Soil Science Society American Journal 54:1784- 1789. Carmer, S.G., Walker, W.M. & Swanson, E.R. 1991. A risk assessement-risk management approach to selecting fertilizer application rates. Journal of Pro- duction Agriculture 4: 67-73. Cerrato, M.E. & Blackmer, A.M. 1990. Comparison of models for describing corn yield response to nitro- gen fertilizer. Agronomy Journal 82: 138-143. Dennis, J.E. jr. & Schnabel, R.B. 1983. Numerical meth- ods for unconstrained optimization and nonlinear equations. Prentice-Hall,lnc., New Jersey. 378 p. Frank, M.D., Beattie, B.R. & Embleton, M.E. 1990. A com- parison of alternative crop response models. Amer- ican Journalof Agricultural Economics 72: 597-603. Keulen, H. van & Stol, W. 1991. Quantitative aspects of nitrogen nutrition in crops. Fertilizer Research 27: 151-160, Kvålseth, T.0.1985. Cautionary note about R 2. The Amer- ican Statistican 39: 279-285. McGuirk, A.M. & Driscoll, P. 1995. The hot air in R2 and consistent measures of explained variation. Ameri- can Journal of Agricultural Economics 77: 319-328. Miettinen, A. 1993. The effectiveness and feasibility of economic incentives of input control in the mitiga- tion of agricultural water pollution. Agricultural Sci- ence in Finland 2: 453-463. Paris, Q. 1992. The von Liebig hypothesis. American Journal of Agricultural Economics 74: 1019-1028. Sumelius, J. 1993. A response analysis of wheat and barley to nitrogen in Finland. Agricultural Science in Finland 2: 465-479. Yli-Halla, M. 1991. Phosphorus supplying capacity of soils previously fertilized with different rates of P. Journal of Agricultural Science in Finland 63: 75-83. 159 AGRICULTURAL AND FOOD SCIENCE IN FINLAND Bäckman, S. T. et al. Fertilizer trials: mathematical response models SELOSTUS Lannoituksen pitkäaikaiset kenttäkokeet: kolmen matemaattisen mallin vertailu Stefan T. Bäckman, S. Vermeulen jaV.-M. Taavitsainen Helsingin yliopisto ja KemiraAgro Oy Vehnän, ohran ja kauran hinta laski Suomessa noin 60 prosenttia vuodesta 1994 vuoteen 1995 Euroopan unioniin liittymisen myötä. Myös lannoitteiden hin- ta laski, mutta huomattavasti vähemmän. Tämä suh- teellisten hintojen muutos merkitsi sitä, että taloudel- lisesti optimaalinen lannoitteiden käyttömäärä piene- ni. Lannoitteiden taloudellisesti optimaalisen käytön määrittäminen edellyttää fyysisten tuotantofunktioi- den tuntemista. Tutkimuksessa estimoitiin kolme fyy- sistä tuotantofunktiota; toisen asteen polynomi-, LRP-ja eksponenttifunktio. Tutkimusaineistona käy- tettiin 21 vuoden kenttäkokeita, joissa oli tutkittu mineraalilannoitteiden vaikutusta viljanviljelyyn. Kokeissa viljeltyjä kasveja olivat ohra, kaura ja veh- nä. Lajike-erot otettiin malleissa huomioon. Mallit eivät poikenneet suuresti toisistaan selitysasteen pe- rusteella ( 0,82-0,90), mutta malleista lasketut talou- delliset optimit poikkesivat toisistaan. 160 AGRICULTURAL AND FOOD SCIENCE IN FINLAND Appendix 1 The DUNLSF algorithm: 1 1 m min-F(x)TF(x)--gf,(x) 2 where min,F:R" -» R m , and is the ith component function of F(x) • From a current point, the algorithm uses the thrust region approach: ™?||f(Xc)+ j(Xc)(x„ -Xc)|| 2 subject to ||x„ -Xc||2 sö c to get a new point x„ which is computed as X„ - Xc ~(J(X c )Tj(Xc) + J(Xc)Tp(Xc) where |». -0 if6. »|[J(x.) T J(x.))_II(X.) TF(xj| l and n c > O F(Xc) and J(Xc) are tne function values and the Jacobian evaluated at the current point xc > respectively. This procedure is repeated until the stopping criteria are satisfied (IMSL MATH/LIBRARY). AGRICULTURAL AND FOOD SCIENCE IN FINLAND