Model Analysis of Corn Response to Applied Nitrogen and Plant Population Density

Communications in Soil Science and Plant Analysis, 37: 1157 117, 006 Copyright # Taylor & Francis Group, LLC ISSN 0010-364 print/153-416 online DOI: 10.1080/001036060063350 Model Analysis of Corn Response to Applied Nitrogen and Plant Population Density Agricultural and Biological Engineering Department, University of Florida, Gainesville, Florida, USA Abstract: The etended logistic model relates seasonal dry matter and plant nutrient uptake to applied nutrient level. It has been shown to apply to data for annuals such as corn (Zea mays L.) and perennials such as bermudagrass (Cynodon dactylon L.) and bahiagrass (Paspalum notatum Flügge). The linear parameters in the model have been shown to depend on water availability and harvest interval (for perennials). Further work is needed to relate model parameters to plant characteristics. In this article, data from a field eperiment with corn at si nitrogen levels (0, 0.5, 1.0,.0, 3.0, and 5.0 g N plant 1 ) and three plant population densities (3, 6, and 9 plants m ; 3, 6, and 9 10 4 plants ha 1 ) are used to provide insight into this question. It turns out that all five model parameters are dependent on plant density, approaching maimum values at 8.3 plants m. Three of the parameters approach zero as density approaches zero, which seems intuitively correct. It is concluded that lower and upper limits of plant nitrogen concentration are independent of population density and are functions of the particular plant species. Detailed procedures are described for estimation of model parameters. Keywords: Corn, models, nitrogen, plant population INTRODUCTION Crop production depends on several factors, including level of applied nutrients [such as nitrogen (N), phosphorus (P), and potassium (K)], water Received 7 July 004, Accepted 5 July 005 Address correspondence to A. R. Overman, Agricultural and Biological Engineering Department, University of Florida, Gainesville, FL 3611-0570, USA. E-mail: aroverman@ifas.ufl.edu 1157

1158 availability (rainfall and/or irrigation), and cultural practices (such as harvest interval for perennials). The etended logistic model was developed to relate seasonal production (dry matter and plant nutrient uptake) to applied nutrients. This analysis focuses on response of corn to applied nitrogen at three plant population densities. Quantitative relations are developed between model parameters and population density. Significance of the relations is discussed. MODEL DESCRIPTION The etended logistic model of seasonal crop response to applied nutrients is based on two postulates (Overman and Scholtz 00): 1) seasonal uptake of a plant nutrient is coupled to applied nutrient through a logistic equation, and ) seasonal plant yield of dry matter is coupled to seasonal plant nutrient uptake through a hyperbolic equation. These postulates can be stated in mathematical terms as N u ¼ A n 1 þ epðb n C n NÞ ð1þ Y ¼ Y mn u ðþ K n þ N u where N is applied nutrient (N, P, or K), kg ha 1 ; N u is seasonal plant nutrient uptake (N, P, or K), kg ha 1 ; Y is seasonal dry matter yield, Mg ha 1 ; A n is maimum plant nutrient uptake at high applied nutrient level, kg ha 1 ; B n is intercept parameter for plant nutrient uptake; C n is response coefficient, ha kg 1 ; Y m is potential maimum dry matter yield, Mg ha 1 ; and K n is nutrient uptake coefficient, kg ha 1. These postulates were formulated after etensive analysis of field data. Equation () is easily rearranged to the linear form N c ¼ N u Y ¼ K n þ 1 N u ð3þ Y m Y m where N c is plant nutrient concentration (N, P, or K), g kg 1. Equations () and (3) constitute the phase relations (Y and N c vs. N u ) for the plant system. It is common practice to refer to N c as concentration, but it should more properly be called specific nutrient, mass of nutrient per unit mass of dry matter, N u /Y. Several consequences follow directly from equations (1) and (). Substitution of equation (1) into equation () leads, after rearrangement and simplification, to the second logistic equation, Y ¼ A y 1 þ epðb y C n NÞ ð4þ

Corn Response to Nitrogen and Plant Density 1159 where A y is maimum dry matter yield at high applied nutrient, Mg ha 1 ; and B y is intercept parameter for dry matter yield. Hyperbolic and logistic model parameters are related by A y Y m ¼ 1 epð DBÞ A n K n ¼ epðdbþ 1 ð5þ ð6þ where DB is defined by DB ¼ B n B y ð7þ Dependence of plant nutrient concentration on applied nutrient is obtained by the combination of equations (1) and (4), N c ¼ N u Y ¼ N 1 þ epðb y C n NÞ cm 1 þ epðb n C n NÞ where N cm ¼ A n /A y is maimum plant nutrient concentration at high applied nutrient, g kg 1. Because in practice B n. B y always holds, it can be shown that N c is bounded by N cl, N c, N cm, where the lower limit of N c for highly depleted soil nutrient level (N 0) is defined from equation (8) as ð8þ N cl ¼ N cm epð DBÞ ð9þ It follows from equation (3) that as N u! 0, N c! N cl ¼ K n /Y m. Substitution of equations (5) and (6) into this ratio leads to equation (9). As defined by the physical system, the quantities Y, N u, N c, A y, A n, and C n are all positive. Parameters B y and B n may be positive, zero, or negative. Convergence of equation (8) requires that B n. B y. It should be noted that N, 0 would correspond to reduced (depleted) soil nutrient by some means; more on this point later. The logistic model has been shown to be mathematically well behaved (Overman 1995). In other words Y, N u, and N c are monotone increasing functions that are bounded by 0, Y, A y, 0, N u, A n, and N cl, N c, N cm. All of these relations seem intuitively correct. In this article, we attempt to clarify dependence of model parameters on plant characteristics. Data from a field study with response of corn to applied nitrogen for three plant-population densities are analyzed. The variables and parameters in the model (with the eception of B y and B n, which are dimensionless) can be defined on a per plant basis and then converted to an area basis. Lowercase symbols are used for the per plant basis.

1160 DATA ANALYSIS Data for this analysis are taken from a field study by Rhoads and Stanley (1979) at Quincy, Florida. Corn (cv Pioneer 3369A) was planted on Ruston loamy fine sand (fine-loamy, siliceous, thermic Typic Paleudult) on March 4, 1977. Plant population densities of 3, 6, and 9 plants m were obtained by varying drill spacing on 76-cm rows. Fertilizer was applied at rates of 0, 0.5, 1.0,.0, 3.0, and 5.0 g plant 1 with a fied ratio of (N P K ¼ 1 0.3 0.8). Irrigation of.5 cm was applied to all plots when the soil moisture tension at the 15-cm depth reached 0 centibars. All plots were replicated four times. Plant samples were collected 87 days after planting. Results are given in Table 1 and shown in Figure 1 for dry matter (y), plant N uptake (n u ), and plant N concentration (n c ) in response to applied nitrogen (n) at the three population densities. Phase plots (y and n c vs. n u ) Table 1. Response of corn to applied nitrogen and plant population density at Quincy, FL (1977) a n (g plant 1 ) y (g plant 1 ) n u (g plant 1 ) n c (g kg 1 ) 3 plants m 0 300 3.07 10. 0.5 60.37 9.1 1 97 3.30 11.1 300 3.73 1.4 3 347 4.57 13. 5 347 5.40 15.6 6 plants m 0 167 1.3 7.4 0.5 183 1.68 9. 1 4.53 10.5 40.78 11.6 3 43 3.40 14.0 5 7 4.10 15.1 9 plants m 0 13 0.98 7.9 0.5 130 1.00 7.7 1 183 1.70 9.3 183.9 1.5 3 193.66 13.7 5 07 3.9 15.9 a Data adapted from Rhoads and Stanley (1979).

Corn Response to Nitrogen and Plant Density 1161 Figure 1. Response of seasonal dry matter yield (y), plant N uptake (n u ), and plant N concentration (n c ) to applied nitrogen (n) at three population densities for corn at Quincy, FL. Data adapted from Rhoads and Stanley (1979). Curves drawn from Equations (1), (4), and (8) with parameters from Table 4. are shown in Figure. The first step is to calibrate equation (3) from the data, which leads to the phase equations: 3 plants m : n c ¼ k n y m þ 1 y m n u ¼ 4:07 þ :10n u ; r ¼ 0:9886 ð10þ

116 Figure. Phase plots (y and n c vs. n u ) at three population densities for corn at Quincy, FL. Data adapted from Rhoads and Stanley (1979). Lines drawn from equations (10), (1), and (14); curves drawn from equations (11), (13), and (15). 6 plants m : 9 plants m : y ¼ y mn u ¼ 0:475n u ð11þ k n þ n u 1:94 þ n u n c ¼ k n y m þ 1 y m n u ¼ 4:3 þ :70n u ; r ¼ 0:9889 ð1þ y ¼ y mn u k n þ n u ¼ 0:370n u 1:57 þ n u ð13þ n c ¼ k n y m þ 1 y m n u ¼ 4:0 þ 3:60n u ; r ¼ 0:990 ð14þ y ¼ y mn u k n þ n u ¼ 0:78n u 1:1 þ n u ð15þ

Corn Response to Nitrogen and Plant Density 1163 where high correlation coefficients, r ffi 0.99, may be noted. The lines in Figure are drawn from equations (10), (1), and (14); the curves are drawn from equations (11), (13), and (15). The model describes the pattern in the data rather well. The intercepts in Figure are very close together (averaging approimately 4.1 g kg 1 ), which represent the value of n cl.by visual inspection of Figure 1 it appears that n cm ffi 17.0 g kg 1. It follows from equation (9) that Db ¼ ln(17.0/4.1) ¼ 1.4. This value is substituted into equations (5) and (6) to obtain estimates of logistic parameters a y and a n. Results of these calculations are listed in Table. The net step is to estimate parameters b n and c n for each population density. Equation (1) can be written in the linearized form z ¼ ln a n 1 ¼ b n c n n n u ð16þ Calculations of z vs. n are listed in Table 3 for each population density. Further analysis then leads to the regression equations: 3 plants m : z ¼ ln 6:09 1 ¼ b n c n n ¼ 0:337 0:467n; n u r ¼ 0:9657 ð17þ 6 plants m : z ¼ ln 4:93 1 ¼ b n c n n ¼ 0:84 0:51n; n n 9 plants m : z ¼ ln 3:51 1 ¼ b n c n n ¼ 1:003 0:745n; n u r ¼ 0:9737 ð18þ r ¼ 0:990 ð19þ Table. Calibration of the logistic model for corn at Quincy, FL Population (plants m ) 3 6 9 n cl (g kg 1 ) 4.1 4.1 4.1 n cm (g kg 1 ) 17.0 17.0 17.0 Db 1.4 1.4 1.4 y m (g plant 1 ) 475 370 78 k n (g plant 1 ) 1.94 1.57 1.1 a y (g plant 1 ) 360 81 11 a n (g plant 1 ) 6.09 4.93 3.51

1164 Table 3. Estimation of logistic model parameters for corn at Quincy, FL 3 plants m 6 plants m 9 plants m n (g plant 1 ) n u (g plant 1 ) z n (g plant 1 ) n u (g plant 1 ) z n (g plant 1 ) n u (g plant 1 ) z 0 3.07 0.016 0 1.3 þ1.10 0 0.98 þ0.948 0.5.37 þ0.451 0.5 1.68 þ0.660 0.5 1.00 þ0.90 1 3.30 0.168 1.53 0.053 1 1.70 þ0.063 3.73 0.458.78 0.57.9 0.630 3 4.57 1.10 3 3.40 0.799 3.66 1.14 5 5.40.06 5 4.10 1.60 5 3.9.71

Corn Response to Nitrogen and Plant Density 1165 It is now convenient to write equation (1) in the equivalent form, n u ¼ a n 1 þ ep½ðn 0 1= nþ=n0 Š ð0þ where the new logistic parameters are defined by n 0 ¼ 1 c n n 0 1= ¼ b n ¼ b n n 0 c n ð1þ ðþ In this case n 0 is characteristic N, g plant 1, and n 0 1/ is applied N to reach 50% of maimum plant N uptake, g plant 1. It follows that applied N to reach 50% of maimum yield, n 1/, is defined by n 1= ¼ b y c n ¼ b y n 0 ð3þ A summary of logistic parameters is listed in Table 4 and plotted in Figure 3. Because parameters a y, a n, and n 0 appear to decrease eponentially, we assume eponential relationships with plant population density, plants m, of the form w ¼ ln a y ¼ 5:93 0:00731 ; r ¼ 0:9949 ð4þ w ¼ ln a n ¼ 1:87 0:00764 ; r ¼ 0:999954 ð5þ w ¼ ln n 0 ¼ 0:855 0:00669 ; r ¼ 0:9817 ð6þ Table 4. Dependence of model parameters on plant population density for corn at Quincy, FL Population (plants m ) 3 6 9 a n (g plant 1 ) 6.09 4.93 3.51 a y (g plant 1 ) 360 81 11 b n 0.34 0.8 1.00 b y 1.08 0.60 0.4 c n (plants g ) 0.467 0.51 0.745 n 0 (g plant 1 ).14 1.95 1.34 n 0 1/ (g plant 1 ) 0.73 1.60 1.34 n 1/ (g plant 1 ).33 1.17 0.56

1166 Figure 3. Dependence of logistic parameters a y, a n,andn 0 on plant population density (, plantsm ) for corn at Quincy, FL. Curves drawn from equations (35) through (37). which leads to a y ¼ 376 ep 1 8:7 a n ¼ 6:51 ep 1 8:09 n 0 ¼ :36 ep 1 8:65 ð7þ ð8þ ð9þ

Corn Response to Nitrogen and Plant Density 1167 Because the eponential coefficients are similar numerically, we now proceed to optimize parameters in equations (7) through (9) as shown in Table 5. The nine decimal fractions, f, are used to estimate an overall eponential coefficient from ln f ¼ 0:000617 0:0071 ; r ¼ 0:9895 ð30þ f ¼ 1:00 ep 1 8:33 ð31þ Intercept values of the parameters (a y0, a n0, n0) 0 are now calculated from a y0 ¼ a y ep þ 1 8:33 ð3þ a n0 ¼ a n ep þ 1 8:33 ð33þ n 0 0 ¼ n0 ep þ 1 8:33 ð34þ Values are then averaged over plant populations as shown in Table 5, which leads to overall equations a y ¼ 376 ep 1 g plant 1 ð35þ 8:33 a n ¼ 6:40 ep 1 g plant 1 ð36þ 8:33 n 0 ¼ :41 ep 1 g plant 1 ð37þ 8:33 Table 5. Optimization of model parameters on plant population density for corn at Quincy, FL Population (plants m ) 3 6 9 avg a n 6.09 4.93 3.51 a y (g plant 1 ) 360 81 11 n 0.14 1.95 1.34 a n /6.51 0.935 0.757 0.539 a y /376 0.957 0.747 0.561 n 0 /.36 0.907 0.86 0.568 a no 6.50 6.39 6.30 6.40 a yo (g plant 1 ) 384 364 379 376 no 0.8.53.41.41

1168 The curves in Figure 3 are drawn from equations (35) through (37). These three parameters can now be converted to an area basis by multiplying by the plant density to obtain A y ¼ a y ¼ 3:76 ep 1 Mg ha 1 ð38þ 8:33 A n ¼ a n ¼ 64:0 ep 1 kg ha 1 ð39þ 8:33 N 0 ¼ n 0 ¼ 4:1 ep 1 kg ha 1 ð40þ 8:33 Results are shown in Figure 4. It can be shown that these curves reach a peak at p ¼ 8.33 plants m. It follows from equations () and (3) that DN 1= N 0 ¼ Dn 1= n 0 ¼ Db ð41þ which leads to DN 1= ¼ N1= 0 N 1= ¼ Db N 0 ¼ 34: ep 1 ð4þ 8:33 Values of N1/ 0 and N 1/ are shown in Figure 5. It appears that both trends converge to 100 kg ha 1 as! 0. To incorporate this fact and to be consistent with equation (4), we choose N1= 0 ¼ 44: ep 1 100 ð43þ 8:33 N 1= ¼ 10:0 ep 1 100 ð44þ 8:33 The curves in Figure 5 are drawn from equations (43) and (44). It follows that DN 1/ is dependent on plant population density, with the value ranging from 0 as! 0 to 17 kg ha 1 for ¼ 8.33 plants m. DISCUSSION The logistic model provides a reasonable description of corn response to applied N at the three plant populations of the study (Figure 1), and the phase plots follow the predicted form of the model rather well (Figure ). Because it appears from the analysis that lower and upper limits of plant N concentration are independent of plant population, it is possible to estimate the linear logistic parameters a y and a n from hyperbolic parameters y m and k n (Table ). The eponential logistic parameters b y, b n, and c n are

Corn Response to Nitrogen and Plant Density 1169 Figure 4. Dependence of logistic parameters A y, A n, and N 0 on plant population density (, plants m ) for corn at Quincy, FL. Curves drawn from equations (38) through (40). estimated from the response data (Table 4). Logistic parameters are then converted from a per plant basis to per area basis (Table 6). A major challenge of this analysis is to relate model parameters to plant population density. Parameters a y,a n, and n 0 all ehibit a decreasing trend with increasing plant population (Figure 3). This clearly represents competition among plants as the number increases. It may be noted that for a row spacing of 76 cm, drill spacing is approimately 45 cm for a population density of 3 plants m and 15 cm for a population density of 9 plants m.

1170 Figure 5. Dependence of logistic parameters N 1/ and N1/ 0 on plant population density (, plants m ) for corn at Quincy, FL. Curves drawn from equations (43) and (44). Therefore, a competition function is needed to provide quantitative coupling of the parameters with plant population. Although many mathematical functions could be chosen for this purpose, we have chosen the Gauss equation: p ¼ p 0 ep 1 ð45þ s where p o ¼ intercept parameter p(a y, a n, or n 0 ) at ¼ 0 and s ¼ spread parameter for the distribution. Note that and s have the same units. It may be noted that s represents the inflection point of the Gaussian curve by including the factor of 1/. Equation (45) is chosen because it reflects low competition at low population densities and remains positive for all, which seems intuitively correct. This relationship provides ecellent correlation of parameters A y, A n, and N 0 with plant population (Figure 4). Peak values occur at ¼ s ¼ 8.33 plants m. This also seems intuitively correct, because it appears likely that production would decrease with overcrowding of plants. Linking intercept parameters N 1/ and N1/ 0 to plant population Table 6. Summary of model parameters on plant population density for corn at Quincy, FL Population (plants m ) A y (Mg ha 1 ) A n (kg ha 1 ) N 0 (kg ha 1 ) N 1/ (kg ha 1 ) N 0 1/ (kg ha 1 ) 3 10.8 183 64. 70 6 16.9 96 117 70 96 9 19.0 316 11 50 11

Corn Response to Nitrogen and Plant Density 1171 proved more difficult because values can be positive, zero, or negative. The lower limit on both parameters proved to be 100 kg ha 1 as! 0 (Figure 5). This value reflects the background level of available soil N, which is believed to derive from grass sod on the field prior to the eperiment of Rhoads and Stanley (1979). The Gaussian function provided reasonable correlation with plant population through equations (43) and (44). It can be concluded that all parameters in the logistic model are dependent on plant population. However, near the optimum population of 8.3 plants m, the parameters are relatively insensitive to changes in plant population (Figures 4 and 5). Greatest sensitivity occurs at populations near zero, as might be epected. Finally, a comment on the issue of validation: It is believed that the etended logistic model has been clearly validated as a useful mathematical description of plant response to nutrient input (Overman and Scholtz 00). On the other hand, particular numbers have not been validated for parameters, as these may depend upon plant and soil characteristics. In this view, there is a wide distinction between the two. Many mathematical models in physics have been shown to give reasonable approimations to observations (Kepler s laws of planetary motion, Galileo s law of falling bodies, Newton s laws of dynamics and of gravitation, Einstein s laws of relativity, Planck s law of black body radiation, Schrödinger s wave equation, etc.). In contrast to this are the universal constants (gravitational constant, G; speed of light, c; charge of the electron, e; Planck s constant of action, h; Boltzmann s constant, k; Avogadro s constant, N 0 ), which have been discussed by Sheldrake (1995, chapter 6). At the present state of knowledge in crop science, the search is more toward laws or models of physical processes than toward universal constants. This article has described the relationship of model parameters to plant characteristics (as measured by population density). Work is under way to relate these parameters to soil characteristics as well. To insist that all of this be accomplished before a given model can be published is to ignore 400 years of history in science. Progress is made on an incremental basis rather than in one grand sweep. The search for the unifying theory of everything is still in progress in physics (Greene 1999; Gribbin 1995). To understand some things in science does not require that we understand everything (Green 004). REFERENCES Greene, B. (1999) The Elegant Universe; W.W. Norton: New York. Greene, B. (004) The Fabric of the Cosmos: Space, Time, and the Teture of Reality; Alfred A. Knopf: New York. Gribbin, J. (1995) Schrödinger s Kittens and the Search for Reality; Little, Brown: New York. Overman, A.R. (1995) Rational basis for the logistic model for forage grasses. Journal of Plant Nutrition, 18: 995 101.

117 Overman, A.R. and Scholtz, R.V., III (00) Mathematical Models of Crop Growth and Yield; Marcel Dekker: New York. Rhoads, F.M. and Stanley, R.L. (1979) Effect of population and fertility on nutrient uptake and yield components of irrigated corn. Soil Crop Science Society of Florida Proceedings, 38: 78 81. Sheldrake, R. (1995) Seven Eperiments that Could Change the World; Riverhead Books: New York.