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PAGE 1 A MEMOIR ON A Model of Growth and Nutrient Uptake by Switchgrass Allen R. Overman Agricultural and Biological Engineering Department University of Florida Gainesville, Florida, USA Copyright 2013 Allen R. Overman PAGE 2 Overman Production of Switchgrass i Key words : Mathematical models, plant growth, plant nutrient upt ake, switchgrass This memoir is focused on a model of plant growth and nutrient uptake by the warmseason perennial switchgrass ( Panicum virgatum L.). Mathematical analys is utilizes the expanded growth model and the extended logistic m odel published previously. The growth model describes accumulation of biomass by photosynthesi s based on the three ba sic processes of (1) seasonal distribution of solar en ergy, (2) partitioning of biomass between lightgathering (leaf) and structural (stems and stal ks) components, and (3) an agi ng function. Plant nutrients are coupled to biomass through a hyperbolic phase relation for the mineral elements nitrogen, phosphorus, potassium, calcium, and magnesium. The logistic model descri bes response of the plant to levels of applied nutrients. Biomass is coupled to applied nutr ients through hyperbolic phase equations. Previous work has focused on annuals such as corn ( Zea mays L.) and forage grasses such as bermudagrass ( Cynodon dactylon L.) and bahiarass ( Paspalum notatum Flgge). Parts 1 and 3 of the memoir includes graphs of response data, while part 2 includes detail instructions on how to plot the 21 graphs but whic h are left for use as exercises to be completed by readers. Switchgrass is sometimes promoted as a kind of miracle grass which can be grown without either supplemental water (irrigation) or added fertilizer (nitrogen). While this can be done, biomass yield and efficiency of nitrogen utilization may be rather low. In fact, in Part 3 of this analysis it is shown that maximum efficiency of nitrogen utilization o ccurs at applied N of approximately 100 kg ha1. The general goal of science is to gain knowledge and understa nding of how nature works. Achievement of this goal requires a comprehensive theory to relate various factors together and among various studies with different crop specie s, soils, and environmental factors (such as rainfall, plant population, cultural practices). The u ltimate test is comparison of predictions of the theory with measurements and observations (fro m field studies in this particular case). Acknowledgement : The author thanks Amy G. Buhler, Engineering Librarian, Marston Science Library, University of Florida, for assist ance with preparation of this memoir. PAGE 3 Overman Production of Switchgrass 1 MODEL RESPONSE OF SWITCHGRASS TO APPLIED NITROGEN AND HARVEST INTERVAL INTRODUCTION Mathematical models are useful for describi ng output (response variables) in response to input (control variables). Extensive field studies have been conducted in various parts of the world over the last 150 years on crop response to a pplied nutrients and other factors, such as crop species, soil type, harvest interv al for perennial grasses, enviro nmental conditions (such as water availability), and plant population. The present analysis will focus on two particular models. The extended logistic model (Overman et al., 1994) de scribes coupling of seasonal biomass yield and plant nitrogen uptake with applied nutrients (such as nitrogen) through logistic equations. Biomass yield is then linked to plant nitrog en uptake through a hyperbol ic phase relation, which predicts a linear relationship between plant nitr ogen concentration and plant nitrogen uptake. The expanded growth model (Overman, 1998) descri bes accumulation of biomass with time through an analytical function which in corporates effects of energy input, partitioning of biomass between lightgathering and structural components, and aging as the plant grows. Plant nutrient and biomass accumulation are coupled through a hyper bolic phase relation. Field data have been used to confirm this model for coastal bermudagrass ( Cynodon dactylon L.) grown in Georgia (Overman and Brock, 2003). The growth model predicts a linearexpone ntial dependence of seasonal yield with a fixed harvest interval for pe rennial grasses. All of the results have been shown to be consistent with field data for nu merous crops, soils, and environmental conditions (Overman and Scholtz, 2002). This analysis will focus on response of the warmseason perennial switchgrass ( Panicum virgatum L.) to applied nitrogen at various harvest intervals. MODEL DESCRIPTION Response of biomass yield and plant nitrogen uptake to applied nitrog en can be described by the extended logistic model given by ) exp( 1 N c b A Yn y y (1) ) exp( 1 N c b A Nn n n u (2) ) exp( 1 ) exp( 1 N c b N c b N Y N Nn n n y cm u c (3) where N is applied nitrogen, kg ha1; Y is seasonal total biomass yield, Mg ha1; Nu is seasonal total plant nitrogen uptake, kg ha1; Nc is plant nitrogen concentration, g kg1; Ay is maximum yield at high N Mg ha1; An is maximum plant nitrogen uptake at high N kg ha1; Ncm = An/Ay is maximum plant nitrogen concentration at high N g kg1; by is intercept parameter for plant yield; bn is intercept parameter for plant nitrogen uptake; and cn is response coefficient for applied PAGE 4 Overman Production of Switchgrass 2 nitrogen, ha kg1. Note that the units on cn are the reciprocal of those for N Equations (1) through (3) are wellbehaved monotone increasing functions. Since variables N, Y, and Nu are defined as positive, parameters Ay, An, and cn must be positive as well, while parameters by and bn can be either positive, zero, or negative. Equations (1) and (2) with common cn can be combined to give the hyperbolic phase relation between Y and Nu u n u mN K N Y Y (4) where the hyperbolic and logistic parameters are coupled through ) exp( 1 b A Yy m (5) 1 ) exp( b A Kn n (6) with the shift in phase parameters defined by y nb b b (7) For Ym and Kn to be positive requires that b > 0. Equation (4) can be rearranged to the linear form u m m n u cN Y Y K Y N N 1 (8) which predicts a linear relationship between Nc and Nu. Dependence of biomass yield and plant nitroge n uptake on harvest inte rval for a perennial grass can be described by t t Yy y exp (9) t t Nn n u exp (10) t t Y N Ny y n n u c (11) where t is a fixed harvest interval, wk; and and are intercept and response parameters, respectively, for yield and plant nitrogen uptake. Parameter is assumed common for yield and plant nitrogen uptake. The expanded growth model for biomass accu mulation with calendar time is described by i iQ A Y (12) PAGE 5 Overman Production of Switchgrass 3 where Yi is biomass yield for the i th time interval, Mg ha1; Qi growth quantifier for the i th time interval; and A is the yield factor, Mg ha1. The growth quantifier is given by i i i i icx x x k x x kx Q2 exp exp exp erf erf 12 2 (13) with dimensionless time, x, defined in terms of calendar time, t, by 2 2 2c t x (14) where model parameters are defined as: is time to the mean of the energy distribution, wk; is time spread of the energy distribution, wk; c is the aging coefficient, wk1; and k is the partition coefficient between lightgatheri ng and structural components of the plant. The error function, erf, in Eq. (13) is defined by xdu u x0 2) exp( 2 erf (15) where u is the variable of integration. Cumulative growth quantifier, Q, and biomass, Y, are then defined by i iQ Q (16) i iY Y (17) It follows from Eq. (12) that cumulative yiel d and growth quantifier are coupled by the linear relationship i i i iAQ Q A Y Y (18) DATA ANALYSIS Data for this analysis are adapted from a field study by Madakadze et al. (1999) at McGill University in Montreal, Quebec, Canada. Sw itchgrass (cv. CaveinRock, Pathfinder, and Sunburst) was grown on St. Bernar d sandy clay loam (Typic Hapl udalf). Nitrogen treatments were 0, 75, and 150 kg ha1 yr1. Harvest intervals were 4, 6, and 15 wk. Measurements included PAGE 6 Overman Production of Switchgrass 4 dry matter yields and plant nitrogen uptake. Surf ace plant residue was re moved at the beginning of each season. Results are summarized in Table 1 and shown in Figure 1 for the 1996 season. The challenge Table 1. Dependence of biomass yield (Y), plant nitrogen uptake (Nu), and plant nitrogen concentration (Nc) on applied nitrogen (N) and harvest interval ( t) for switchgrass grown at Montreal, Quebec, Canada (1996). Data are averages of three cultivars.1 N Y Nu Nc Y Nu Nc Y Nu Nc kg ha1 Mg ha1 kg ha1 g kg1 Mg ha1 kg ha1 g kg1 Mg ha1 kg ha1 g kg1 4 wk 6 wk 15 wk 0 4.87 69.1 14.2 6.01 69.0 11.5 8.30 42.2 5.08 75 6.84 108.6 15.9 7.80 106.6 13.7 9.66 52.1 5.39 150 8.75 155.1 17.7 10.01 157.1 15.7 11.25 66.9 5.95 1Data adapted from Madakadze et al. (1999). is to estimate parameters in Eqs. (1) through (3) from the data. Results for 4and 6wk harvest intervals are averaged for this purpose. Equati ons (1) and (2) can be linearized to the forms N r N N c b Y Zn y y0100 0 22 0 99918 0 00975 0 22 0 1 00 12 ln (12) N r N N c b N Zn n u n0100 0 98 0 9971 0 00981 0 98 0 1 250 ln (13) where Ay = 12.00 Mg ha1 and An = 250 kg ha1 have been chosen to give equal values of cn. The next step is to estimate Ay and An for each harvest interval. On the assumption that by, bn, and cn are common for all harvest intervals, standardized yields ( Y* ) and plant N uptake ( Nu* ) can be calculated, respectively, from yA N Y Y 0100 0 22 0 exp 1 (14) n u uA N N N 0100 0 98 0 exp 1 (15) for each nitrogen level and each harvest interval as shown in Table 2. This leads to estimates of Ay(4 wk) = 10.99, Ay(6 wk) = 12.89, and Ay(15 wk) = 16.12 Mg ha1; An(4 wk) = 248, An(6 wk) = 248, and An(15 wk) = 127 kg ha1; and Ncm(4 wk) = 22.6, Ncm(6 wk) = 19.2, and Ncm(15 wk) = 7.88 g kg1. The yield and plant N res ponse equations now become PAGE 7 Overman Production of Switchgrass 5 Table 2. Dependence of standardized yield ( Y* ) and standardized pl ant nitrogen uptake ( Nu* ) on applied nitrogen ( N ) and harvest interval ( t ) for switchgrass at Montreal, Quebec, Canada. N Y* Nu* kg ha1 Mg ha1 kg ha1 4 wk 6 wk 15 wk 4 wk 6 wk 15 wk 0 10.94 13.50 18.64 253 253 155 75 10.85 12.39 15.35 245 241 118 150 11.17 12.79 14.38 247 250 107 avg 10.99 12.89 16.12 248 248 127 ) 0100 0 22 0 exp( 1 N A Yy (16) ) 0100 0 98 0 exp( 1 N A Nn u (17) ) 0100 0 98 0 exp( 1 ) 0100 0 22 0 exp( 1 N N N Y N Ncm u c (18) Curves in Figure 1 are drawn from Eqs. (16) through (18) with appropriate values for Ay, An, and Ncm. The next step is to estimate hyperbolic parame ters for the phase relations. These are given by t = 4 wk: 1 1ha kg 218 1 ) 76 0 exp( 248 ha Mg 64 20 ) 76 0 exp( 1 99 10 n mK Y (19) u uN N Y 218 64 20 (20) u cN N 0485 0 56 10 (21) t = 6 wk: 1 1ha kg 218 1 ) 76 0 exp( 248 ha Mg 21 24 ) 76 0 exp( 1 89 12 n mK Y (22) u uN N Y 218 21 24 (23) u cN N 0413 0 00 9 (24) t = 15 wk: 1 1ha kg 112 1 ) 76 0 exp( 127 ha Mg 28 30 ) 76 0 exp( 1 12 16 n mK Y (25) u uN N Y 112 28 30 (26) u cN N 0330 0 70 3 (27) PAGE 8 Overman Production of Switchgrass 6 Phase plots are shown in Figure 2, where curves are drawn from Eqs. (20), (23), and (26) and lines from Eqs. (21), (24), and (27). The second challenge is to determine response to harvest interval. This is equivalent to determining dependence of Ay and An on harvest interval. It is assumed that these relations are given by t t Ay y y exp (28) t t An n n exp (29) Following analysis by Overman and Scholtz (2002) for coastal bermudagrass, it is assumed that = 0.075 wk1. It follows that standardized Ay, ( Ay* ), and An, ( An* ), can be calculated from t t A Ay y y y ) 075 0 exp( (30) t t A An n n n ) 075 0 exp( (31) for each harvest interval. Results are given in Tabl e 3. This leads to the standardized equations Table 3. Dependence of sta ndardized yield parameter ( Ay* ) and standardized plant N parameter ( An* ) on harvest interval ( t ) for switchgrass at Montreal, Quebec, Canada. t Ay Ay* An An* Ncm wk Mg ha1 Mg ha1 kg ha1 kg ha1 g kg1 4 10.99 14.83 248 335 22.6 6 12.89 20.22 248 389 19.2 15 16.12 49.65 127 391 7.88 99963 0 20 3 59 1 r t Ay (32) t An 0 20 267 (33) and to dependence of Ay, An, and Ncm on harvest interval of t t Ay 075 0 exp 20 3 59 1 (34) t t An 075 0 exp 0 20 267 (35) t t A A Ny n cm 20 3 59 1 0 20 267 (36) Results are shown in Figure 3, where the curves ar e drawn from Eqs. (34) through (36). It can be shown that peak harvest interval for yield ( tpy) and plant N uptake ( tpn) are determined by 8 12 20 3 59 1 075 0 1 1 y y pyt wk (37) PAGE 9 Overman Production of Switchgrass 7 0 0 0 20 267 075 0 1 1 n n pnt wk (38) Values of n andn have been chosen to make tpn = 0, consistent with the data. Finally, accumulation of plant nitrogen and bi omass can be coupled. Data are given in Table 4 and shown in Figure 4. The data appear to fo llow straight lines for the constant harvest Table 4. Coupling of biomass yield ( Y ) and plant nitrogen accumulation ( Nu) with calendar time ( t ) and applied nitrogen ( N ) for switchgrass grown at Montreal, Quebec, Canada (1996).1 t t Y Y Nu Nu Y Y Nu Nu Y Y Nu Nu wk wk Mg ha1 kg ha1 Mg ha1 kg ha1 Mg ha1 kg ha1 N = 0 N = 75 kg ha1 N = 150 kg ha1 4 23.1 0 0 0 0 0 0 2.54 37.8 3.21 48.3 3.87 66.7 27.1 2.54 37.8 3.21 48.3 3.87 66.7 1.36 21.2 2.22 42.1 3.08 62.3 31.1 3.90 59.0 5.43 90.4 6.95 129.0 0.97 10.1 1.41 18.2 1.80 26.1 35.4 4.87 69.1 6.84 108.6 8.75 155.1 6 23.1 0 0 0 0 0 0 3.86 42.0 4.46 60.3 5.98 89.0 29.1 3.86 42.0 4.46 60.3 5.98 89.0 2.15 26.9 3.34 46.3 4.03 68.1 35.3 6.01 68.9 7.80 106.6 10.01 157.1 15 23.1 0 0 0 0 0 0 8.30 42.2 9.66 52.1 11.25 66.9 38.0 8.30 42.2 9.66 52.1 11.25 66.9 1Data adapted from Madakadze et al. (1999). intervals. Since the lines s hould exhibit intercepts of ( Y, Nu) = (0, 0) on intuitive grounds, simple moment analysis leads to the lines shown in Figure 5 given by t = 4 wk: N = 0: Y Y Nu6 14 38 45 6 662 (39) N = 75 kg ha1: Y Y Nu0 16 57 86 1389 (40) N = 150 kg ha1: Y Y Nu0 18 8 139 2511 (41) PAGE 10 Overman Production of Switchgrass 8 t = 6 wk: N = 0: Y Y Nu3 11 02 51 2 576 (42) N = 75 kg ha1: Y Y Nu6 13 73 80 1100 (43) N = 150 kg ha1: Y Y Nu5 15 0 136 2105 (44) t = 15 wk: N = 0: Y Y Nu08 5 9 68 3 350 (45) N = 75 kg ha1: Y Y Nu39 5 3 93 3 503 (46) N = 150 kg ha1: Y Y Nu95 5 6 126 6 752 (47) It may be noted that the slopes represent plant N concentrations ( Nc). This is illustrated in Figure 5, where the curves are drawn from Eq. (18) with Ncm(4 wk) = 22.6, Ncm(6 wk) = 19.2, and Ncm(15 wk) = 7.88 g kg1 as estimated previously. Results by two different approaches are consistent. DISCUSSION The extended logistic model appears to give reasonable correlation of yield, plant N uptake, and plant N concentration with a pplied N (Figure 1) for switchgra ss grown in Canada. This leads to excellent phase plots of yiel d vs. plant N uptake (Figure 2). Pr ediction of a linear relationship between plant N concentration and plant N uptake is confirmed. Dependence of the phase plots on harvest interval is illustrate d. Dependence of yield, plant N uptake, and plant N concentration parameters on harvest interval is described adeq uately by the expanded grow th model (Figure 3). Several inferences follow from this analys is. The point of maximum slope of the yield response curves is determined by 22 0100 0 22 02 / 1 n yc b N N kg ha1, 22 / 1yA Y (48) with the corresponding value of the yield dependi ng on harvest interval. The point of maximum slope of the plant N respons e curves is determined by 98 0100 0 98 02 / 1 n nc b N Nkg ha1, 22 / 1 n uA N (49) with the corresponding value of plant N uptak e depending on harvest interval. Differential response of plant N uptake at this point is given by PAGE 11 Overman Production of Switchgrass 9 62 0 4 ) 0100 0 )( 248 ( 4max n n uc A dN dN (50) for harvest intervals of 4 and 6 wk. This means th at maximum efficiency of nitrogen recovery is 62% for this harvest interval. Actual efficiency of plant N recovery, E is defined by N N N Eu u 0 (51) where Nu 0 is plant N uptake at N = 0. Peak nitrogen recovery, Ep, can be estimated from ) exp( 1 1 ) 5 0 exp( 1 1 5 1 4 4 5 1n n n n n p n n pb b b c A E c b N (52) For harvest intervals of 4 and 6 wk these values become Np = 147 kg ha1 and Ep = 0.585 = 58.5%. A dramatic decrease in maximum plant N concentration ( Ncm) with increased harvest interval ( t ) may be noted from Figure 3. This occurs be cause as the plant ages the fraction of lightgathering component (high N con centration) decreases in relation to the st ructural component (low N concentration) of the plant. The expa nded growth model describes this phenomenon quite well. The lower limit of plant N concentration, Ncl, at reduced soil nitrogen ( N <<0) can be estimated by combining Eqs. (5), (6), and (8) to obtain ) exp(b N Y K Ncm m n cl (53) For a harvest interval of 4 wk, this value is 10.6 g kg1. At this harvest interval the variables are bounded by 0 < Y < 10.99 Mg ha1, 0 < Nu < 248 kg ha1, and 10.6 < Nc < 22.6 g kg1. Actual values of the variables depend on the level of applied N. REFERENCES Madakadze, I.C., K.A. Stewart, P.R. Peterson, B.E. Coulman, and D.L. Smith. 1999. Cutting frequency and nitrogen fertil ization effects on yield and nitrogen concentration of switchgrass in a short season area. Crop Science 39:552557. Overman, A.R. 1998. An expanded growth model for grasses. Communications in Soil Science and Plant Analysis 29:6785. Overman, A.R. and K.H. Brock. 2003. Confirma tion of the expanded growth model for a warmseason perennial grass. Communications in Soil Science and Plant Analysis 34:11051117. Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield Taylor & Francis. Philadelphia, PA. 328 p. Overman, A.R., S.R. Wilkinson, and D.M. W ilson. 1994. An extended model of forage grass response to applied nitrogen. Agronomy J. 86:617620. PAGE 12 Overman Production of Switchgrass 10 List of Figures Figure 1. Response of biomass yield ( Y ), plant N uptake ( Nu), and plant N concentration ( Nc) to applied nitrogen ( N ) and harvest interval ( t ) for switchgrass grown at Montreal, Quebec, Canada. Data adapted from Madakadze et al. (1999) Curves drawn from Eqs. (16) through (18) with Ay(4 wk) = 10.99, Ay(6 wk) = 12.89, and Ay(15 wk) = 16.12 Mg ha1; An(4 wk) = 248, An(6 wk) = 248, and An(15 wk) = 127 kg ha1; and Ncm(4 wk) = 22.6, Ncm(6 wk) = 19.2, and Ncm(15 wk) = 7.88 g kg1. Figure 2. Phase plots of biomass yield ( Y ) and plant N concentration ( Nc) vs. plant N uptake ( Nu) for three harvest intervals ( t ) for switchgrass grown at Montreal Quebec, Canada. Data adapted from Madakadze et al. (1999). Curves drawn from Eqs. (20), (23), and (26); lines drawn from Eqs. (21), (24), and (27). Figure 3. Dependence of parameters for yield ( Ay), plant N uptake ( An), and plant N concentration ( Nc) on harvest interval ( t ) for switchgrass grown at Montreal, Quebec, Canada. Curves drawn from Eqs. (34) through (36). Figure 4. Correlation of cumulative plant N uptake ( Nu) with cumulative biomass ( Y ) for applied nitrogen ( N ) and harvest interval ( t ) for switchgrass grown at Mont real, Quebec, Canada. Data adapted from Madakadze et al. (1999). Li nes drawn from Eqs. (39) through (47). Figure 5. Dependence of plant N concentration ( Nc) on applied nitrogen ( N ) and harvest interval ( t ) for switchgrass grown at Montreal, Quebec, Ca nada. Data taken from Table 4. Curves drawn from Eq. (18) with Ncm(4 wk) = 22.6, Ncm(6 wk) = 19.2, and Ncm(15 wk) = 7.88 g kg1. PAGE 13 Overman Production of Switchgrass 11 Figure 1. Response of biomass yield ( Y ), plant N uptake ( Nu), and plant N concentration ( Nc) to applied nitrogen ( N ) and harvest interval ( t ) for switchgrass grown at Montreal, Quebec, Canada. Data adapted from Madakadze et al. (1999) Curves drawn from Eqs. (16) through (18) with Ay(4 wk) = 10.99, Ay(6 wk) = 12.89, and Ay(15 wk) = 16.12 Mg ha1; An(4 wk) = 248, An(6wk) = 248, and An(15 wk) = 127 kg ha1; and Ncm(4 wk) = 22.6, Ncm(6 wk) = 19.2, and Ncm(15 wk) = 7.88 g kg1. PAGE 14 Overman Production of Switchgrass 12 Figure 2. Phase plots of biomass yield ( Y ) and plant N concentration ( Nc) vs. plant N uptake ( Nu) for three harvest intervals ( t ) for switchgrass grown at Montreal Quebec, Canada. Data adapted from Madakadze et al. (1999). Curves drawn from Eqs. (20), (23), and (26); lines drawn from Eqs. (21), (24), and (27). PAGE 15 Overman Production of Switchgrass 13 Figure 3. Dependence of parameters for yield ( Ay), plant N uptake ( An), and plant N concentration ( Nc) on harvest interval ( t ) for switchgrass grown at Montreal, Quebec, Canada. Curves drawn from Eqs. (34) through (36). PAGE 16 Overman Production of Switchgrass 14 Figure 4. Correlation of cumulative plant N uptake ( Nu) with cumulative biomass ( Y ) for applied nitrogen ( N ) and harvest interval ( t ) for switchgrass grown at Mont real, Quebec, Canada. Data adapted from Madakadze et al. (1999). Li nes drawn from Eqs. (39) through (47). PAGE 17 Overman Production of Switchgrass 15 Figure 5. Dependence of plant N concentration ( Nc) on applied nitrogen ( N ) and harvest interval ( t ) for switchgrass grown at Montreal, Quebec, Ca nada. Data taken from Table 4. Curves drawn from Eq. (18) with Ncm(4 wk) = 22.6, Ncm(6 wk) = 19.2, and Ncm(15 wk) = 7.88 g kg1. PAGE 18 Overman Production of Switchgrass 16 Assume the parameters: ti, = 21.8 wk, 5 wk 15 0 wk, 00 8 2 wk, 0 261 k c These values lead to dimensionless time ( x ) and growth quantifier ( Q ) given by 00 8 2 21 600 0 00 8 0 26 2 2 2 t t c t x xi = 0.075 ( ) 094 1 9943 0 exp 821 2 085 0 erf 625 0 2 exp exp exp erf erf 12 2 2 x x x c x x k x x kx Qi i i i ( ) Values of the variables along with cu mulative biomass are listed in Table Table Correlation of cumulative biomass ( Y ) and the growth quantifier ( Q ) with calendar time ( t ) for switchgrass grown at Montreal, Quebec, Canada (1996).1 t x erf x exp( x2) Q Y wk Mg ha1 N kg ha1 0 75 150 avg 21.8 0.0750 0.085 0.9943 0.00 27.1 0.7375 0.702 0.580 1.71 2.54 3.21 3.87 3.21 31.1 1.2375 0.921 0.216 2.97 3.90 5.43 6.95 5.43 35.7 1.8125 0.9896 0.0374 3.57 4.87 6.84 8.75 6.82 1.0000 0.0000 3.69 1Yield data adapted from Madakadze et al. (1999). Correlation of biomass yields with growth quantifier is gi ven by the regression equations N = 0: Q Y219 1 422 0 r = 0.9942 ( ) N = 75 kg ha1: Q Y911 1 089 0 r = 0.9971 ( ) N = 150 kg ha1: Q Y596 2 616 0 r = 0.9987 ( ) Avg: Q Y913 1 108 0 r = 0.9975 ( ) PAGE 19 Overman Production of Switchgrass 17 Table Coupling of biomass yield (Y) and plant nitrog en accumulation (Nu) with calendar time (t) and applied nitrogen (N) for switchgrass grown at Montreal, Quebec, Canada (1996).1 t Y Y Nu Nu Y Y Nu Nu Y Y Nu Nu wk Mg ha1 kg ha1 Mg ha1 kg ha1 Mg ha1 kg ha1 N = 0 N = 75 kg ha1 N = 150 kg ha1 21.8 0 0 0 0 0 0 2.54 37.8 3.21 48.9 3.87 66.7 27.1 2.54 37.8 3.21 48.4 3.87 66.7 1.36 21.2 2.22 42.1 3.08 62.3 31.1 3.90 59.0 5.43 90.4 6.95 129.0 0.97 10.1 1.41 18.2 1.80 26.1 35.4 4.87 69.1 6.84 108.6 8.75 155.1 1Data adapted from Madakadze et al. (1999). N = 0: Y Y Nu6 14 38 45 6 662 ( ) N = 75 kg ha1: Y Y Nu0 16 73 80 1389 ( ) N = 150 kg ha1: Y Y Nu0 18 8 139 2511 ( ) PAGE 20 Overman Production of Switchgrass 18 Assume the parameters: ti, = 23.0 wk, 5 wk 15 0 wk, 00 8 2 wk, 0 261 k c These values lead to dimensionless time (x) and growth quantifier (Q) given by 00 8 2 21 600 0 00 8 0 26 2 2 2 t t c t x xi = 0.225 ( ) 310 1 9506 0 exp 821 2 250 0 erf 125 0 2 exp exp exp erf erf 12 2 2 x x x c x x k x x kx Qi i i i ( ) Table Estimation of biomass yield (Y), plant N uptake (Nu), and plant N concentration (Nc) with calendar time (t) from the expanded growth model for switchgrass grown at Montreal, Quebec, Canada (1996). t x erf x exp(x2) Q Y wk Mg ha1 N, kg ha1 0 75 150 23.0 0.225 0.250 0.9506 0.000 0.00 0.00 24 0.350 0.379 0.8847 0.222 0.51 0.59 0.72 25 0.475 0.498 0.798 0.523 1.20 1.40 1.69 26 0.600 0.604 0.698 0.876 2.01 2.35 2.83 27 0.725 0.694 0.591 1.256 2.88 3.37 4.06 27.1 0.7375 0.703 0.580 1.295 2.97 3.47 4.18 28 0.850 0.771 0.486 1.633 3.74 4.38 5.37 29 0.975 0.832 0.386 1.991 4.56 5.34 6.43 29.1 0.9875 0.838 0.377 2.023 4.63 5.42 6.53 30 1.100 0.880 0.298 2.309 5.29 6.19 7.46 31 1.225 0.9163 0.223 2.580 5.91 6.91 8.33 32 1.350 0.9438 0.162 2.801 6.41 7.51 9.05 33 1.475 0.9632 0.114 2.975 6.81 7.97 9.61 34 1.600 0.9763 0.0773 3.108 7.12 8.33 10.04 35 1.725 0.9852 0.0510 3.204 7.34 8.59 10.35 36 1.850 0.9911 0.0326 3.271 7.49 8.77 10.57 37 1.975 0.9948 0.0202 3.316 7.59 8.89 10.71 38 2.100 0.9970 0.0122 3.346 7.66 8.97 10.81 39 2.225 0.9983 0.00708 3.364 7.70 9.02 10.87 40 2.350 0.9991 0.00400 3.376 7.73 9.05 10.90 42 2.600 0.99976 0.00116 3.386 7.75 9.07 10.94 44 2.850 0.99996 0.00030 3.389 7.76 9.08 10.95 1.00000 0.00000 3.390 7.76 9.09 10.95 PAGE 21 Overman Production of Switchgrass 19 Q Y Mg ha1 N, kg ha1 0 75 150 1.295 2.54 3.21 3.87 2.023 3.86 4.46 5.98 3.346 8.30 9.66 11.25 N = 0: Q Q Q Q QY Y29 2 96 16 87 38 2 N = 75 kg ha1: Q Q Q Q QY Y68 2 96 16 50 45 2 N = 150 kg ha1: Q Q Q Q QY Y23 3 96 16 75 54 2 PAGE 22 Overman Production of Switchgrass 20 Assume the parameters: 5 wk 15 0 wk, 00 8 2 wk, 0 261 k c These values lead to dimensionless time (x) and growth quantifier (Q) given by 00 8 2 21 600 0 00 8 0 26 2 2 2 t t c t x ( ) i i i i i i i i ix x x x x x x c x x k x x kx Q20 1 exp exp exp 821 2 erf erf 5 1 2 exp exp exp erf erf 12 2 2 2 () Table Estimates of the growth quantifier (Q) with calendar time (t) for the expanded growth model. t x erf x exp(x2) Qi Q wk 23.1 0.2375 0.264 0.9452 0 1.259 27.1 0.7375 0.703 0.5805 1.259 1.078 31.1 1.2375 0.9198 0.2162 2.337 0.598 35.4 1.7750 0.9880 0.0428 2.935 23.1 0.2375 0.264 0.9452 0 1.988 29.1 0.9875 0.837 0.3771 1.988 1.140 35.4 1.7750 0.9880 0.0428 3.128 23.1 0.2375 0.264 0.9452 0 3.317 38.0 2.100 0.9970 0.0122 3.317 PAGE 23 Overman Production of Switchgrass 21 Table Dependence of biomass accumulation (Y) on the growth quantifier (Q) and applied nitrogen (N) for switchgrass grown at Montreal, Quebec, Canada.1 t Q Y wk Mg ha1 N, kg ha1 0 75 150 23.1 0 0 0 0 27.1 1.26 2.54 3.21 3.87 31.1 2.34 3.90 5.43 6.95 35.4 2.94 4.87 6.84 8.75 23.1 0 0 0 0 29.1 1.99 3.86 4.46 5.98 35.4 3.13 6.01 7.80 10.01 23.1 0 0 0 0 38.0 3.32 8.30 9.66 11.25 1Yield data adapted from Madakadze et al. (1999). t = 4 wk N = 0: Q Q Q Q QY Y70 1 71 15 64 26 2 N = 75 kg ha1: Q Q Q Q QY Y35 2 71 15 86 36 2 N = 150 kg ha1: Q Q Q Q QY Y98 2 71 15 86 46 2 t = 6 wk N = 0: Q Q Q Q QY Y93 1 76 13 49 26 2 N = 75 kg ha1: Q Q Q Q QY Y42 2 76 13 29 33 2 N = 150 kg ha1: Q Q Q Q QY Y14 3 76 13 23 43 2 t = 15 wk N = 0: Q Q Q Q QY Y50 2 02 11 56 27 2 N = 75 kg ha1: Q Q Q Q QY Y91 2 02 11 07 32 2 PAGE 24 Overman Production of Switchgrass 22 N = 150 kg ha1: Q Q Q Q QY Y39 3 02 11 35 37 2 PAGE 25 Overman Production of Switchgrass 23 MODEL RESPONSE OF SWITCHGRASS TO APPLIED NITROGEN AND HARVEST INTERVAL ALLEN R. OVERMAN Agricultural and Biological Engineering Department, University of Florida, Gainesville, FL 326110570 INTRODUCTION Mathematical models are useful for describi ng output (response variables) in response to input (control variables). Extensive field studies have been conducted in various parts of the world over the past 150 years on crop response to applied nutrients and other factors, such as crop species, soil type, harvest interval for pere nnial grasses, environmen tal conditions (such as water availability), and plant population. The present analysis will focus on two particular models. The extended logistic model (Overman et al., 1994) describes coupling of seasonal biomass yield and plant nitrogen uptake with applied nutrients (suc h as nitrogen) through logistic equations. Biomass yield is then linked to pl ant nitrogen uptake th rough a hyperbolic phase relation, which predicts a linear relationship between plant nitrogen concentration and plant nitrogen uptake. The expanded growth mode l (Overman, 1998) describes accumulation of biomass with time through an analytical function which incorporates effects of energy input, partitioning of biomass between lightgathering and structural components, and aging as the plant grows. Plant nutrient and biomass accumulation are coupled through a hyperbolic phase relation. Field data have been used to c onfirm this model for coastal bermudagrass (Cynodon dactylon L.) grown in Georgia (Overman and Brock, 2003). The growth mode l predicts a linearexponential dependence of seasonal yi eld with a fixed harvest interv al for perennial grasses. All of the results have been shown to be consistent with field data for numerous crops, soils, and environmental conditions (Overman and Scholtz, 2002). This analysis will focus on response of the warmseason perennial switchgrass (Panicum virgatum L.) to applied nitrogen at various harvest intervals. MODEL DESCRIPTION Response of biomass yield and plant nitrogen uptake to applied nitrog en can be described by the extended logistic model given by ) exp( 1N c b A Yn y y (1) ) exp( 1N c b A Nn n n u (2) ) exp( 1 ) exp( 1N c b N c b N Y N Nn n n y cm u c (3) PAGE 26 Overman Production of Switchgrass 24 where N is applied nitrogen, kg ha1; Y is seasonal total biomass yield, Mg ha1; Nu is seasonal total plant nitrogen uptake, kg ha1; Nc is plant nitrogen concentration, g kg1; Ay is maximum yield at high N, Mg ha1; An is maximum plant nitrogen uptake at high N, kg ha1; Ncm = An/Ay is maximum plant nitrogen concentration at high N, g kg1; by is intercept parameter for plant yield; bn is intercept parameter for plant nitrogen uptake; and cn is response coefficient for applied nitrogen, ha kg1. Note that the units on cn are the reciprocal of those for N. Equations (1) through (3) are wellbehaved monotone increasing functions. Since variables N, Y, and Nu are defined as positive, parameters Ay, An, and cn must be positive as well, while parameters by and bn can be either positive, zero, or negative. Equations (1) and (2) with common cn can be combined to give the hyperbolic phase relation between Y and Nu u n u mN K N Y Y (4) where the hyperbolic and logistic parameters are coupled through ) exp( 1b A Yy m (5) 1 ) exp( b A Kn n (6) with the shift in intercept parameters defined by y nb b b (7) For Ym and Kn to be positive requires that b > 0. Equation (4) can be rearranged to the linear form u m m n u cN Y Y K Y N N1 (8) which predicts a linear relationship between Nc and Nu. Dependence of biomass yield and plant nitroge n uptake on harvest inte rval for a perennial grass can be described by t t Yy y exp (9) t t Nn n u exp (10) t t Y N Ny y n n u c (11) PAGE 27 Overman Production of Switchgrass 25 where t is a fixed harvest interval, wk; and and are intercept and response parameters, respectively, for yield and plant nitrogen uptake. Parameter is a plant aging factor and is assumed common for yield a nd plant nitrogen uptake. The expanded growth model for biomass accumulation (Y) with calendar time (t) from Jan. 1 for a perennial grass is described by i iQ A Y (12) where Yi is biomass yield for the i th growth interval, Mg ha1; Qi is growth quantifier for the i th growth interval; and A is the yield factor, Mg ha1. The growth quantifier is defined by i i i i icx x x k x x kx Q2 exp exp exp erf erf 12 2 (13) with dimensionless time, x, defined in terms of calendar time, t, by 2 2 2c t x (14) where model parameters are defined by: is time to the mean of the energy distribution, wk; is time spread of the energy distribution, wk; c is the aging coefficient, wk1; and k is the partition coefficient between lightgathering and struct ural components of the plant. Note that xi corresponds to the time of initiation of growth, ti. The error function, erf, in Eq. (13) is defined by xdu u x0 2) exp( 2 erf (15) where u is the variable of integration. Values of the error function can be obtained from mathematical tables (Abramowitz and St egun, 1965). Cumulative growth quantifier, Q, and biomass, Y, are then given by i iQ Q (16) i iY Y (17) It follows from Eq. (12) that cumulative yiel d and growth quantifier are coupled by the linear relationship i i i iAQ Q A Y Y (18) PAGE 28 Overman Production of Switchgrass 26 DATA ANALYSIS Data for this analysis are adapted from a field study by Madakadze et al. (1999) at McGill University in Montreal, Quebec, Canada. Switc hgrass (cv. CaveinRock, Pathfinder, and Sunburst) was grown on St. Bernar d sandy clay loam (Typic Hapludalf), a free draining soil. Nitrogen treatments were 0, 75, and 150 kg ha1 yr1 applied as ammonium nitrate. Harvest intervals were 4, 6, and 15 wk. Measurements included dry matter yields and plant nitrogen uptake. Surface plant residue was remove d at the beginning of each season. Results are summarized in Table 1 and shown in Figure 1 for the 1996 season. The challenge is estimation of parameters in Eqs. (1) through (3 ) from the data. Results for 4and 6wk harvest intervals are averaged for this purpose. Equati ons (1) and (2) can be linearized to the forms N r N N c b Y Zn y y0098 0 22 0 9971 0 00975 0 22 0 1 00 12 ln (19) N r N N c b N Zn n u n0098 0 98 0 99917 0 00981 0 98 0 1 250 ln (20) where Ay = 12.00 Mg ha1 and An = 250 kg ha1 have been chosen to give equal values of cn. Note the high correlation coefficients, r. The next step is to estimate Ay and An for each harvest interval. On the assumption that by, bn, and cn are common for all harvest intervals, standardized yields (Y*) and standardized plant N uptake (Nu*) can be calculated, respectively, from yA N Y Y 0098 0 22 0 exp 1 (21) n u uA N N N 0098 0 98 0 exp 1 (22) for each nitrogen level and each harvest interval as shown in Table 2. This leads to estimates of Ay(4 wk) = 11.04, Ay(6 wk) = 12.95, and Ay(15 wk) = 16.18 Mg ha1; An(4 wk) = 250, An(6 wk) = 250, and An(15 wk) = 127 kg ha1; and Ncm(4 wk) = 22.6, Ncm(6 wk) = 19.3, and Ncm(15 wk) = 7.85 g kg1. The yield and plant N res ponse equations now become ) 0098 0 22 0 exp( 1N A Yy (23) ) 0098 0 98 0 exp( 1N A Nn u (24) ) 0098 0 98 0 exp( 1 ) 0098 0 22 0 exp( 1N N N Y N Ncm u c (25) Curves in Figure 1 are drawn from Eqs. (23) through (25) with appropriate values for Ay, An, and Ncm. The next step is to estimate hyperbolic parame ters for the phase relations. These are given by PAGE 29 Overman Production of Switchgrass 27 t = 4 wk: 1 1ha kg 220 1 ) 76 0 exp( 250 ha Mg 74 20 ) 76 0 exp( 1 04 11 n mK Y (26) u uN N Y 220 74 20 (27) u cN N0482 0 61 10 (28) t = 6 wk: 1 1ha kg 220 1 ) 76 0 exp( 250 ha Mg 33 24 ) 76 0 exp( 1 95 12 n mK Y (29) u uN N Y 220 33 24 (30) u cN N0411 0 04 9 (31) t = 15 wk: 1 1ha kg 112 1 ) 76 0 exp( 127 ha Mg 39 30 ) 76 0 exp( 1 18 16 n mK Y (32) u uN N Y 112 39 30 (33) u cN N0329 0 69 3 (34) Phase plots are shown in Figure 2, where curves are drawn from Eqs. (27), (30), and (33) and lines from Eqs. (28), (31), and (34). The second challenge is to determine response to harvest interval. This is equivalent to determining dependence of Ay and An on harvest interval. It is assumed that these relations are given by t t Ay y y exp (35) t t An n n exp (36) Following analysis by Overman and Scholtz (2002) for coastal bermudagrass, it is assumed that = 0.075 wk1. It follows that standardized Ay, (yA), and An, (nA), can be calculated from t t A Ay y y y ) 075 0 exp( (37) t t A An n n n ) 075 0 exp( (38) for each harvest interval. Results are given in Tabl e 3. This leads to the standardized equations 99963 0 21 3 61 1 r t Ay (39) t An 0 20 267 (40) and to dependence of Ay, An, and Ncm on harvest interval of t t Ay 075 0 exp 21 3 61 1 (41) PAGE 30 Overman Production of Switchgrass 28 t t An 075 0 exp 0 20 267 (42) t t A A Ny n cm 21 3 61 1 0 20 267 (43) Results are shown in Figure 3, where the curves ar e drawn from Eqs. (41) through (43). It can be shown that peak harvest interval for yield ( tpy) and plant N uptake ( tpn) are determined by 8 12 21 3 61 1 075 0 1 1 y y pyt wk (44) 0 0 0 20 267 075 0 1 1 n n pnt wk (45) Values of n andn have been chosen to make tpn 0, consistent with the data. Accumulation of plant nitrogen and biomass can be coupled Data are given in Table 4 and shown in Figure 4. The data appear to follow stra ight lines for the constant harvest intervals. Since the lines should ex hibit intercepts of (Y, Nu) = (0, 0) on intuitive grounds, simple moment analysis leads to the lines shown in Figure 5 given by t = 4 wk: N = 0: Y Y Y YN Nu u6 14 38 45 6 6622 (46) N = 75 kg ha1: Y Y Nu0 16 57 86 1389 (47) N = 150 kg ha1: Y Y Nu0 18 8 139 2511 (48) t = 6 wk: N = 0: Y Y Nu3 11 02 51 2 576 (49) N = 75 kg ha1: Y Y Nu6 13 73 80 1100 (50) N = 150 kg ha1: Y Y Nu5 15 0 136 2105 (51) t = 15 wk: N = 0: Y Y Nu08 5 9 68 3 350 (52) N = 75 kg ha1: Y Y Nu39 5 3 93 3 503 (53) N = 150 kg ha1: Y Y Nu95 5 6 126 6 752 (54) It may be noted that the slopes represent plant N concentrations (Nc). This is illustrated in Figure 5, where the curves are drawn from Eq. (25) with Ncm(4 wk) = 22.6, Ncm(6 wk) = 19.3, and PAGE 31 Overman Production of Switchgrass 29 Ncm(15 wk) = 7.85 g kg1 as estimated previously. Results by two different approaches are consistent. Finally, biomass accumulation with time can be described by the expanded growth model. Assume the parameters: 5 wk 15 0 wk, 00 8 2 wk, 0 261 k c These values lead to dimensionless time (x) and growth quantifier ( Qi) given by 00 8 2 21 600 0 00 8 0 26 2 2 2 t t c t x (55) i i i i i i i i ix x x x x x x c x x k x x kx Q20 1 exp exp exp 821 2 erf erf 5 1 2 exp exp exp erf erf 12 2 2 2 (56) Estimates of the cumulative growth quantifier (Q) with calendar time (t) are given in Table 5 for the three harvest intervals. Note that time of initiation (ti) must be reset for each growth interval. The growing season is assumed to begin at t = 23.1 wk. The next step is to couple cumulative biomass yield (Y) with cumulative growth quantifier (Q) as given in Table 6. Results are also shown in Figure 6. Due to the limited number of points, slopes of th e lines in Figure 6 are estimated by simple moment analysis to be t = 4 wk N = 0: Q Q Q Q QY Y70 1 71 15 64 26 2 (57) N = 75 kg ha1: Q Q Y35 2 71 15 86 36 (58) N = 150 kg ha1: Q Q Y98 2 71 15 86 46 (59) t = 6 wk N = 0: Q Q Y93 1 76 13 49 26 (60) N = 75 kg ha1: Q Q Y42 2 76 13 29 33 (61) N = 150 kg ha1: Q Q Y14 3 76 13 23 43 (62) t = 15 wk N = 0: Q Q Y50 2 02 11 56 27 (63) N = 75 kg ha1: Q Q Y91 2 02 11 07 32 (64) N = 150 kg ha1: Q Q Y39 3 02 11 35 37 (65) PAGE 32 Overman Production of Switchgrass 30 Correlation of yield factor (A) with applied nitrogen (N) is shown in Figure 7. It seems logical that this correlation should be described by the logistic model ) 0098 0 22 0 exp( 1 ) exp( 1N A N c b A An y (66) where A is maximum A at high N, Mg ha1. It follows from Eq. (66) that standardized A, (A), can be calculated for each nitrogen level and harvest interval from A N A A) 0098 0 22 0 exp( 1 (67) Results are given in Table 7. It follows from Table 7 that response of parameter A to applied nitrogen is given by Eq. (66) with A(4 wk) = 3.80, A(6 wk) = 4.08, A(15 wk) = 4.88 Mg ha1. The curves in Figure 7 are drawn from Eq. (66) with appr opriate values of A. Results are very similar to those for Y vs. N in Figure 1. DISCUSSION The extended logistic model appears to give reasonable correlation of yield, plant N uptake, and plant N concentration with applied N (Figure 1) for switchgrass grown in Canada. This leads to excellent phase plots of yiel d vs. plant N uptake (Figure 2). Pr ediction of a linear relationship between plant N concentration and plant N uptake is confirmed. Dependence of the phase plots on harvest interval is illustrate d. Dependence of yield, plant N uptake, and plant N concentration parameters on harvest interval is described adeq uately by the expanded grow th model (Figure 3). Several inferences follow from this analys is. The point of maximum slope of the yield response curves is determined by 22 0100 0 22 02 / 1 n yc b N Nkg ha1, 22 / 1yA Y (68) with the corresponding value of the yield dependi ng on harvest interval. The point of maximum slope of the plant N respons e curves is determined by 98 0100 0 98 02 / 1 n nc b N Nkg ha1, 22 / 1n uA N (69) with the corresponding value of plant N uptak e depending on harvest interval. Differential response of plant N uptake at this point is given by 625 0 4 ) 0100 0 )( 250 ( 4max n n uc A dN dN (70) for harvest intervals of 4 and 6 wk. This means th at maximum efficiency of nitrogen recovery is 62.5% for this harvest interval. Actu al efficiency of plant N recovery, E, is defined by PAGE 33 Overman Production of Switchgrass 31 N N N Eu u0 (71) where Nu 0 is plant N uptake at N = 0. Peak nitrogen recovery, Ep, can be estimated from (Overman, 2006a) ) exp( 1 1 ) 5 0 exp( 1 1 5 1 4 4 5 1n n n n n p n n pb b b c A E c b N (72) For harvest intervals of 4 and 6 wk these values become Np = 147 kg ha1 and Ep = 0.585 = 58.5%. A dramatic decrease in maximum plant N concentration (Ncm) with increased harvest interval ( t) may be noted from Figure 3. This occurs be cause as the plant ages the fraction of lightgathering component (higher N conc entration) decreases in relati on to the structural component (lower N concentration) of th e plant. The expanded growth model describes this phenomenon quite well. The lower limit of plant N concentration, Ncl, at reduced soil nitrogen (N <<0) can be estimated by combining Eqs. (5), (6), and (8) to obtain ) exp( b N Y K Ncm m n cl (73) For a harvest interval of 4 wk, this value is 10.6 g kg1. At this harvest interval the variables are bounded by 0 < Y < 11.04 Mg ha1, 0 < Nu < 250 kg ha1, and 10.6 < Nc < 22.6 g kg1. Actual values of the variables depend on the level of applied N. Attention is now focused in greater detail on the growth curves, particularly between harvests. Calculations are given in Table 8 for a harvest interval of t = 4 wk. Equations (55) and (56) are used for this purpose. Calculations c over the entire calendar year for completeness. The slight jump in time step at t = 35.4 wk is made to conform to sampling times. Because of the linear relationship between biomass accumulation ( Yi) and growth quantifier ( Qi) given by Eq. (12), it is sufficient to focus on dependence of Qi on calendar time (t). Resulting curves are shown in Figure 8. Note that the first point in each growth increment represents the reference point at which the growth quantifier is set to zero. It should also be noted that for t < 26 wk growth curves exhibit upward curvature, whereas for t > 26 wk the curves exhibit downward curvature. This is due to the Gaussian distribution of the ener gy driving function in the model. Overman and Scholtz (2002) have shown that the terminus points in Figure 8 follow a Gaussian distribution with calendar time. Th ese values are listed in Table 9. Cumulative growth quantifier (Q) is given as the third column in Table 9. Column 4 lists the normalized values (F), which is obtained by dividing each value of Q by the maximum value of 4.586. Now if the distribution of Qi is Gaussian, then it follows that F vs. t should be described by the probability function 2 erf 1 2 1 t F (74) PAGE 34 Overman Production of Switchgrass 32 To test this hypothesis it is convenient to write Eq. (74) in linearized form t F Z 2 1 2 1 2 erf1 (75) where erf 1 designates the inverse of the error function. It should be not ed that erf (0) = 0, erf ( ) = 1, and erf (x) = erf (+x). Values of Z are listed in Table 9. Linear regression of Z vs. t leads to t = 4 wk: t Z 1267 0 267 3 r = 0.999928 (76) with a correlation coefficient of r = 0.999928. The distribution of Z vs. t is shown in Figure 9, where the line is drawn from Eq. (76). It follows from Eqs. (75) and (76) that 00 8 894 7 2 wk and 0 26 79 25 wk as assumed for the expanded growth model. Estimates of Q are then obtained from t = 4 wk: 00 8 0 26 erf 1 2 586 4 t Q (77) as given by the last column in Table 9. The distribution of Q vs. t is shown in Figure 10, where the curve is drawn from Eq. (77). Other harvest intervals are now examined. Results for t = 6 wk are given in Tables 10 and 11, and shown in Figures 11 and 12. The curves in Figure 11 are drawn from Table 10, while the line in Figure 12 is drawn from t = 6 wk: t Z1235 0 200 3 r = 0.99973 (78) which leads to wk 0 26 91 25 and wk 00 8 096 8 2 It follows that Q vs. t is described by t = 6 wk: 00 8 0 26 erf 1 2 028 5 t Q (79) Results for t = 8 wk are given in Tables 12 and 13, and shown in Figures 13 through 15. The curves in Figure 13 are drawn from Table 12, while the line in Figur e 14 is drawn from t = 8 wk: t Z1267 0 268 3 r = 0.9999954 (80) which leads to wk 0 26 79 25 and wk 00 8 891 7 2 It follows that Q vs. t is described by PAGE 35 Overman Production of Switchgrass 33 t = 8 wk: 00 8 0 26 erf 1 2 024 5 t Q (81) as shown in Figure 15. Results for t = 12 wk are given in Tabl es 14 and 15, and shown in Figures 16 through 18. The line in Figure 17 is drawn from t = 12 wk: t Z1285 0 343 3 r = 1 (82) which leads to wk 0 26 01 26 and wk 00 8 779 7 2 Distribution of Q vs. t is described by t = 12 wk: 00 8 0 26 erf 1 2 868 4 t Q (83) as shown in Figure 18. Results for t = 16 wk are given in Table 16, without any graphs since this would be redundant for one point. Finally, results are given in Table 17 for t = 2 wk, and shown in Figures 19 and 20. The line in Figure 19 is drawn from t = 2 wk: t Z1236 0 214 3 r = 0.99924 (84) which leads to wk 0 26 01 26 and wk 00 8 093 8 2 Distribution of Q vs. t is described by t = 2 wk: 00 8 0 26 erf 1 2 000 4 t Q (85) as shown in Figure 20. The question now occurs as to depe ndence of seasonal growth quantifier (Q) on harvest interval ( t). A summary of values is listed in Table 18 from the various simulations. It can be shown from the expanded growth mode l (Overman and Scholtz, 2002) that t t Q exp (86) where and are to be determined by regre ssion analysis. It was shown that 075 0 2 / 15 0 2 / c wk1. It follows that Eq. (86) can be linearized to the form t t t Q Q 709 0 40 3 075 0 exp r = 0.99920 (87) The prediction equation then becomes t t t t Q 075 0 exp 709 0 40 3 exp (88) PAGE 36 Overman Production of Switchgrass 34 Results are shown in Figure 21, where the line and curve are drawn from Eqs. (87) and (88), respectively. This confirms the linearexponential rela tionship between Q and t predicted by the expanded growth model. It can be shown from calculus that peak harvest interval ( tp) for maximum Q can be estimated from 54 8 709 0 40 3 075 0 1 1 pt wk (89) which is confirmed in Figure 21. Response is relatively insensitive in the range 7 < t < 10 wk. CONCLUSIONS The extended logistic model of seasonal response of biomass yield, plant N uptake, and plant N concentration to applied N has been shown to describe field data of switchgrass grown at Montreal, Quebec, Canada rather well. The model predicts a linear rela tionship between plant N concentration and plant N uptake, whic h has been confirmed from the data. The expanded growth model has been shown to describe accumulation of biomass with calendar time by the process of photosynthesi s. A linear relationship between biomass accumulation and the growth quantifier defined in the model has been confirmed. Parameter values appear to be the same as those for a different grass grown at Tifton, GA. The model assumes a linear relationship between bioma ss/plant N ratio and biomass accumulation with calendar time, which has been confirmed from the data. This phase relation implies that biomass accumulation by photosynthesis is the rate limiting st ep in plant growth, and that plant nitrogen accumulates in virtual equilibrium with biomass. This is consistent with results for other crops as well. Growth curves have been generated for vari ous harvest intervals fr om the expanded growth model (Figures 8, 11, 13, and 16). Distribu tion of dimensionless probability function (Z) vs. calendar time (t) generated straight lines with 26 wk and 8 2 wk as assumed in the expanded growth model (Figures 9, 12, 14, 17, and 19). Cumulative growth quantifier (Q) vs. calendar time (t) followed the simple probability function (Figures 10, 15, 18, and 20) as predicted by the theory. Depende nce of seasonal growth quantifie r on harvest interval followed a linearexponential functi on, as also predicted by the theory. These results are consistent with results obtained for the warmseason perennial coastal bermudagrass (Overman and Scholtz, 2002). Maximum biomass production occurs for a harves t interval of approximately 12 wk (Figure 3). Other characteristics of the growth model have been discussed by Overman (2006b). REFERENCES Abramowitz, M. and I.A. Stegun. 1965. Handbook of Mathematical Functions. Dover Publications. New York, NY. 1046 p. Madakadze, I.C., K.A. Stewart, P.R. Peterson, B.E. Coulman, and D.L. Smith. 1999. Cutting frequency and nitrogen fertil ization effects on yield and nitrogen concentration of switchgrass in a short season area. Crop Science 39:552557. Overman, A.R. 1998. An expanded growth model for grasses. Communications in Soil Science and Plant Analysis 29:6785. PAGE 37 Overman Production of Switchgrass 35 Overman, A. R. 2006a. A Memoir on Crop Yield and Nutrient Uptake. University of Florida. Gainesville, FL. 116 p. (46 Tables and 62 Figures). http://www.uflib.ufl.edu/UFDC/UFDC.aspx?g=all&b=UF00072010&v=00001 Overman, A. R. 2006b. A Memoir on Crop Growth: Accumulation of Biomass and Mineral Elements. University of Florida. Gainesville, FL 386 p. (84 Tables and 184 Figures). http://www.uflib.ufl.edu/UFDC/UFDC.aspx?g=all&b=UF00072283&v=00001 Overman, A.R. and K.H. Brock. 2003. Confirma tion of the expanded growth model for a warmseason perennial grass. Communications in Soil Science and Plant Analysis 34:11051117. Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield. Taylor & Francis. Philadelphia, PA. 328 p. Overman, A.R., S.R. Wilkinson, and D.M. W ilson. 1994. An extended model of forage grass response to applied nitrogen. Agronomy J. 86:617620. PAGE 38 Overman Production of Switchgrass 36 Table 1. Dependence of seasonal biomass yield (Y), plant nitrogen uptake (Nu), and plant nitrogen concentration (Nc) on applied nitrogen (N) and harvest interval ( t) for switchgrass grown at Montreal, Quebec, Canada (1996) Data are averages of three cultivars.1 N Y Nu Nc Y Nu Nc Y Nu Nc kg ha1 Mg ha1 kg ha1 g kg1 Mg ha1 kg ha1 g kg1 Mg ha1 kg ha1 g kg1 t = 4 wk t = 6 wk t = 15 wk 0 4.87 69.1 14.2 6.01 69.0 11.5 8.30 42.2 5.08 75 6.84 108.6 15.9 7.80 106.6 13.7 9.66 52.1 5.39 150 8.75 155.1 17.7 10.01 157.1 15.7 11.25 66.9 5.95 1Data adapted from Madakadze et al. (1999). Table 2. Dependence of standardized yield (Y*) and standardized pl ant nitrogen uptake (Nu*) on applied nitrogen (N) and harvest interval ( t) for switchgrass at Montreal, Quebec, Canada. N Y* Nu* kg ha1 Mg ha1 kg ha1 t = 4 wk t = 6 wk t = 15 wk t = 4 wk t = 6 wk t = 15 wk 0 10.94 13.50 18.64 253 253 155 75 10.93 12.46 15.43 247 243 119 150 11.26 12.88 14.47 250 253 108 avg (std dev) 11.04 (0.19) 12.95 (0.52) 16.18 (2.18) 250 (3.0) 250 (5.8) 127 (24.6) Table 3. Dependence of sta ndardized yield parameter (Ay*) and plant N parameter (An*) on harvest interval ( t) for switchgrass at Montreal, Quebec, Canada. t Ay Ay* An An* Ncm wk Mg ha1 Mg ha1 kg ha1 kg ha1 g kg1 4 11.04 14.90 250 337 22.6 6 12.95 20.31 250 392 19.3 15 16.18 49.84 127 391 7.85 PAGE 39 Overman Production of Switchgrass 37 Table 4. Coupling of biomass yield (Y) and plant nitrogen accumulation (Nu) with calendar time (t) and applied nitrogen (N) for switchgrass grown at Montreal, Quebec, Canada (1996).1 t t Y Y Nu Nu Y Y Nu Nu Y Y Nu Nu wk wk Mg ha1 kg ha1 Mg ha1 kg ha1 Mg ha1 kg ha1 N = 0 N = 75 kg ha1 N = 150 kg ha1 4 23.1 0 0 0 0 0 0 2.54 37.8 3.21 48.3 3.87 66.7 27.1 2.54 37.8 3.21 48.3 3.87 66.7 1.36 21.2 2.22 42.1 3.08 62.3 31.1 3.90 59.0 5.43 90.4 6.95 129.0 0.97 10.1 1.41 18.2 1.80 26.1 35.4 4.87 69.1 6.84 108.6 8.75 155.1 6 23.1 0 0 0 0 0 0 3.86 42.0 4.46 60.3 5.98 89.0 29.1 3.86 42.0 4.46 60.3 5.98 89.0 2.15 26.9 3.34 46.3 4.03 68.1 35.3 6.01 68.9 7.80 106.6 10.01 157.1 15 23.1 0 0 0 0 0 0 8.30 42.2 9.66 52.1 11.25 66.9 38.0 8.30 42.2 9.66 52.1 11.25 66.9 1Data adapted from Madakadze et al. (1999). Table 5. Estimates of the growth quantifier (Q) with calendar time (t) for the expanded growth model. t x erf x exp(x2) Qi Q wk 23.1 0.2375 0.264 0.9452 0 1.259 27.1 0.7375 0.703 0.5805 1.259 1.078 31.1 1.2375 0.9198 0.2162 2.337 0.598 35.4 1.7750 0.9880 0.0428 2.935 23.1 0.2375 0.264 0.9452 0 1.988 29.1 0.9875 0.837 0.3771 1.988 1.140 35.4 1.7750 0.9880 0.0428 3.128 23.1 0.2375 0.264 0.9452 0 3.317 38.0 2.100 0.9970 0.0122 3.317 PAGE 40 Overman Production of Switchgrass 38 Table 6. Dependence of biomass accumulation (Y) on the growth quantifier (Q), applied nitrogen (N) and harvest interval ( t) for switchgrass grown at Montreal, Quebec, Canada.1 t t Q Y wk wk Mg ha1 N, kg ha1 0 75 150 4 23.1 0 0 0 0 27.1 1.26 2.54 3.21 3.87 31.1 2.34 3.90 5.43 6.95 35.4 2.94 4.87 6.84 8.75 6 23.1 0 0 0 0 29.1 1.99 3.86 4.46 5.98 35.4 3.13 6.01 7.80 10.01 15 23.1 0 0 0 0 38.0 3.32 8.30 9.66 11.25 1Yield data adapted from Madakadze et al. (1999). Table 7. Estimates for sta ndardized yield factor (A*) for each yield factor (A) at applied nitrogen (N) and harvest interval ( t) for switchgrass grown at Montreal, Quebec, Canada. N A A* kg ha1 Mg ha1 Mg ha1 t, wk t, wk 4 6 15 4 6 15 0 1.70 1.93 2.50 3.82 4.33 5.62 75 2.35 2.42 2.91 3.75 3.87 4.65 150 2.98 3.14 3.39 3.83 4.04 4.36 avg (std dev) 3.80 (0.04) 4.08 (0.23) 4.88 (0.66) PAGE 41 Overman Production of Switchgrass 39 Table 8. Simulation of the growth quantifier ( Qi) with calendar time (t) for the expanded growth model for harvest interval t = 4 wk. t x erf x exp(x2) Qi wk 3.1 2.2625 0.9985 0.00598 0.0000 4.1 2.1375 0.9973 0.0104 0.0002 5.1 2.0125 0.9954 0.0174 0.0004 6.1 1.8875 0.9924 0.0284 0.0008 7.1 1.7625 0.9873 0.0448 0.0019 7.1 1.7625 0.9873 0.0448 0.000 8.1 1.6375 0.9794 0.0685 0.001 9.1 1.5125 0.9676 0.1015 0.004 10.1 1.3875 0.9503 0.1458 0.009 11.1 1.2625 0.9258 0.2031 0.019 11.1 1.2625 0.9258 0.2031 0.000 12.1 1.1375 0.8923 0.2742 0.010 13.1 1.0125 0.8478 0.3587 0.029 14.1 0.8875 0.7905 0.4549 0.061 15.1 0.7625 0.7191 0.5591 0.111 15.1 0.7625 0.7191 0.5591 0.000 16.1 0.6375 0.6327 0.6660 0.046 17.1 0.5125 0.5313 0.7690 0.125 18.1 0.3875 0.4163 0.8606 0.243 19.1 0.2625 0.2895 0.9334 0.405 19.1 0.2625 0.2895 0.9334 0.000 20.1 0.1375 0.1542 0.9813 0.130 21.1 0.0125 0.0140 0.9998 0.328 22.1 0.1125 0.1264 0.9874 0.591 23.1 0.2375 0.2625 0.9452 0.907 23.1 0.2375 0.2625 0.9452 0.000 24.1 0.3625 0.3918 0.8769 0.224 25.1 0.4875 0.5095 0.7885 0.526 26.1 0.6125 0.6136 0.6872 0.880 27.1 0.7375 0.7030 0.5805 1.258 27.1 0.7375 0.7030 0.5805 0.000 28.1 0.8625 0.7774 0.4752 0.235 29.1 0.9875 0.8374 0.3771 0.515 30.1 1.1125 0.8843 0.2901 0.804 31.1 1.2375 0.9199 0.2162 1.078 PAGE 42 Overman Production of Switchgrass 40 Table 8 (continued). 31.1 1.2375 0.9199 0.2162 0.000 32.1 1.3625 0.9460 0.1562 0.150 33.1 1.4875 0.9646 0.1094 0.306 34.1 1.6125 0.9774 0.07426 0.451 35.4 1.7750 0.9879 0.04283 0.602 35.4 1.7750 0.9879 0.04283 0.000 36.4 1.9000 0.9928 0.02705 0.050 37.4 2.0250 0.9957 0.01656 0.107 38.4 2.1500 0.9975 0.00983 0.147 39.4 2.2750 0.9987 0.00565 0.167 39.4 2.2750 0.9987 0.00565 0.000 40.4 2.4000 0.99931 0.00315 0.011 41.4 2.5250 0.99968 0.00170 0.015 42.4 2.6500 0.99981 0.00089 0.029 43.4 2.7750 0.99990 0.00045 0.034 43.4 2.7750 0.999910 0.000453 0.0000 44.4 2.9000 0.999959 0.000223 0.0005 45.4 3.0250 0.999983 0.000106 0.0011 46.4 3.1500 0.999993 0.000049 0.0020 47.4 3.2750 0.999999 0.0000220 0.0022 PAGE 43 Overman Production of Switchgrass 41 Table 9. Distribution of Z vs. t for harvest interval t = 4 wk. t Qi Q F Z Q wk 3.1 0.000 0.00000 0.0001 0.002 7.1 0.002 0.00044 2.380 0.0021 0.019 11.1 0.021 0.00458 1.842 0.019 0.111 15.1 0.132 0.0288 1.346 0.124 0.405 19.1 0.537 0.117 0.842 0.511 0.907 23.1 1.444 0.315 0.341 1.396 1.258 27.1 2.702 0.589 0.159 2.653 1.078 31.1 3.780 0.824 0.658 3.742 0.602 35.4 4.382 0.9555 1.203 4.364 0.167 39.4 4.549 0.99193 1.702 4.545 0.034 43.4 4.583 0.99935 2.270 4.581 4.586 1 4.586 PAGE 44 Overman Production of Switchgrass 42 Table 10. Simulation of the growth quantifier ( Qi) with calendar time (t) for the expanded growth model for harvest interval of t = 6 wk. t x erf x exp(x2) Qi wk 5.1 2.0125 0.9954 0.0174 0.0000 6.1 1.8875 0.9924 0.0284 0.0002 7.1 1.7625 0.9873 0.0448 0.0011 8.1 1.6375 0.9794 0.0685 0.003 9.1 1.5125 0.9676 0.1015 0.006 10.1 1.3875 0.9503 0.1458 0.012 11.1 1.2625 0.9258 0.2031 0.022 11.1 1.2625 0.9258 0.2031 0.000 12.1 1.1375 0.8923 0.2742 0.010 13.1 1.0125 0.8478 0.3587 0.029 14.1 0.8875 0.7905 0.4549 0.061 15.1 0.7625 0.7191 0.5591 0.111 16.1 0.6375 0.6327 0.6660 0.184 17.1 0.5125 0.5313 0.7690 0.283 17.1 0.5125 0.5313 0.7690 0.000 18.1 0.3875 0.4163 0.8606 0.082 19.1 0.2625 0.2895 0.9334 0.215 20.1 0.1375 0.1542 0.9813 0.403 21.1 0.0125 0.0140 0.9998 0.644 22.1 0.1125 0.1264 0.9874 0.934 23.1 0.2375 0.2625 0.9452 1.260 23.1 0.2375 0.2625 0.9452 0.000 24.1 0.3625 0.3918 0.8769 0.224 25.1 0.4875 0.5095 0.7885 0.639 26.1 0.6125 0.6136 0.6872 0.993 27.1 0.7375 0.7030 0.5805 1.371 28.1 0.8625 0.7774 0.4752 1.747 29.1 0.9875 0.8374 0.3771 2.100 29.1 0.9875 0.8374 0.3771 0.000 30.1 1.1125 0.8843 0.2901 0.199 31.1 1.2375 0.9199 0.2162 0.422 32.1 1.3625 0.9460 0.1562 0.640 33.1 1.4875 0.9646 0.1094 0.832 34.1 1.6125 0.9774 0.07426 0.991 35.4 1.7750 0.9879 0.04283 1.146 PAGE 45 Overman Production of Switchgrass 43 Table 10 (continued). 35.4 1.7750 0.9879 0.04283 0.000 36.4 1.9000 0.9928 0.02705 0.050 37.4 2.0250 0.9957 0.01656 0.107 38.4 2.1500 0.9975 0.00983 0.147 39.4 2.2750 0.9987 0.00565 0.167 40.4 2.4000 0.99931 0.00315 0.186 41.4 2.5250 0.99968 0.00170 0.196 41.4 2.5250 0.99968 0.00170 0.000 42.4 2.6500 0.99981 0.00089 0.016 43.4 2.7750 0.999910 0.000453 0.017 44.4 2.9000 0.999959 0.000223 0.019 45.4 3.0250 0.999983 0.000106 0.020 46.4 3.1500 0.999993 0.000049 0.021 47.4 3.2750 0.999999 0.0000220 0.021 Table 11. Distribution of Z vs. t for harvest interval t = 6 wk. t Qi Q F Z Q wk 5.1 0.000 0.000 0.001 0.022 11.1 0.022 0.0044 1.842 0.021 0.283 17.1 0.305 0.0606 1.097 0.289 1.260 23.1 1.565 0.3111 0.349 1.528 2.100 29.1 3.665 0.7286 0.430 3.565 1.146 35.4 4.811 0.9565 1.210 4.784 0.196 41.4 5.007 0.9958 1.865 5.012 0.021 47.4 5.028 1 5.028 PAGE 46 Overman Production of Switchgrass 44 Table 12. Simulation of the growth quantifier ( Qi) with calendar time (t) for the expanded growth model for harvest interval of t = 8 wk. t x erf x exp(x2) Qi wk 7.1 1.7625 0.9873 0.0448 0.000 8.1 1.6375 0.9794 0.0685 0.001 9.1 1.5125 0.9676 0.1015 0.004 10.1 1.3875 0.9503 0.1458 0.009 11.1 1.2625 0.9258 0.2031 0.019 12.1 1.1375 0.8923 0.2742 0.034 13.1 1.0125 0.8478 0.3587 0.058 14.1 0.8875 0.7905 0.4549 0.093 15.1 0.7625 0.7191 0.5591 0.142 15.1 0.7625 0.7191 0.5591 0.000 16.1 0.6375 0.6327 0.6660 0.046 17.1 0.5125 0.5313 0.7690 0.125 18.1 0.3875 0.4163 0.8606 0.243 19.1 0.2625 0.2895 0.9334 0.405 20.1 0.1375 0.1542 0.9813 0.612 21.1 0.0125 0.0140 0.9998 0.861 22.1 0.1125 0.1264 0.9874 1.146 23.1 0.2375 0.2625 0.9452 1.456 23.1 0.2375 0.2625 0.9452 0.000 24.1 0.3625 0.3918 0.8769 0.224 25.1 0.4875 0.5095 0.7885 0.526 26.1 0.6125 0.6136 0.6872 0.880 27.1 0.7375 0.7030 0.5805 1.258 28.1 0.8625 0.7774 0.4752 1.635 29.1 0.9875 0.8374 0.3771 1.988 30.1 1.1125 0.8843 0.2901 2.302 31.1 1.2375 0.9199 0.2162 2.571 31.1 1.2375 0.9199 0.2162 0.000 32.1 1.3625 0.9460 0.1562 0.150 33.1 1.4875 0.9646 0.1094 0.306 34.1 1.6125 0.9774 0.07426 0.451 35.4 1.7750 0.9879 0.04283 0.602 36.4 1.9000 0.9928 0.02705 0.686 37.4 2.0250 0.9957 0.01656 0.750 38.4 2.1500 0.9975 0.00983 0.793 39.4 2.2750 0.9987 0.00565 0.818 PAGE 47 Overman Production of Switchgrass 45 Table 12 (continued). 39.4 2.2750 0.9987 0.00565 0.000 40.4 2.4000 0.99931 0.00315 0.011 41.4 2.5250 0.99968 0.00170 0.015 42.4 2.6500 0.99981 0.000892 0.029 43.4 2.7750 0.99991 0.000453 0.032 44.4 2.9000 0.999959 0.000223 0.034 45.4 3.0250 0.999983 0.000106 0.036 46.4 3.1500 0.999993 0.000049 0.037 47.4 3.2750 0.999999 0.0000220 0.037 Table 13. Distribution of Z vs. t for harvest interval t = 8 wk. t Qi Q F Z Q wk 7.1 0.000 0.000 0.0025 0.142 15.1 0.142 0.0283 1.358 0.136 1.456 23.1 1.598 0.3181 0.335 1.530 2.571 31.1 4.169 0.8298 0.674 4.092 0.818 39.4 4.987 0.9926 1.723 4.979 0.037 47.4 5.024 1 5.024 PAGE 48 Overman Production of Switchgrass 46 Table 14. Simulation of the growth quantifier ( Qi) with calendar time (t) for the expanded growth model for harvest interval of t = 12 wk. t x erf x exp(x2) Qi wk 11.1 1.2625 0.9258 0.2031 0.000 12.1 1.1375 0.8923 0.2742 0.010 13.1 1.0125 0.8478 0.3587 0.029 14.1 0.8875 0.7905 0.4549 0.061 15.1 0.7625 0.7191 0.5591 0.111 16.1 0.6375 0.6327 0.6660 0.184 17.1 0.5125 0.5313 0.7690 0.283 18.1 0.3875 0.4163 0.8606 0.411 19.1 0.2625 0.2895 0.9334 0.570 20.1 0.1375 0.1542 0.9813 0.758 21.1 0.0125 0.0140 0.9998 0.972 22.1 0.1125 0.1264 0.9874 1.205 23.1 0.2375 0.2625 0.9452 1.450 23.1 0.2375 0.2625 0.9452 0.000 24.1 0.3625 0.3918 0.8769 0.224 25.1 0.4875 0.5095 0.7885 0.526 26.1 0.6125 0.6136 0.6872 0.880 27.1 0.7375 0.7030 0.5805 1.258 28.1 0.8625 0.7774 0.4752 1.635 29.1 0.9875 0.8374 0.3771 1.988 30.1 1.1125 0.8843 0.2901 2.302 31.1 1.2375 0.9199 0.2162 2.571 32.1 1.3625 0.9460 0.1562 2.789 33.1 1.4875 0.9646 0.1094 2.960 34.1 1.6125 0.9774 0.07426 3.089 35.4 1.7750 0.9879 0.04283 3.204 35.4 1.7750 0.9879 0.04283 0.000 36.4 1.9000 0.9928 0.02705 0.050 37.4 2.0250 0.9957 0.01656 0.107 38.4 2.1500 0.9975 0.00983 0.147 39.4 2.2750 0.9987 0.00565 0.167 40.4 2.4000 0.99931 0.00315 0.186 41.4 2.5250 0.99968 0.00170 0.196 42.4 2.6500 0.99981 0.00089 0.206 43.4 2.7750 0.99991 0.000453 0.210 44.4 2.9000 0.999959 0.000223 0.212 45.4 3.0250 0.999983 0.000106 0.213 46.4 3.1500 0.999993 0.000049 0.214 47.4 3.2750 0.999999 0.0000220 0.214 PAGE 49 Overman Production of Switchgrass 47 Table 15. Distribution of Z vs. t for harvest interval t = 12 wk. t Qi Q F Z Q wk 11.1 0.000 0.0000 0.020 1.450 23.1 1.450 0.2979 0.374 1.480 3.204 35.4 4.654 0.9560 1.207 4.633 0.214 47.4 4.868 1 4.868 PAGE 50 Overman Production of Switchgrass 48 Table 16. Simulation of the growth quantifier ( Qi) with calendar time (t) for the expanded growth model for harvest interval of t = 16 wk. t x erf x exp(x2) Qi wk 15.1 0.7625 0.7191 0.5591 0.000 16.1 0.6375 0.6327 0.6660 0.046 17.1 0.5125 0.5313 0.7690 0.125 18.1 0.3875 0.4163 0.8606 0.243 19.1 0.2625 0.2895 0.9334 0.405 20.1 0.1375 0.1542 0.9813 0.612 21.1 0.0125 0.0140 0.9998 0.861 22.1 0.1125 0.1264 0.9874 1.146 23.1 0.2375 0.2625 0.9452 1.456 24.1 0.3625 0.3918 0.8769 1.782 25.1 0.4875 0.5095 0.7885 2.109 26.1 0.6125 0.6136 0.6872 2.424 27.1 0.7375 0.7030 0.5805 2.717 28.1 0.8625 0.7774 0.4752 2.979 29.1 0.9875 0.8374 0.3771 3.206 30.1 1.1125 0.8843 0.2901 3.394 31.1 1.2375 0.9199 0.2162 3.547 31.1 1.2375 0.9199 0.2162 0.000 32.1 1.3625 0.9460 0.1562 0.150 33.1 1.4875 0.9646 0.1094 0.306 34.1 1.6125 0.9774 0.07426 0.451 35.4 1.7750 0.9879 0.04283 0.602 36.4 1.9000 0.9928 0.02705 0.686 37.4 2.0250 0.9957 0.01656 0.750 38.4 2.1500 0.9975 0.00983 0.793 39.4 2.2750 0.9987 0.00565 0.818 40.4 2.4000 0.99931 0.00315 0.835 41.4 2.5250 0.99968 0.00170 0.844 42.4 2.6500 0.99981 0.00089 0.851 43.4 2.7750 0.99991 0.000453 0.855 44.4 2.9000 0.999959 0.000223 0.856 45.4 3.0250 0.999983 0.000106 0.857 46.4 3.1500 0.999993 0.000049 0.858 47.4 3.2750 0.999999 0.0000220 0.858 PAGE 51 Overman Production of Switchgrass 49 Table 17. Simulation of the growth quantifier ( Qi) with calendar time (t) for the expanded growth model for harvest interval of t = 2 wk. t x erf x exp(x2) Qi Q F Z Q wk 1.1 2.5125 0.99951 0.00181 0.0000 3.1 2.2625 0.9985 0.00598 0.0001 0. 0001 5.1 2.0125 0.9954 0.0174 0.0004 0.0005 0.00012 2.600 0.0004 7.1 1.7625 0.9873 0.0448 0.0011 0.0016 0.00040 2.480 0.0016 9.1 1.5125 0.9676 0.1015 0.0040 0.0056 0.00140 2.120 0.0058 11.1 1.2625 0.9258 0.2031 0.0116 0.0172 0.00430 1.858 0.0170 13.1 1.0125 0.8478 0.3587 0.0289 0.0461 0.0115 1.618 0.0452 15.1 0.7625 0.7191 0.5591 0.111 0.157 0.0392 1.245 0.108 17.1 0.5125 0.5313 0.7690 0.125 0.282 0.0705 1.040 0.232 19.1 0.2625 0.2895 0.9334 0.215 0.497 0.124 0.817 0.446 21.1 0.0125 0.0140 0.9998 0.328 0.825 0.206 0.580 0.774 23.1 0.2375 0.2625 0.9452 0.441 1.266 0.316 0.339 1.218 25.1 0.4875 0.5095 0.7885 0.526 1.792 0.448 0.093 1.726 27.1 0.7375 0.7030 0.5805 0.554 2.346 0.586 0.154 2.310 29.1 0.9875 0.8374 0.3771 0.515 2.861 0.715 0.401 2.832 31.1 1.2375 0.9199 0.2162 0.422 3.283 0.821 0.650 3.266 33.1 1.4875 0.9646 0.1094 0.306 3.589 0.897 0.895 3.580 35.4 1.7750 0.9879 0.04283 0.225 3.814 0.9535 1.188 3.807 37.4 2.0250 0.9957 0.01656 0.107 3.921 0.9802 1.453 3.912 39.4 2.2750 0.9987 0.00565 0.039 3.960 0.9900 1.645 3.964 41.4 2.5250 0.99968 0.00170 0.015 3.975 0.99375 1.778 3.987 43.4 2.7750 0.99991 0.000453 0.017 3.992 0.99800 2.040 3.996 45.4 3.0250 0.999984 0.000106 0.007 3.999 0.99975 2.440 3.999 47.4 3.2750 0.999999 0.0000220 0.001 4.000 1 4.000 Table 18. Dependence of the s easonal growth quantifier (Q) on harvest interval ( t) for switchgrass grown at Montreal Quebec, Canada. t Q Q Q wk 2 4.000 4.647 4.155 4 4.586 6.190 4.627 6 5.028 7.885 4.887 8 5.024 9.154 4.984 12 4.868 11.973 4.845 16 4.405 14.625 4.444 PAGE 52 Overman Production of Switchgrass 50 List of Figures Figure 1. Response of biomass yield (Y), plant N uptake (Nu), and plant N concentration (Nc) to applied nitrogen (N) and harvest interval ( t) for switchgrass grown at Montreal, Quebec, Canada. Data adapted from Madakadze et al. (1999) Curves drawn from Eqs. (23) through (25) with Ay(4 wk) = 11.04, Ay(6 wk) = 12.95, and Ay(15 wk) = 16.18 Mg ha1; An(4 wk) = 250, An(6 wk) = 250, and An(15 wk) = 127 kg ha1; and Ncm(4 wk) = 22.6, Ncm(6 wk) = 19.3, and Ncm(15 wk) = 7.85 g kg1. Figure 2. Phase plots of biomass yield (Y) and plant N concentration (Nc) vs. plant N uptake (Nu) for three harvest intervals ( t) for switchgrass grown at Montreal Quebec, Canada. Data adapted from Madakadze et al. (1999). Curves drawn from Eqs. (27), (30), and (33); lines drawn from Eqs. (28), (31), and (34). Figure 3. Dependence of parameters for yield (Ay), plant N uptake (An), and plant N concentration (Ncm) on harvest interval ( t) for switchgrass grown at Montreal, Quebec, Canada. Curves drawn from Eqs. (41) through (43). Figure 4. Correlation of cumulative plant N uptake (Nu) with cumulative biomass (Y) for applied nitrogen (N) and harvest interval ( t) for switchgrass grown at Mont real, Quebec, Canada. Data adapted from Madakadze et al. (1999). Li nes drawn from Eqs. (46) through (54). Figure 5. Dependence of plant N concentration (Nc) on applied nitrogen (N) and harvest interval ( t) for switchgrass grown at Montreal, Quebec, Ca nada. Data are from Table 4. Curves drawn from Eq. (25) with Ncm(4 wk) = 22.6, Ncm(6 wk) = 19.3, and Ncm(15 wk) = 7.85 g kg1. Figure 6. Correlation of cumulative biomass (Y) with cumulative growth quantifier (Q) for applied nitrogen (N) and harvest interval ( t) for switchgrass grown at Montreal, Quebec, Canada. Data adapted from Madakadze et al. (1999) Lines drawn from Eqs. (57) through (65). Figure 7. Dependence of yield factor (A) on applied nitrogen (N) and harvest interval ( t) for switchgrass grown at Montreal, Qu ebec, Canada. Data are from Table 7. Curves drawn from Eq. (66) with A(4 wk) = 3.80, A(6 wk) = 4.08, A(15 wk) = 4.88 Mg ha1. Figure 8. Dependence of the growth quantifier ( Q) on calendar time (t) for harvest interval ( t) of 4 wk. Curves drawn from Table 8. Figure 9. Distribution of dime nsionless probability function (Z) vs. calendar time (t) for harvest interval ( t) of 4 wk. Values from Table 9; line drawn from Eq. (76). Figure 10. Distribution of seasonal growth quantifier (Q) vs. calendar time (t) for harvest interval ( t) = 4 wk. Values from Table 10; curve drawn from Eq. (77). Figure 11. Dependence of the growth quantifier ( Q) on calendar time (t) for harvest interval ( t) of 6 wk. Curves drawn from Table 11. PAGE 53 Overman Production of Switchgrass 51 Figure 12. Distribution of dime nsionless probability function (Z) vs. calendar time (t) for harvest interval ( t) of 6 wk. Values from Table 11; line drawn from Eq. (78). Figure 13. Dependence of the growth quantifier ( Q) on calendar time (t) for harvest interval ( t) of 8 wk. Curves drawn from Table 12. Figure 14. Distribution of dime nsionless probability function (Z) vs. calendar time (t) for harvest interval ( t) of 8 wk. Values from Table 13; line drawn from Eq. (80). Figure 15. Distribution of seasonal growth quantifier (Q) vs. calendar time (t) for harvest interval ( t) = 8 wk. Values from Table 13; curve drawn from Eq. (81). Figure 16. Dependence of the growth quantifier ( Q) on calendar time (t) for harvest interval ( t) of 12 wk. Curves drawn from Table 14. Figure 17. Distribution of dime nsionless probability function (Z) vs. calendar time (t) for harvest interval ( t) of 12 wk. Values from Table 15; line drawn from Eq. (82). Figure 18. Distribution of seasonal growth quantifier (Q) vs. calendar time (t) for harvest interval ( t) = 12 wk. Values from Table 15; curve drawn from Eq. (83). Figure 19. Distribution of dime nsionless probability function (Z) vs. calendar time (t) for harvest interval ( t) of 2 wk. Values from Table 17; line drawn from Eq. (84). Figure 20. Distribution of seasonal growth quantifier (Q) vs. calendar time (t) for harvest interval ( t) = 2 wk. Values from Table 18; curve drawn from Eq. (85). Figure 21. Dependence of seasonal growth quantifier (Q) and transformed growth quantifier (Q*) on harvest interval ( t). Values from Table 19. Line an d curve drawn from Eqs. (87) and (88), respectively. PAGE 54 Overman Production of Switchgrass 52 The correlation coefficient is given by 969 0 8170 0 04932 0 1 ) ( ) ( 12 / 1 2 / 1 2 2 Q Q Q Q r () PAGE 55 Overman Production of Switchgrass 53 Table Dependence of seasonal biomass yield ( Y ), plant nitrogen uptake ( Nu), and plant nitrogen concentration ( Nc) on applied nitrogen ( N ) and harvest interval ( t ) for switchgrass grown at Montreal, Quebec, Canada (1996). Data are averages of three cultivars.1 N Y Nu Nc Y Nu Nc Y Nu Nc Zy Zn kg ha1 Mg ha1 kg ha1 g kg1 Mg ha1 kg ha1 g kg1 Mg ha1 kg ha1 g kg1 t = 4 wk t = 6 wk avg 0 4.87 69.1 14.2 6.01 69.0 11.5 5.44 69.0 12.7 0.187 0.964 75 6.84 108.6 15.9 7.80 106.6 13.7 7.32 107.6 14.7 0.447 0.280 150 8.75 155.1 17.7 10.01 157.1 15.7 9.38 156.1 16.6 1.275 0.508 1Crop data adapted from Ma dakadze et al. (1999). PAGE 56 Overman Production of Switchgrass 54 MODEL RESPONSE OF SWITCHGRASS TO APPLIED NITROGEN AND CALENDAR TIME ALLEN R. OVERMAN Agricultural and Biological Engineering Department, University of Florida, Gainesville, FL 326110570 INTRODUCTION Mathematical models are useful for describi ng output (response variables) in response to input (control variables). Extensive field studies have been conducted in various parts of the world over the past 150 years on crop response to applied nutrients and other factors, such as crop species, soil type, harvest interval for pere nnial grasses, environmen tal conditions (such as water availability), and plant population. The present analysis will focus on two particular models. The extended logistic model (Overman et al., 1994) describes coupling of seasonal biomass yield and plant nutrient uptake with applie d nutrients (such as nitr ogen) through logistic equations. Biomass yield is then linked to pl ant nutrient uptake th rough a hyperbolic phase relation, which predicts a linear relationship be tween plant nutrient concentration and plant nutrient uptake. The expanded growth model (Overman, 1998) describes accumulation of biomass with time through an analytical function which incorporates effects of energy input, partitioning of biomass between lightgathering and structural components, and aging as the plant grows. Plant nutrient and plant bioma ss accumulation are coupled through a hyperbolic phase relation. Field data have been used to confirm this model for coastal bermudagrass ( Cynodon dactylon L.) grown in Georgia (Overman a nd Brock, 2003). The growth model predicts a linearexponential depe ndence of seasonal yield with a fixed harvest interval for perennial grasses. All of the results have been shown to be consistent with field data for numerous crops, soils, and environmental conditions (Overman and Scholtz, 2002). This analysis will focus on response of the warmseason perennial switchgrass ( Panicum virgatum L.) to applied nitrogen and with calendar time. MODEL DESCRIPTION Response of biomass yield and plant nitrogen uptake to applied nitrogen can be described by the extended logistic model given by ) exp( 1 N c b A Yn y y (1) ) exp( 1 N c b A Nn n n u (2) ) exp( 1 ) exp( 1 N c b N c b N Y N Nn n n y cm u c (3) PAGE 57 Overman Production of Switchgrass 55 where N is applied nitrogen, kg ha1; Y is seasonal total biomass yield, Mg ha1; Nu is seasonal total plant nitrogen uptake, kg ha1; Nc is plant nitrogen concentration, g kg1; Ay is maximum yield at high N Mg ha1; An is maximum plant nitrogen uptake at high N kg ha1; Ncm = An/Ay is maximum plant nitrogen concentration at high N g kg1; by is intercept parameter for plant yield; bn is intercept parameter for plant nitrogen uptake; and cn is response coefficient for applied nitrogen, ha kg1. Note that the units on cn are the reciprocal of those for N Equations (1) through (3) are wellbehaved monotone increasing functions. Since variables N, Y, and Nu are defined as positive, parameters Ay, An, and cn must be positive as well, while parameters by and bn can be either positive, zero, or negative. Equations (1) and (2) with common cn can be combined to give the hyperbolic phase relation between Y and Nu u n u mN K N Y Y (4) where hyperbolic and logistic pa rameters are coupled through ) exp( 1 b A Yy m (5) 1 ) exp( b A Kn n (6) with the shift in intercept parameters defined by y nb b b (7) For Ym and Kn to be positive requires that b > 0. Equation (4) can be rearranged to the linear form u m m n u cN Y Y K Y N N 1 (8) which predicts a linear relationship between Nc and Nu. This prediction is easily tested from data. The expanded growth model for biomass accumulation ( Y ) with calendar time ( t ) from Jan. 1 is described by AQ Y (9) where Y is accumulated biomass, Mg ha1; Q is accumulated growth quantifier; and A is the yield factor, Mg ha1. The growth quantifier is defined by i i i icx x x k x x kx Q2 exp exp exp erf erf 12 2 (10) PAGE 58 Overman Production of Switchgrass 56 with dimensionless time, x defined in terms of calendar time, t by 2 2 2 c t x (11) where model parameters are defined by: is time to the mean of the energy distribution, wk; is time spread of the energy distribution, wk; c is the aging coefficient, wk1; and k is the partition coefficient between lightgathering and struct ural components of the plant. Note that xi corresponds to the time of initiation of growth, ti. The error function, erf, in Eq. (10) is defined by xdu u x0 2) exp( 2 erf (12) where u is the variable of integration. Values of the error function can be obtained from mathematical tables (Abramowitz and Stegun, 1965). DATA ANALYSIS Data for this analysis are adapted from field studies with switchgrass (cv. CaveinRock) at Meade, NE and Ames, IA by Vogel et al. (2002). Applied nitrogen levels were 0, 60, 120, 180, 240, and 300 kg ha1. Measurements included biomass (dry matter) and plant nitrogen uptake. Samples were also collected periodically to m easure growth from the beginning of the season. This analysis will focus on the study at Ames, IA where the soil was a WebsterNicollet complex (fineloamy, mixed, mesic T ypic HaplaquollAquic Hapludoll). No supplemental irrigation was provided. Nitrogen response is given in Table 1 and show n in Figure 1 for the first cutting only. The increase of all three measurements with increas e in applied N may be noted. The procedure for evaluation of model parameters is now discussed. Equations (1) and (2) can be linearized to the forms N N c b Y Zn y y0130 0 39 0 1 60 11 ln r = 0.9982 (13) N N c b N Zn n u n0131 0 89 0 1 152 ln r = 0.9971 (14) where Ay = 11.60 Mg ha1 and An = 152 kg ha1 have been chosen to give the same value of cn. Note the high correlation coefficients ( r > 0.99). The response equations now become ) 0130 0 39 0 exp( 1 60 11 N Y (15) ) 0130 0 89 0 exp( 1 152 N Nu (16) PAGE 59 Overman Production of Switchgrass 57 ) 0130 0 89 0 exp( 1 ) 0130 0 39 0 exp( 1 1 13 N N Y N Nu c (17) Curves in Figure 1 are drawn from Eqs. (15) th rough (17). Hyperbolic phase parameters can be estimated from 07 16 ) 28 1 exp( 1 60 11 ) exp( 1 b A Yy m Mg ha1 (18) 5 58 1 ) 28 1 exp( 152 1 ) exp( b A Kn n kg ha1 (19) which leads to the phase relations u u u n u mN N N K N Y Y 5 58 07 16 (20) u u m m n cN N Y Y K N 0622 0 64 3 1 (21) Phase plots are shown in Figure 2, where the curv e and line are drawn from Eqs. (20) and (21), respectively. The model describes the seasonal data rather well. Growth data are given in Table 2 and shown in Figure 3. The first step in the growth analysis is to calculate the growth quantifier ( Q ) as a function of calendar time ( t ) from Jan. 1. Model parameters are assumed to be: 5 wk 15 0 wk, 00 8 2 wk, 0 261 k c Model parameters are the same as those used for coastal bermudagrass ( Cynodon dactylon L.) by Overman and Scholtz (2002). Time of initiation is assumed to be ti = 21.8 wk. This leads to the dimensionless time ( x ) and growth quantifier equations 00 8 2 21 600 0 00 8 0 26 2 2 2 t t c t x xi = 0.075 (22) 094 1 9944 0 exp 821 2 085 0 erf 625 0 2 exp exp exp erf erf 12 2 2 x x cx x x k x x kx Qi i i i (23) Values of model variables in Table 2 are calcul ated from Eqs. (22) a nd (23). Correlation of biomass ( Y ) with the growth quantifier is shown in Figure 4, where the line is drawn from Q Y 97 3 069 0 r = 0.980 (24) with an intercept of essentia lly zero, in accordance with the growth model. In fact ti = 21.8 wk has been chosen to make the intercept zero. The phase relations for growth are given by PAGE 60 Overman Production of Switchgrass 58 Y Y N N K N Yum um y u00679 0 0208 0 1 r = 0.957 (25) Y Y Y K Y N Ny um u 06 3 147 (26) Results are shown in Figure 5, where the line and curve are drawn from Eqs. (25) and (26), respectively. It follows from Eq. (26) that plant N concentration is given by Y Y N Nu c 06 3 147 (27) The growth response curves in Figure 3 are draw n from Eqs. (22) thr ough (27), with model estimates listed in Table 3. Maximum plant N concentration at t = 21.8 wk where Y 0 is Nc = 147/3.06 = 48.0 g kg1. The model describes results reasonably well. DISCUSSION The extended logistic model appears to give r easonable correlation of yield, plant N uptake, and plant N concentration with applied N (Figur e 1) for switchgrass grown at Ames, IA. This leads to excellent phase plots of yield vs. plant N uptake (Fig ure 2). Prediction of a linear relationship between plant N concentration and plant N uptake is confirmed. Dependence of yield, plant N uptake, and plant N concentrati on on calendar time is described adequately by the expanded growth model (Figure 3). The linear re lationship between the growth quantifier and biomass accumulation with time is confirmed (Figure 4). The linear relationship between yield/plant N uptake ratio and biomass accumulati on predicted by the model is also confirmed (Figure 5). Several inferences follow from this analys is. The point of maximum slope of the yield response curve is determined by 30 0130 0 39 02 / 1 n yc b N N kg ha1, 8 5 22 / 1 yA Y Mg ha1 (28) The negative value indicates that more than enou gh nitrogen is already present in the soil to reach 50% of maximum yield, which may be noted in Figure 1. The point of maximum slope of the plant N response curve is determined by 68 0130 0 89 02 / 1 n nc b N Nkg ha1, 76 22 / 1 n uA N kg ha1 (29) Differential response of plant N upt ake at this point is given by 49 0 4 ) 0130 0 )( 152 ( 4max n n uc A dN dN (30) PAGE 61 Overman Production of Switchgrass 59 This means that maximum efficiency of nitrogen recovery is 49%. Actual efficiency of plant N recovery, E is defined by N N N Eu u 0 (31) where Nu 0 is plant N uptake at N = 0. Peak nitrogen recovery, Ep, can be estimated from (Overman, 2006a) ) exp( 1 1 ) 5 0 exp( 1 1 5 1 4 4 5 1n n n n n p n n pb b b c A E c b N (32) For the Ames, IA study, these values become Np = 102 kg ha1 and Ep = 0.47 = 47%. The logistic model can be used to establis h bounds on the response va riables. According to Eq. (17) maximum plant N concentration is Ncm = 13.1 g kg1. Now the lower limit, Ncl, can be estimated from Eqs. (5), (6), and (8) as 64 3 28 1 exp 1 13 exp b N Y K Ncm m n cl g kg1 (33) which is also the intercept in Eq. (21). It follows that response variables are bounded by 0 < Y < 11.6 Mg ha1, 0 < Nu < 152 kg ha1, and 3.64 < Nc < 13.1 g kg1. The expanded growth model can be used to establish bounds on th e growth variables. According to Eq. (23) the upper bound on the growth quantifier is Q 3.695. From Eqs. (24) and (26) the upper bounds on biom ass and plant N uptake are Y 14.60 Mg ha1 and Nu 122 kg ha1. The lower bound on plant N concentration is Nc = Nu/Y 8.3 g kg1, compared to an upper bound of Nc 147/3.06 = 48.0 g kg1. It follows that the growth variables are bounded by 0 < Y < 14.60 Mg ha1, 0 < Nu < 122 kg ha1, and 48.0 > Nc > 8.3 g kg1. This behavior may be noted in Figure 3. The decline in plant N concentr ation with age is due to a shift from dominance by the lightgathering component (higher plant N concentration) to that of the structural component (lower plant N concen tration) as the plant ages. Since Ncl and Ncm are characteristics of the plant, it follows from Eq. (33) that b is characteristic of the plant. There is some indication that parameter c may relate to availability of nutrients in the soil (Overman, 2006b). Parameters Ay and An have been shown to relate to water availability and to plant population. Furt hermore, it appears from the definition Ncm = An/Ay that this ratio is a characteristic of the plant. Some investigators use the quadratic model for data analysis. This is due in part because it represents a linear model in the regression sense (linear in the m odel parameters). Data from the Ames, IA study are now analyzed by this approa ch. Assume yield and pl ant N uptake response to applied nitrogen are describe d by the quadratic equations. 2 2 1 0N a N a a Y (34) 2 2 1 0N a N a a Nu (35) PAGE 62 Overman Production of Switchgrass 60 2 2 1 0 2 2 1 0N a N a a N a N a a Y N Nu c (36) Equation (36) is defined automatically from Eqs. (34) and (35). The next step is to evaluate model parameters by regression analysis. Yield response Regression analysis by the least squares criter ion leads to the three simultaneous equations for yield 2 2 1 0a N a N na Y 2 3 1 2 0a N a N a N NY (37) 2 4 1 3 0 2 2a N a N a N Y N where n is the number of observations and the su ms are over the six observations. Since the equations are linear in the parameters (2 1 0, a a a ) the procedure for ev aluation is called linear regression The system of equations can be written in matrix form as Y N NY Y a a a N N N N N N N N n2 2 1 0 4 3 2 3 2 2 (38) This represents three simultaneo us equations in three unknowns (2 1 0, ,a a a). Note that elements of the coefficient matrix and the righthand vector can be calculated dire ctly from the data. The system can be solved provided that the determinan t of the coefficient matrix does not vanish. For the present set of data Eq. ( 38) becomes (with scaling of N/100) 968 220 604 98 48 59 8784 126 6 48 8 19 6 48 8 19 00 9 8 19 00 9 62 1 0a a a (39) We can solve for parameters 2 1 0and ,a a a by Cramers rule using determinants (Ayers, 1962) 89 182 8784 126 6 48 8 19 6 48 8 19 00 9 8 19 00 9 64 3 2 3 2 2 N N N N N N N N n D (40) PAGE 63 Overman Production of Switchgrass 61 12 1247 8784 126 6 48 968 220 6 48 8 19 604 98 8 19 00 9 48 594 3 2 3 2 20 N N Y N N N NY N N Y Da (41) 69 665 8784 126 968 220 8 19 6 48 604 98 00 9 8 19 48 59 64 2 2 3 21 N Y N N N NY N N Y n Da (42) 09 131 968 220 6 48 8 19 604 98 8 19 00 9 48 59 00 9 62 3 2 22 Y N N N NY N N Y N n Da (43) Parameters are now estimated from 82 6 89 182 12 124700 D D aa (44) 0364 0 10 89 182 69 665 102 2 11 D D aa (45) 0000717 0 10 89 182 09 131 104 4 22 D D aa (46) The regression equation for yield becomes 20000717 0 0364 0 82 6 N N Y (47) Plant N uptake response Regression analysis leads to the three simultaneous equations for plant N uptake 2 2 1 0a N a N a n Nu 2 3 1 2 0a N a N a N NNu (48) 2 4 1 3 0 2 2a N a N a N N Nu which can be written in matrix form as u u uN N NN N a a a N N N N N N N N n2 2 1 0 4 3 2 3 2 2 (49) For the present set of data Eq. (49) becomes (with scaling of N /100 and Nu/100) PAGE 64 Overman Production of Switchgrass 62 6245 26 494 11 235 6 8784 126 6 48 8 19 6 48 8 19 00 9 8 19 00 9 62 1 0a a a (50) We can solve for 2 1 0and , a a a by Cramers rule using determinants 89 182 8784 126 6 48 8 19 6 48 8 19 00 9 8 19 00 9 64 3 2 3 2 2 N N N N N N N N n D (51) 769 79 8784 126 6 48 6245 26 6 48 8 19 494 11 8 19 00 9 235 64 3 2 3 2 20 N N N N N N NN N N N Du u u a (52) 75 104 8784 126 6245 26 8 19 6 48 494 11 00 9 8 19 235 6 64 2 2 3 21 N N N N N NN N N N n Du u u a (53) 194 14 6245 26 6 48 8 19 494 11 8 19 00 9 235 6 00 9 62 3 2 22 u u u aN N N N NN N N N N n D (54) Parameters are now estimated from 6 43 10 89 182 769 792 0 D D aa (55) 573 0 89 182 75 104 10 102 2 1 D D ab (56) 000776 0 10 89 182 194 14 10 102 4 2 2 D D ac (57) The regression equation for plant N uptake becomes 2000776 0 573 0 6 43 N N Nu (58) It follows that plant N concentration is described by 2 20000717 0 0364 0 82 6 000776 0 573 0 6 43 N N N N Y N Nu c (59) PAGE 65 Overman Production of Switchgrass 63 Response plots are shown in Figure 6, where th e curves are drawn from Eqs. (47), (58), and (59). The correlation coe fficients for yield ( Y ), plant N uptake ( Nu), and plant N concentration ( Nc) are r = 0.9912, 0.99971, and 0.9936, respectively. It is obvious that the quadratic model fits the data very well. However, for extrapolations above N = 300 kg ha1 the curves exhibit patterns which are not consistent with data from other studies at higher applied N values, where the curves all tend to approach plateaus (Overman, 2006a). For N < 0 (reduced soil N) the curves go negative, which is incompatible with the definitio ns of positive response variables. In fact values go negative at Y = 0 N = 145, 650 kg ha1 and Nu = 0 N = 70, 810 kg ha1. The phase plots ( Y and Nc vs. Nu) are shown in Figure 7, wher e the curves are again drawn from Eqs. (47), (58), and (59). While the curves pass through the data poin ts rather well, these exhibit very strange behavior out side the range of applied N used in the experiments. Overman and Scholtz (2003) have provided a comparison between linear and nonlinear regression models. The expanded growth model can be used to estimate partitioning of biomass into lightgathering and structural components. For the data from Ames, IA the growth quantifier is given by Eq. (23). It follows that the lightgathering growth quantifier ( QL) and structural growth quantifier ( QS) are defined by 094 1 085 0 erf 2 exp erf erf x cx x x Qi i L (60) 094 1 9944 0 exp 821 2 085 0 erf 375 0 2 exp exp exp erf erf2 2 2 x x cx x x k x x kx Qi i i i S (61) The lightgathering fraction ( fL) is then defined by Q Q fL L (62) Values are listed in Table 4 and shown in Figure 8. This allows simulation of biomass for lightgathering (LY ) and structural (SY ) components from Y f YL L (63) Y f YL S 1 (64) as given in Table 5. In order to simulate plant nitroge n partitioning it is nece ssary to make some assumption about plant N concentration. The choice is made to assume that plant N concentration in the structural component (cSN ) remains constant at 6.0 g kg1 during growth. Plant N uptake in the structural (uSN ) and lightgathering (uLN ) components are estimated by S S cS uSY Y N N 0 6 (65) uS u uLN N N (66) PAGE 66 Overman Production of Switchgrass 64 as given in Table 5. Estimates of plant N concentration in the lightgathering component (cLN ) are made from L uL cLY N N (67) which is listed as the last column in Table 5. Results are shown in Figur e 9. It should be noted that NcL declines as the plant ages. The last step s hould be considered as somewhat speculative at this point. CONCLUSIONS The extended logistic model of seasonal response of biomass yield, plant N uptake, and plant N concentration to applied N has been shown to describe field data of switchgrass grown at Ames, IA rather well. The model predicts a li near relationship between plant N concentration and plant N uptake, which has been confirmed fr om the data. While a quadratic model can be used to describe response data, it leads to inc onsistencies with patterns observed from studies with other grasses. A proper model should desc ribe connections among variables in a given study, but should be consistent among locations, times (years), and environmental conditions. Model parameters may change with physical conditions, but the form of the model should remain unchanged. This has been a guiding pr inciple in physics for over four hundred years. The expanded growth model has been shown to describe accumulation of biomass with calendar time by the process of photosynthesi s. A linear relationship between biomass accumulation and the growth quantifier defined in the model has been confirmed. Parameter values appear to be the same as those for a different grass grown at Tifton, GA. The model assumes a linear relationship between bioma ss/plant N ratio and biomass accumulation with calendar time, which has been confirmed from the data. This phase relation implies that biomass accumulation by photosynthesis is the rate limiting st ep in plant growth, and that plant nitrogen accumulates in virtual equilibrium with biomass. Th is is consistent with results for other crops as well. It should be noted that this anal ysis has utilized analytical func tions to describe the data. This offers distinct advantages over fi nite difference procedures in th at the models are well behaved and avoid problems with numerical stability in simulations. Rucker (1987) has pointed out the limitations of a truncated power series. It has be en shown previously that effects due to water availability (rainfall or irriga tion) are accounted for in the linea r model parameters (Overman and Scholtz, 2002). This greatly simplifies model application. Peak efficiency of plant N utilization can be estimated from Eq. (32) where values of model parameters An, bn, and cn are available. Results are dependent on plant species, soil characteristics, and water availability. T ypical values range between 50% and 80%. PAGE 67 Overman Production of Switchgrass 65 REFERENCES Abramowitz, M. and I.A. Stegun. 1965. Handbook of Mathematical Functions. Dover Publications. New York, NY. 1046 p. Ayers, F. 1962. Theory and Problems of Matrices. McGrawHill. New York, NY. 219 p. Overman, A.R. 1998. An expanded growth model for grasses. Communications in Soil Science and Plant Analysis 29:6785. Overman, A. R. 2006a. A Memoir on Crop Yield and Nutrient Uptake. University of Florida. Gainesville, FL. 116 p. (46 Tables and 62 Figures). http://www.uflib.ufl.edu/UFDC/UFDC.aspx?g=all&b=UF00072010&v=00001 Overman, A. R. 2006b. A Memoir on Chemical Transport: Application to Soils and Crops. University of Florida. Gainesville, FL. 364 p. (13 Tables and 34 Figures). http://www.uflib.ufl.edu/UFDC/UFDC.aspx?g=all&b=UF00072282&v=00001 Overman, A.R. and K.H. Brock. 2003. Confirma tion of the expanded growth model for a warmseason perennial grass. Communications in Soil Science and Plant Analysis 34:11051117. Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield Taylor & Francis. Philadelphia, PA. 328 p. Overman, A.R. and R.V. Scholtz III. 2003. In de fense of the extended logistic model of crop production. Communications in Soil Science and Plant Analysis 34:851864. Overman, A.R., S.R. Wilkinson, and D.M. W ilson. 1994. An extended model of forage grass response to applied nitrogen. Agronomy J. 86:617620. Rucker, R. 1987. Mind Tools: The Five Levels of Mathematical Reality. Houghton Mifflin Co. Boston, MA. 328 p. Vogel, K.P., J.J. Brejda, D.T. Walters, and D.R. Buxton. 2002. Switchgrass biomass production in the Midwest USA: Harvest and nitrogen management. Agronomy J. 94:413420. PAGE 68 Overman Production of Switchgrass 66 Table 1. Response of biomass yield ( Y ), plant nitrogen uptake ( Nu), and plant nitrogen concentration ( Nc) to applied nitrogen ( N ) by switchgrass grown at Ames, IA. Averaged over 1994 and 1995.1 N Y Nu Nc kg ha1 Mg ha1 kg ha1 g kg1 0 6.59 43.8 6.65 60 9.15 74.7 8.16 120 10.16 101 9.94 180 10.89 123 11.3 240 11.25 135 12.0 300 11.44 146 12.8 1Data adapted from Vogel et al. (2002). Table 2. Accumulation of biomass ( Y ), plant nitrogen ( Nu), and plant nitroge n concentration ( Nc) with calendar time ( t ) from Jan. 1 by switchgrass grown at Ames, IA. Averaged over 1994 and 1995 and applied nitrogen.1 t x erf x exp( x2) Q Y Nu Nc wk Mg ha1 kg ha1 g kg1 21.8 0.0750 0.085 0.9944 0.000 25.8 0.5750 0.584 0.718 1.194 4.6 85 18.5 27.7 0.8125 0.750 0.517 1.928 6.7 97 14.5 28.8 0.9500 0.821 0.406 2.320 9.9 120 12.1 29.8 1.0750 0.872 0.315 2.635 11.1 129 11.6 30.8 1.2000 0.910 0.237 2.902 11.7 117 10.0 32.3 1.3875 0.9503 0.146 3.210 12.6 118 9.37 33.7 1.5625 0.9728 0.0870 3.408 12.8 108 8.44 1Crop data adapted from Vogel et al. (2002). PAGE 69 Overman Production of Switchgrass 67 Table 3. Simulation of biomass ( Y ), plant nitrogen ( Nu), and plant nitrogen concentration ( Nc) with calendar time ( t ) from Jan. 1 by switchgrass grown at Ames, IA. t x erf x exp( x2) Q Y uN cN wk Mg ha1 kg ha1 g kg1 21.8 0.0750 0.085 0.9944 0.000 0.00 0.0 48.0 23 0.225 0.250 0.9506 0.248 0.916 33.9 37.0 24 0.350 0.379 0.8847 0.540 2.08 59.5 28.6 25 0.475 0.499 0.798 0.889 3.46 78.0 22.5 26 0.600 0.604 0.698 1.270 4.97 91.0 18.3 27 0.725 0.695 0.591 1.662 6.53 100.1 15.3 28 0.850 0.771 0.486 2.038 8.02 106.4 13.3 29 0.975 0.832 0.386 2.389 9.42 111.0 11.8 30 1.100 0.880 0.298 2.693 10.62 114.1 10.7 31 1.225 0.9173 0.223 2.950 11.64 116.4 10.0 32 1.350 0.9438 0.162 3.157 12.46 118.0 9.47 33 1.475 0.9630 0.114 3.318 13.10 119.2 9.10 34 1.600 0.9763 0.0773 3.440 13.59 120.0 8.83 35 1.725 0.9853 0.0510 3.528 13.94 120.5 8.65 36 1.850 0.9911 0.0326 3.588 14.18 120.9 8.53 38 2.100 0.9970 0.0122 3.655 14.44 121.3 8.40 40 2.350 0.9991 0.0040 3.682 14.55 121.46 8.35 1. 0. 3.695 14.60 121.53 8.32 Table 4. Simulation of growth quantifier ( Q ), lightgathering component ( QL), structural component ( QS), and ligtgathering fraction ( fL) with calendar time ( t ) from Jan. 1 by switchgrass grown at Ames, IA. t x erf x exp( x2) Q QL QS fL wk 21.8 0.0750 0.085 0.9944 0.000 0.000 0.000 23 0.225 0.250 0.9506 0.248 0.181 0.067 0.730 24 0.350 0.379 0.8847 0.540 0.322 0.218 0.596 25 0.475 0.499 0.798 0.889 0.453 0.436 0.510 26 0.600 0.604 0.698 1.270 0.568 0.702 0.447 27 0.725 0.695 0.591 1.662 0.667 0.995 0.401 28 0.850 0.771 0.486 2.038 0.750 1.288 0.368 29 0.975 0.832 0.386 2.389 0.817 1.572 0.342 30 1.100 0.880 0.298 2.693 0.870 1.823 0.323 31 1.225 0.9173 0.223 2.950 0.910 2.040 0.308 32 1.350 0.9438 0.162 3.157 0.940 2.217 0.298 33 1.475 0.9630 0.114 3.318 0.961 2.357 0.290 34 1.600 0.9763 0.0773 3.440 0.975 2.465 0.283 35 1.725 0.9853 0.0510 3.528 0.985 2.543 0.279 36 1.850 0.9911 0.0326 3.588 0.991 2.597 0.276 38 2.100 0.9970 0.0122 3.655 0.998 2.657 0.273 40 2.350 0.9991 0.0040 3.682 1.000 2.682 0.272 1. 0. 3.695 1.001 2.694 0.271 PAGE 70 Overman Production of Switchgrass 68 Table 5. Simulation of biomass (LY and SY ), plant N uptake (uLN and uSN ), and plant N concentration (cLN and cSN ) with calendar time ( t ) from Jan. 1 for lightg athering and structural components, respectively. t fL Y LY SY uN uSN uLN cLN wk Mg ha1 Mg ha1 Mg ha1 kg ha1 kg ha1 kg ha1 g kg1 21.8 0.00 0.00 0.00 0.0 0.0 0.0 23 0.730 0.916 0.669 0.247 33.9 1.5 32.4 48.4 24 0.596 2.08 1.24 0.84 59.5 5.0 54.5 43.9 25 0.510 3.46 1.76 1.70 78.0 10.2 67.8 38.5 26 0.447 4.97 2.22 2.75 91.0 16.5 74.5 33.6 27 0.401 6.53 2.62 3.91 100.1 23.5 76.6 29.2 28 0.368 8.02 2.95 5.07 106.4 30.4 76.0 25.8 29 0.342 9.42 3.22 6.20 111.0 37.2 73.8 22.9 30 0.323 10.62 3.43 7.19 114.1 43.1 71.0 20.7 31 0.308 11.64 3.59 8.05 116.4 48.3 68.1 19.0 32 0.298 12.46 3.71 8.75 118.0 52.5 65.5 17.7 33 0.290 13.10 3.80 9.30 119.2 55.8 63.4 16.7 34 0.283 13.59 3.85 9.74 120.0 58.4 61.6 16.0 35 0.279 13.94 3.89 10.05 120.5 60.3 60.2 15.5 36 0.276 14.18 3.91 10.27 120.9 61.6 59.3 15.2 38 0.273 14.44 3.94 10.50 121.3 63.0 58.3 14.8 40 0.272 14.55 3.96 10.59 121.46 63.5 58.0 14.6 0.271 14.60 3.96 10.64 121.53 63.8 57.7 14.6 PAGE 71 Overman Production of Switchgrass 69 List of Figures Figure 1. Response of biomass yield ( Y ), plant nitrogen uptake ( Nu), and plant nitrogen concentration ( Nc) to applied nitrogen ( N ) for switchgrass grown at Ames, IA. Data adapted from Vogel et al. (2002). Curves draw n from Eqs. (15) through (17). Figure 2. Phase plots between biomass yield ( Y ) and plant nitrogen concentration ( Nc) and plant nitrogen uptake ( Nu) for switchgrass grown at Ames, IA. Data adapted from Vogel et al. (2002). Curve and line drawn from Eqs. (20) and (21), respectively. Figure 3. Accumulation of biomass yield ( Y ), plant nitrogen ( Nu), and plant nitrogen concentration ( Nc) with calendar time ( t ) for switchgrass grown at Ames, IA. Data adapted from Vogel et al. (2002). Curves drawn from Eqs. (22) through (27). Figure 4. Correlation of biomass ( Y ) with growth quantifier ( Q ) for switchgrass grown at Ames, IA. Yield data adapted from Vogel et al. (2002). Line draw n from Eq. (24). Figure 5. Phase plots of plant nitrogen uptake ( Nu) and yield/plant N uptake ratio ( Y/Nu) vs. biomass ( Y ) for switchgrass grown at Ames IA. Data adapted from Voge l et al. (2002). Line and curve drawn from Eqs. (25) and (26), respectively. Figure 6. Response of biomass yield ( Y ), plant nitrogen uptake ( Nu), and plant nitrogen concentration ( Nc) to applied nitrogen ( N ) for switchgrass grown at Ames, IA. Data adapted from Vogel et al. (2002). Curves drawn from Eqs. ( 47), (58), and (59) for the quadratic model. Figure 7. Phase plots between biomass yield ( Y ) and plant nitrogen concentration ( Nc) and plant nitrogen uptake ( Nu) for switchgrass grown at Ames, IA. Data adapted from Vogel et al. (2002). Curves drawn from Eqs. (47), (58), and (59) for the quadratic model. Figure 8. Simulation of the growth quantifier for lightgathering ( QL) and structural ( QS) components, and lightgathering fraction ( fL) with calendar time ( t ) from Jan. 1 for switchgrass grown at Ames, IA. Curves are dr awn from values in Table 4. Figure 9. Simulation of biomass (LY and SY ), plant N uptake (uLN and uSN ), and plant N concentration (cLN and cSN ) with calendar time ( t ) from Jan. 1 for lightg athering and structural components, respectively. Curves are drawn from Table 5. PAGE 72 Overman Production of Switchgrass 70 Figure 1. Response of biomass yield ( Y ), plant nitrogen uptake ( Nu), and plant nitrogen concentration ( Nc) to applied nitrogen ( N ) for switchgrass grown at Ames, IA. Data adapted from Vogel et al. (2002). Curves draw n from Eqs. (15) through (17). PAGE 73 Overman Production of Switchgrass 71 Figure 2. Phase plots between biomass yield ( Y ) and plant nitrogen concentration ( Nc) and plant nitrogen uptake ( Nu) for switchgrass grown at Ames, IA. Data adapted from Vogel et al. (2002). Curve and line drawn from Eqs. (20) and (21), respectively. PAGE 74 Overman Production of Switchgrass 72 Figure 3. Accumulation of biomass yield ( Y ), plant nitrogen ( Nu), and plant nitrogen concentration ( Nc) with calendar time ( t ) for switchgrass grown at Ames, IA. Data adapted from Vogel et al. (2002). Curves drawn from Eqs. (22) through (27). PAGE 75 Overman Production of Switchgrass 73 Figure 4. Correlation of biomass ( Y ) with growth quantifier ( Q ) for switchgrass grown at Ames, IA. Yield data adapted from Vogel et al. (2002). Line draw n from Eq. (24). PAGE 76 Overman Production of Switchgrass 74 Figure 5. Phase plots of plant nitrogen uptake ( Nu) and yield/plant N uptake ratio ( Y/Nu) vs. biomass ( Y ) for switchgrass grown at Ames IA. Data adapted from Voge l et al. (2002). Line and curve drawn from Eqs. (25) and (26), respectively. PAGE 77 Overman Production of Switchgrass 75 Figure 6. Response of biomass yield ( Y ), plant nitrogen uptake ( Nu), and plant nitrogen concentration ( Nc) to applied nitrogen ( N ) for switchgrass grown at Ames, IA. Data adapted from Vogel et al. (2002). Curves drawn from Eqs. ( 47), (58), and (59) for the quadratic model. PAGE 78 Overman Production of Switchgrass 76 Figure 7. Phase plots between biomass yield ( Y ) and plant nitrogen concentration ( Nc) and plant nitrogen uptake ( Nu) for switchgrass grown at Ames, IA. Data adapted from Vogel et al. (2002). Curves drawn from Eqs. (47), (58), and (59) for the quadratic model. PAGE 79 Overman Production of Switchgrass 77 Figure 8. Simulation of the growth quantifier for lightgathering ( QL) and structural ( QS) components, and lightgathering fraction ( fL) with calendar time ( t ) from Jan. 1 for switchgrass grown at Ames, IA. Curves are dr awn from values in Table 4. PAGE 80 Overman Production of Switchgrass 78 Figure 9. Simulation of biomass (LY and SY ), plant N uptake (uLN and uSN ), and plant N concentration (cLN and cSN ) with calendar time ( t ) from Jan. 1 for lightg athering and structural components, respectively. Curves are drawn from Table 5. PAGE 81 Overman Production of Switchgrass 79 Table Response of switchgr ass to applied nitrogen.1 N Nu Y Nc kg ha1 kg ha1 Mg ha1 g kg1 0 43.8 6.59 6.65 60 74.7 9.15 8.16 120 101 10.16 9.94 180 123 10.89 11.3 240 135 11.25 12.0 300 146 11.44 12.8 1Data adapted from Vogel et al. (2002). Table Estimated response of sw itchgrass to app lied nitrogen. N uN Y cN kg ha1 kg ha1 Mg ha1 g kg1 65 3.08 4.15 0.742 50 13.0 4.82 2.70 25 28.8 5.86 4.91 0 43.6 6.82 6.39 25 57.4 7.69 7.46 50 70.3 8.46 8.31 75 82.2 9.15 8.98 100 93.1 9.74 9.56 150 112.1 10.67 10.51 200 127.2 11.24 11.32 300 145.7 11.30 12.89 350 149.1 10.78 13.83 365 149.4 10.55 14.16 375 149.4 10.39 14.38 400 148.6 9.91 15.00 500 136.1 7.12 19.12 550 124.0 5.15 24.08 600 108.0 2.87 37.63 620 100.6 1.83 55.0 650 88.2 0.187 472 PAGE 82 Overman Production of Switchgrass 80 Table Regression analysis of yiel d response to applied nitrogen. N Y Y Y Y Y Y kg ha1 Mg ha1 Mg ha1 Mg ha1 Mg ha1 0 6.59 6.82 0.23 3.32 60 9.15 8.75 0.40 0.76 120 10.16 10.16 0.00 0.25 180 10.89 11.05 0.16 0.98 240 11.25 11.43 0.18 1.34 300 11.44 11.29 0.15 1.53 avg 9.91 9912 0 9825 0 7594 16 2934 0 1 12 2 2 r Y Y Y Y r Table Regression analysis of plan t N response to applied nitrogen. N Nu uN u uN N u uN N kg ha1 Mg ha1 kg ha1 kg ha1 kg ha1 0 43.8 43.6 0.2 60.2 60 74.7 75.2 0.5 29.3 120 101 101.2 0.2 3.0 180 123 121.6 1.4 19.0 240 135 136.4 1.4 31.0 300 146 145.7 0.3 42.0 avg 104 99971 0 99943 0 5 7577 34 4 1 12 2 2 r N N N N ru u u u Table Regression analysis of plant N con centration response to applied nitrogen. N Nc cN c cN N c cN N kg ha1 g kg1 g kg1 g kg1 g kg1 0 6.65 6.39 0.26 3.49 60 8.16 8.60 0.44 1.98 120 9.94 9.96 0.02 0.20 180 11.3 11.00 0.30 1.16 240 12.0 11.94 0.06 1.86 300 12.8 12.91 0.11 2.66 avg 10.14 9934 0 9869 0 0213 28 3673 0 1 12 2 2 r N N N N rc c c c PAGE 83 Overman Production of Switchgrass 81 Solution of quadratic eq uations by elimination 968 220 8784 126 6 48 8 19 604 98 6 48 8 19 9 48 59 8 19 00 9 62 1 0 2 1 0 2 1 0 a a a a a a a a a 1600 11 408 6 4545 2 9560 10 40 5 20 2 9133 9 30 3 50 12 1 0 2 1 0 2 1 0 a a a a a a a a a 2467 1 108 3 9545 0 10427 1 100 2 70 02 1 2 1 a a a a 3061 1 256 3 4896 1 000 32 1 2 1 a a a a 717 0 1835 0 256 02 2 a a 64 3 ) 717 0 ( 000 3 4896 11 a 82 6 ) 717 0 ( 30 3 ) 64 3 ( 50 1 9133 90 a 20000717 0 0364 0 82 6 N N Y 6245 26 8784 126 6 48 8 19 494 11 6 48 8 19 9 235 6 8 19 00 9 62 1 0 2 1 0 2 1 0 a a a a a a a a a 34467 1 408 6 454545 2 27711 1 40 5 20 2 03917 1 30 3 50 12 1 0 2 1 0 2 1 0 a a a a a a a a a 30550 0 1081 3 954545 0 23794 0 100 2 70 02 1 2 1 a a a a 320048 0 25611 3 339914 0 000 32 1 2 1 a a a a 0776 0 663 0198 0 25611 02 2 a a 573 0 ) 0776 0 ( 000 3 339914 01 a PAGE 84 Overman Production of Switchgrass 82 436 0 ) 0776 0 ( 30 3 ) 573 0 ( 50 1 0392 10 a 2000776 0 573 0 6 43 N N Nu These coefficients are the same as by Cramers rule. PAGE 85 Overman Production of Switchgrass 83 Table Response of biomass yield ( Y ), plant nitrogen uptake ( Nu), and plant nitrogen concentration ( Nc) to applied nitrogen ( N ) by switchgrass grown at Ames, IA. Averaged over 1994 and 1995.1 N Y Nu Nc Zy Zn kg ha1 Mg ha1 kg ha1 g kg1 0 6.59 43.8 6.65 0.27 0.90 60 9.15 74.7 8.16 1.32 0.03 120 10.16 101 9.94 1.95 0.68 180 10.89 123 11.3 2.73 1.44 240 11.25 135 12.0 3.47 2.07 300 11.44 146 12.8 4.27 3.19 1Crop data adapted from Vogel et al. (2002). 