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PAGE 1 ACCUMULATION AND PARTITIONI NG OF BIOMASS BY SOYBEAN Application of the Expanded Growth Model Allen R. Overman Richard V. Scholtz III Agricultural & Biological Engineering University of Florida Gainesville, FL 32611 Copyright 2012 Allen R. Overman PAGE 2 Overman & Scholtz Soybean i ABSTRACT The general goal of this work is to develop a theory of crop grow th with calendar time which has broad application for a wide rang e of plant species and environm ental conditions. In the present case the control variable is sampling time (refere nced to Jan. 1 for the northern hemisphere, wk) while the response variables are accumulation of biomass yield ( Y Mg ha1) and mineral elements such as nitrogen, phosphorus, and potassium ( Nu, kg ha1). The expanded growth model previously developed by the authors is used in this analysis. The model incorporates three fundamental processes: (1) a Gaussian energy dr iving function which desc ribes the distribution of solar energy over the season, (2) a linear partition function between lightgathering and structural components of the pl ants, and (3) an expone ntial aging function. The model contains six parameters: is calendar time to the mean of the solar energy distribution, wk; is time spread of the solar en ergy distribution, wk; k is the partition coeffici ent between lightgathering and structural components of the plants; c is the aging coefficient, wk1; A is the yield factor, Mg ha1; and ti is time of initiation of significan t growth, wk. Plant nutrient uptake ( Nu) is related to biomass yield ( Y ) through a hyperbolic phase relation. This hypothesis can be tested directly from field data for a particular plant species. Th e present analysis is fo cused on data from field studies at Ames, Iowa and Clay ton, North Carolina for soybean ( Glycine max L. Merr.). It is shown that all difference between parameters in the two studies is accounted for in parameters ti and A with parameters , k, and c common between the two. Results highlight the role of soil characteristics and seasonal rainfall on system response. Key Words: Mathematical Models, Biom ass Production, Soybean Acknowledgment: The authors express appr eciation to Amy G. Buhler, Associate University Librarian, University of Florid a, for assistance with preparation of this memoir as part of the UF digital library. PAGE 3 Overman & Scholtz Soybean 1 Accumulation and Partitioning of Biomass by Soybean Application of the Expanded Growth Model Allen R. Overman and Richard V. Scholtz III Agricultural and Biological Engine ering, University of Florida, Gainesville, FL 326110570 INTRODUCTION The mathematical foundation fo r this analysis is based on the expanded growth model of Overman and Scholtz (2002, section 3.5) and the simplified theory of biomass accumulation by photosynthesis by Overman and Scholtz (2010). A very large data base exists of research on agricultural crops, beginning w ith the famous work at Rothamsted, England initiated around 1850 by Gilbert and Laws (Hall, 1905). The first mathematical modeling appears to date from 1909 by E. A. Mitsherlich (Russell, 1937). Resear ch on the warmseason perennial bermudagrass ( Cynodon dactylon L.) has been reported by Prine and Bu rton (1956) and by Burton et al. (1963). Field research on the warmseason perennial bahiagrass ( Paspalum notatum Flgge) have been reported by Beaty at el. (1963) a nd by Blue (1987). Many annuals have been studied as well. These include corn ( Zea mays L.) by Kamprath (1986) and by Ka rlen et al. (1987), as well as cotton ( Gossypium hirsutum L.) by Fritschi et al. (2003) and by Mullins and Burmester (1991). These references provide just a sampling of the vast literature on agricultural crops. MODEL DESCRIPTION The expanded growth model is de scribed by the linear relationship AQ Y (1) in which Y is accumulated biomass with calendar time, Mg ha1; A is yield factor, Mg ha1; and Q is the growth quantifier defined by i i i icx x x k x x kx Q2 exp exp exp erf erf 12 2 (2) with dimensionless time x defined by 2 2 2 c t x (3) in which t is calendar time (referenced to Jan. 1), wk; is calendar time to the mean of the solar energy distribution, wk; is the time spread of the solar energy distribution, wk; k is the partition PAGE 4 Overman & Scholtz Soybean 2 coefficient between lightgathering and st ructural components of the plant; and c is the aging coefficient, wk1. It follows from Eq. (3) that reference time is defined by 2 2 2 c t xi i (4) with ti calendar time to significant plant growt h, wk. In Eq. (2) the error function, erf x is defined by xdu u x0 2exp 2 erf (5) with u as the variable of integrati on for the Gaussian distribution ) exp(2u Values of the error function can be obtained from mathematical ta bles (cf. Abramowitz and Stegun, 1965, chapter 7). Characteristics of th e error function include erf (0) = 0, erf () = 1, ) erf( ) erf( x x 1 ) erf( DATA ANALYSIS Data for these analyses are taken fr om two field studies with soybean ( Glycine max L. Merr.), one from Iowa USA and th e other from North Carolina USA. Iowa USA Study Data for this analysis are taken from a fi eld study with soybean (cv Richland) by Hammond et al. (1951) at Ames, IA. The soils were Webs ter silt loam (fineloamy, mixed, superactive, mesic Typic Endoaquoll) and Clarion loam (f ineloamy, mixed, superactive, mesic Typic Hapludoll). The former was flat, but welldrained and uniform; th e latter area was located on a small knoll where the slope varied from 1 to 4%, with variable surface soil. Both soils had a pH of approximately 7. However the Webster soil showed higher fertility than the Clarion as evidenced by cation exchange capacity, adsorb ed phosphorus, and inorganic nitrogen. Planting date was 24 May 1946 ( t = 20.4 wk). Beginning on 10 August ( t = 31.9 wk) plant samples were divided into vegetative (leaves stems, and stalks) and fruit (seeds + pods) components. No fertilizers were applied during the experiment. Results for soybean vegetation are listed in Table 1. Model parameters are chosen as: = 26.0 wk, 00 8 2 wk, c = 0.050 wk1, k = 5, and ti = 25.3 wk. Dimensionless time and growth quantifier are given by 00 8 4 24 2 ) 050 0 )( 00 8 ( 00 8 0 26 2 2 2 t t c t x xi = 0.112 (6) PAGE 5 Overman & Scholtz Soybean 3 046 1 987 0 exp 821 2 126 0 erf 438 0 2 exp exp exp erf erf 12 2 2 x x cx x x k x x kx Qi i i i (7) Values of x and Q in Table 1 are calculated from Eqs. (6 ) and (7), respectively. Correlation of Y on Q is shown in Figure 1, where the lines are drawn from the regression equations Webster soil: Q Y 37 2 14 0 r = 0.9913 (8) Clarion soil: Q Y 61 1 08 0 r = 0.9949 (9) with correlation coefficients of r = 0.9913 and 0.9949, respectively. Data for 9 33 3 26 t wk have been used for calibration. Biomass accumula tion for the vegetative component is shown in Figure 2, where the curves are dr awn from Eqs. (6) through (9). The next step is to analyze da ta for fruit (seeds + pods). Results are listed in Table 2. Time of initiation of significant plant growth is assumed to be ti = 31.5 wk with other parameters the same as above. Dimensionless time and growth quantif ier are given by 00 8 4 24 2 ) 050 0 )( 00 8 ( 00 8 0 26 2 2 2 t t c t x xi = 0.8875 (10) 426 1 455 0 exp 821 2 791 0 erf 438 3 2 exp exp exp erf erf 12 2 2 x x cx x x k x x kx Qi i i i (11) Values of x and Q in Table 2 are calculated from Eq s. (10) and (11). Correlation of Y with Q is shown in Figure 3, where the lines are drawn from Webster soil: Q Y 74 4 011 0 r = 0.9974 (12) Clarion soil: Q Y 50 3 082 0 r = 0.9796 (13) Accumulation of fruit biomass w ith calendar time is shown in Figure 4, where the curves are drawn from Eqs. (10) through (13). The expanded growth model provides reasona ble description of biomass accumulation with calendar time for both vegetative and fruit comp onents on the two soils. Linear correlation of biomass with the growth quan tifier is confirmed for both plant components. Production was higher for the Webster compared to the Clari on soil. Final grain yi elds were 2.70 Mg ha1 for Webster compared to 1.62 Mg ha1 for Clarion. Furthermore, tota l cation exchange capacity was 427 meq kg1 vs. 160 meq kg1 for Webster compared to Clarion soils. Based on these factors, production would be expected to be higher for Webster compared to Clarion soil. Overman and PAGE 6 Overman & Scholtz Soybean 4 Wilkinson (2003) have shown that soil erosion on hillsides can reduce soil fertility for bermudagrass. North Carolina USA Study Data for this analysis are taken from a fiel d study with soybean (c v Lee) during the period 1966 and 1967 by Henderson and Kamprath (1970) at Clayton, NC. The soil was Norfolk loamy sand (fineloamy, kaolinitic, thermic Typic Ka ndiudult). Planting date was approximately 10 May ( t = 18.7 wk). Plant samples were collected every 10 days from June through September. Starting 110 d after planting, plan ts were divided into fractions labeled vegetative and seeds & pods. For the sake of brevity the la tter is referred to as fruit. Results for the vegetative component are listed in Table 3. Time of initiation of significant growth is chosen as ti = 23.6 wk. Model parameters are c hosen as the same as for Iowa. Dimensionless time and gr owth quantifier become 00 8 4 24 2 ) 050 0 )( 00 8 ( 00 8 0 26 2 2 2 t t c t x 100 0 ix (14) 9608 0 9900 0 exp 821 2 112 0 erf 500 1 2 exp exp exp erf erf 12 2 2 x x cx x x k x x kx Qi i i i (15) Values of x and Q in Table 3 are calculated from Eqs. (14) and (15), respectively. Correlation of Y vs. Q for 4 34 4 24 twk is shown for Figure 5, wh ere the lines are drawn from 1966: Q Y 60 1 008 0 r = 0.9933 (16) 1967: Q Y 82 2 082 0 r = 0.9906 (17) Accumulation of vegetative biomass with calendar time is shown in Figure 6, where the curves are drawn from Eqs. (14) through (17). The drop in biomass at th e latter sampling times reflects shedding of leaves by the plants. The next step is to analyze da ta for fruit (seeds + pods). Results are listed in Table 4. Time of initiation of significant growth is assumed to be ti = 33.3 wk with other parameters the same as above. Dimensionless time and gr owth quantifier are given by 00 8 4 24 2 ) 050 0 )( 00 8 ( 00 8 0 26 2 2 2 t t c t x xi = 1.112 (18) 560 1 290 0 exp 821 2 884 0 erf 562 4 2 exp exp exp erf erf 12 2 2 x x cx x x k x x kx Qi i i i (19) PAGE 7 Overman & Scholtz Soybean 5 Values of x and Q in Table 3 are calculated from Eq s. (18) and (19). Correlation of Y with Q is shown in Figure 7, where th e lines are drawn from 1966: Q Y 76 9 307 0 r = 0.9986 (20) 1967: Q Y 3 23 042 0 r = 0.9959 (21) Accumulation of fruit biomass w ith calendar time is shown in Figure 8, where the curves are drawn from Eqs. (18) through (21). The question naturally occurs as to the basis fo r the difference in respon se of fruit with time between the two years. One possibility is differen ce in water availability between the two years. Seasonal rainfall was 34.4 cm for 1966 and 55. 6 cm for 1967. Overman and Scholtz (2002) assumed an exponential relationship between water availability and yield for corn of the form R R R Y Ym 0exp 1 (22) in which Y is biomass yield, R is seasonal rainfall, R0 is intercept value of R for Y = 0, R is characteristic rainfall, and Ym is maximum yield at high R Soybean fruit yields at calendar time t = 40 wk are estimated as 4.15 Mg ha1 (1966) and 9.22 Mg ha1 (1967) from Figure 6. This result is shown in Figure 9. We estimate R0 = 25 cm, 25 Rcm, and Ym = 13.0 Mg ha1 for this case. The curve in Figure 9 is drawn from 25 25 exp 1 0 13 R Y (23) While estimates used here are very approximate, this result does emphasize the sensitivity of crop production to water availabi lity, and that it can be accoun ted for in the yield factor A Estimates of partitioning of biomass between lightgathering and structural components can be made from the expanded growth model. The dimensionless time and growth quantifiers become 00 8 4 24 2 ) 050 0 )( 00 8 ( 00 8 0 26 2 2 2 t t c t x 100 0 ix (24) ` 9608 0 112 0 erf 2 exp erf erf x cx x x Qi i L (25) 9608 0 9900 0 exp 821 2 112 0 erf 500 0 2 exp exp exp erf erf2 2 2 x x cx x x k x x kx Qi i i i S (26) PAGE 8 Overman & Scholtz Soybean 6 The lightgathering fraction is then given by S L L S L L S L L LQ Q Q AQ AQ AQ Y Y Y f (27) Values for partitioning of biomass are listed in Table 5 and shown in Figure 10. Note that fL drops to approximately 25% at t = 40 wk. This occurs as plants age and the structural component becomes more dominant. DISCUSSION The first thing to note is th e quality of correlation between Y and Q for the two studies. For the Iowa study we obtained r = 0.9913 and 0.9949 for Webster a nd Clarion soils from Eqs. (8) and (9), respectively. Similarly, from the North Carolina study we obtained r = 0.9933 and 0.9906 for 1966 and 1967 from Eqs. (12) a nd (13), respectively. Note that r has been reported to four digits in all cases. Since pe rfect correlation would correspond to r = 1, these values actually differ from 1 by only two digits The real point is the mean ing of the high values of r The correlation coefficient is a measure of th e magnitude of the response variable ( Y ) to the control variable ( Q ) and is influenced by scatte r in the data used in the analysis. Apparently the expanded growth model and the pa rameters chosen provide excelle nt description of the plant system. A low value in r could indicate either an incorrect theory or model parameters, or excessive scatter in data. The theo ry also provided excellent desc ription for the fruit fraction of the soybean crop with r = 0.9974 and 0.9796 for the two soils from Eqs. (12) and (13) and r = 0.9986 and 0.9959 for 1966 and 1967 from Eqs. (16) a nd (17), respectively. In all the regression equations the intercepts are essentially zero so th at the slopes are measures of the yield factor A A general goal of this work has been to develop a comprehensive theory of crop growth which has broad application to a wide variet y of crop species and environmental conditions. Details lie in the choice of model parameters Management factors which influence crop production include sampling time, harv est interval (for pere nnials), water availa bility (rainfall or irrigation), level of applied nutrients (such as nitrogen, phosphorus, and potassium), plant population, soil type, and cu ltural practices (such as tillage, weed control, in sect control, etc.). What is the need for a mathematical theory of anything? Why not simp ly examine results of measurements and observations until the meaning becomes clear? The need for a conceptual framework for interpretation has been clear for t housands of years reaching back to Pythagoras, Aristotle, Euclid, the Greeks, and the Babyloni ans. Recall the atomic hypothesis of matter of Democritus and the later work of John Dalton an d the law of combining proportions (atoms and molecules), which led eventually to the period ic table of the elements by Mendeleev. In 1932 three major discoveries were made (Tomonaga 1997, p. x): James Chadwick proposed the concept of the neutron (a nuclear particle with mass but without ch arge), Carl Anderson confirmed the existence of the positron (twin to the electron but with positive charge) first proposed by Paul Dirac, and Haro ld Urey identified isotopes of hydrogen with charge of +1 and mass numbers of 1, 2, and 3. These discoveries represented major advances in physics which continue to the present. This is not to underestimate the importance of observation. In the end science values theories which explain things over thos e which are deemed mathematic ally beautiful (to paraphrase PAGE 9 Overman & Scholtz Soybean 7 Richard Feynman). To quote Feynman further, For technology to succeed reality must prevail over public relations, for in the end nature cannot be fooled. (Gleick, 1993, p. 428). This may be interpreted to mean that the goa l of science is the search fo r knowledge and understanding of how nature works. The curious reader is referred to books about the life and work of mathematicians (Dunham, 1994) and the life and work of physicists (Penrose, 2004). A broad overview of science in the 20th century is provided by Gerard Piel, the former publisher of Scientific American (Piel, 2001). Finally, it is well to remember the purpose of mathematical analysis of data (including statistics) to draw inference about meaning of the physical, chemical or biological system under investigation (Fisher Box, 1978, chapter 17). We have tried to follow th is strategy in all of our work. REFERENCES Abramowitz M., Stegun I. A. (1965). Handbook of Mathemat ical Functions. Dover. NY. Beaty E. R., Powell J. D., Brown R. H., Ethredge W. J. (1963). Effect of nitr ogen rate and clipping frequency on yield of Pensacola bahiagrass. Agronomy J. 55:34. Blue W. G. (1987). Response of Pensacola bahiagrass ( Paspalum notatum Flgge) to fertilizer nitrogen on an Entisol and a Spodosol in north Florida. So il and Crop Science Society of Florida Proc. 47:135139. Burton G. W., Jackson J. E., and Hart R. H. ( 1963). Effects of cutting frequency and nitrogen on yield, in vitro digestibility, and protein, fiber, and carotene content of coastal bermudagrass. Agronomy J. 55:500502. Dunham W. (1994). The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities. John Wiley & Sons. NY. Fisher Box J. (1978). R. A. Fisher: The Life of a Scientist. John Wiley & Sons. NY. Fritschi F. B., Roberts B. A., Tr avis R. L., Rains D. W., Hutm acher R. B. (2003). Response of irrigated Acala and Pima cotton to nitrogen fe rtilization: growth, dry matter partitioning, and yield. Agronomy J. 95:133146. Gleick J. (1993). Genius: The Life and Science of Richard Feynman. Vintage Books. NY. Hammond L. C., Black C. A., Norman A. G. (1951). Nutrient Uptake by Soybeans on Two Iowa Soils. Research Bulletin No. 384. Iowa State Agri cultural Experiment Station. Ames, IA. 461512. Henderson J. B., Kamprath E. J. (1970). Nutrient and Dry Matter Accumulation by Soybeans. Technical Bulletin No. 197. North Carolina Agricultural Experiment Station. Raleigh, NC. 127. PAGE 10 Overman & Scholtz Soybean 8 Kamprath E. J. (1986). Nitrogen studies with co rn on Coastal plain soils Technical Bulletin 282. North Carolina Agricultural Research Service. North Carolina State University. Raleigh, NC. Karlen D. L., Sadler E. J., Camp C. R. ( 1987). Dry matter, nitrogen, phosphorus, and potassium accumulation rates by corn on Norfolk loamy sand. Agronomy J. 79:649656. Hall A. D. (1905). The Book of the Rot hamsted Experiments John Murray. London. Mullins G. L., Burmester C. H. (1991). Dry matter, nitrogen, phosphorus, and potassium accumulation by four cotton vari eties. Agronomy J. 82:729736. Overman A. R., Scholtz R. V. III. (2002). Mathematical Models of Crop Growth and Yield. Taylor & Francis. NY. Overman A. R., Scholtz R. V. III. (2010). A Memoir on A Simplified Theory of Biomass Production by Photosynthesis. University of Florida. Gainesville FL 19 p. Overman A. R.; Wilkinson S. R. (2003). Extended logistic model of forage grass response to applied nitrogen as affected by soil erosion. Trans. American Society of Agricultural Engineers 46:13751380. Penrose R. (2004). The Road to Reality: A Complete Gu ide to the Laws of the Universe. Alfred A. Knopf. NY. Piel G. (2001). The Age of Science: What Scientists Learned in the Twentieth Century. Basic Books. NY. Prine G. M., Burton, G. W. (1956). The effect of nitrogen rate and clipping frequency upon the yield, protein content and cer tain morphological characteris tics of coastal bermudagrass [ Cynodon dactylon (L.) Pers.]. Agronomy J. 48:296301. Russell E. J. (1937). Soil Conditions and Plant Growth. 7th Ed. Longmans, Green & Co. London. Tomonaga, S. (1997). The Story of Spin. The University of Chicago Press. IL. PAGE 11 Overman & Scholtz Soybean 9 Table 1. Accumulation of biomass ( Y ) with calendar time ( t ) by soybean vegetation on two soils at Ames, IA (1946).1 t x erf x 2exp x Q Y wk Mg ha1 Webster Clarion 20.4 planting 23.6 0.113 0.110 24.9 0.226 0.228 25.3 0.112 0.126 0.987 0.000 26.3 0.238 0.264 0.945 0.187 0.614 0.535 26.9 0.312 0.341 0.907 0.335 0.785 0.737 27.9 0.438 0.464 0.826 0.630 1.41 1.05 28.9 0.562 0.573 0.729 0.966 1.93 1.40 29.9 0.688 0.670 0.623 1.323 2.63 2.00 30.9 0.812 0.749 0.517 1.672 3.44 2.82 31.9 0.938 0.815 0.415 2.004 4.69 3.40 32.9 1.062 0.867 0.323 2.300 5.43 3.88 33.9 1.188 0.907 0.244 2.550 6.25 4.21 34.9 1.312 0.9364 0.179 2.756 6.93 5.58 35.9 1.438 0.9580 0.127 2.919 6.69 4.24 36.9 1.562 0.9728 0.0870 3.044 5.69 3.62 37.9 1.688 0.9830 0.0580 3.134 5.51 3.39 38.9 1.812 0.9896 0.0374 3.198 5.67 39.7 1.912 0.9931 0.0258 3.234 5.34 1.0000 0.0000 3.313 1Crop data adapted from Hammond et al. (1951). Table 2. Accumulation of biomass ( Y ) with calendar time ( t ) by soybean fruit (seeds + pods) on two soils at Ames, IA (1946).1 t x erf x 2exp x Q Y wk Mg ha1 Webster Clarion 31.5 0.8875 0.791 0.455 0.000 31.9 0.938 0.815 0.415 0.043 0.208 0.286 32.9 1.062 0.867 0.323 0.158 0.759 0.500 33.9 1.188 0.907 0.244 0.280 1.48 0.932 34.9 1.312 0.9364 0.179 0.395 1.80 1.68 35.9 1.438 0.9580 0.127 0.505 2.26 2.00 36.9 1.562 0.9728 0.0870 0.589 2.76 2.21 37.9 1.688 0.9830 0.0580 0.656 3.20 2.16 38.9 1.812 0.9896 0.0374 0.706 3.41 39.7 1.912 0.9931 0.0258 0.736 3.50 1.0000 0.0000 0.806 1Crop data adapted from Hammond et al. (1951). PAGE 12 Overman & Scholtz Soybean 10 Table 3. Accumulation of biomass ( Y ) with calendar time ( t ) by soybean vegetation for two years at Clayton, NC.1 t x erf x 2exp x Q Y wk Mg ha1 1966 1967 18.7 planting 23.6 0.100 0.112 0.9900 0.000 24.4 0.000 0.000 1.000 0.134 0.445 0.457 25.9 0.188 0.210 0.965 0.532 0.943 1.72 27.3 0.362 0.391 0.877 1.03 1.55 2.59 28.7 0.538 0.553 0.749 1.61 2.52 5.19 30.1 0.712 0.686 0.602 2.20 3.25 6.50 31.6 0.900 0.797 0.445 2.79 4.27 7.09 33.0 1.075 0.872 0.315 3.25 5.15 8.99 34.4 1.250 0.923 0.210 3.61 6.24 10.89 35.9 1.438 0.9580 0.127 3.88 6.31 8.37 37.4 1.625 0.9784 0.0713 4.06 5.69 7.02 38.7 1.788 0.9884 0.0410 4.16 4.75 5.33 1.0000 0.0000 4.29 1Crop data adapted from He nderson and Kamprath (1970). Table 4. Accumulation of biomass ( Y ) with calendar time ( t ) by soybean fruit (seeds + pods) for two years at Clayton, NC.1 t x erf x 2exp x Q Y wk Mg ha1 1966 1967 33.3 1.112 0.884 0.290 0.000 34.4 1.250 0.923 0.210 0.075 1.09 1.90 35.9 1.438 0.9580 0.127 0.191 2.09 4.43 37.4 1.625 0.9784 0.0713 0.291 3.13 6.50 38.7 1.788 0.9884 0.0410 0.353 3.80 8.55 1.0000 0.0000 0.451 1Crop data adapted from He nderson and Kamprath (1970). PAGE 13 Overman & Scholtz Soybean 11 Table 5. Estimated partitioning of biomass be tween lightgathering and structural components for soybean (vegetation) at Cl ayton, NC USA (1970). t x erf x 2exp x QL QS Q fL wk 23.6 0.100 0.112 0.9900 0.000 0.000 0.000 1.000 25 0.075 0.085 0.9944 0.189 0.083 0.272 0.695 26 0.200 0.223 0.961 0.322 0.240 0.562 0.573 27 0.325 0.354 0.900 0.448 0.468 0.916 0.489 28 0.450 0.475 0.817 0.564 0.751 1.315 0.429 30 0.700 0.678 0.613 0.759 1.401 2.160 0.351 32 0.950 0.821 0.406 0.896 2.031 2.927 0.306 34 1.200 0.910 0.237 0.982 2.532 3.514 0.279 36 1.450 0.960 0.122 1.030 2.868 3.898 0.264 38 1.700 0.984 0.0556 1.053 3.059 4.112 0.256 40 1.950 0.9942 0.0223 1.063 3.154 4.217 0.252 42 2.200 0.9981 0.00791 1.067 3.195 4.262 0.250 44 2.450 0.9995 0.00247 1.068 3.211 4.279 0.250 1 0 1.068 3.218 4.286 0.249 PAGE 14 Overman & Scholtz Soybean 12 Figure 1. Correlation of biomass yield ( Y ) with the growth quantifier ( Q ) for soybean (vegetation) on two soils at Am es, IA, USA. Data adapted from Hammond et al. (1951). Lines drawn from Eqs. (8) and (9). PAGE 15 Overman & Scholtz Soybean 13 Figure 2. Accumulation of biomass yield ( Y ) with calendar time ( t ) for soybean (vegetation) on two soils at Ames, IA, USA. Data adapted from Hammond et al. (1951). Cu rves drawn from Eqs. (6) through (9). PAGE 16 Overman & Scholtz Soybean 14 Figure 3. Correlation of biomass yield ( Y ) with the growth quantifier ( Q ) for soybean (seeds + pods) on two soils at Ames, IA, USA. Data ad apted from Hammond et al. (1951). Lines drawn from Eqs. (12) and (13). PAGE 17 Overman & Scholtz Soybean 15 Figure 4. Accumulation of biomass yield ( Y ) with calendar time ( t ) for soybean (seeds + pods) on two soils at Ames, IA, USA. Data adapted from Hammond et al. (1951). Cu rves drawn from Eqs. (10) through (13). PAGE 18 Overman & Scholtz Soybean 16 Figure 5. Correlation of biomass yield ( Y ) with the growth quantifier ( Q ) for soybean (vegetation) for two years at Clayton, NC, US A. Data adapted from Henderson and Kamprath. (1970). Lines drawn from Eqs. (16) and (17). PAGE 19 Overman & Scholtz Soybean 17 Figure 6. Accumulation of biomass yield ( Y ) with calendar time ( t ) for soybean (vegetation) for two years at Clayton, NC, USA. Data adapte d from Henderson and Kamprath (1970). Curves drawn from Eqs. (14) through (17). PAGE 20 Overman & Scholtz Soybean 18 Figure 7. Correlation of biomass yield ( Y ) with the growth quantifier ( Q ) for soybean (seeds + pods) for two years at Clayton, NC, USA. Da ta adapted from Henderson and Kamprath(1970). Lines drawn from Eqs. (20) and (21). PAGE 21 Overman & Scholtz Soybean 19 Figure 8. Accumulation of biomass yield ( Y ) with calendar time ( t ) for soybean (seeds + pods) for two years at Clayton, NC, USA. Data adapted from Hender son and Kamprath(1970). Curves drawn from Eqs. (18) through (21). PAGE 22 Overman & Scholtz Soybean 20 Figure 9. Dependence of biomass yield ( Y ) on seasonal rainfall ( R ) for soybean (vegetation) for 1966 and 1967 at Clayton, NC, USA. Data adap ted from Henderson and Kamprath (1970). Curve drawn from Eq. (23). PAGE 23 Overman & Scholtz Soybean 21 Figure 10. Partitioning of biomass yield ( Y ) for soybean (vegetation) at Clayton, NC, USA. Curve for leaf fraction ( fL) vs. calendar time ( t ) drawn from Eq. (27). 