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A MEMOIR ON A Simplified Theory ofBiomass Production by Photosynthesis Allen R. Overman and Richard V. Scholtz III Agricultural and Biological Engineering University of Florida Copyright 2010 Allen R. Overman A Simplified Crop Growth Model Key words: Plant growth, mathematical model, photosynthesis This memoir is focused on a simplified theory of biomass production by photosynthesis. It describes accumulation of biomass with calendar time. The theory is structured on a rigorous mathematical framework and a sound empirical foundation using data from the literature. Particular focus in on the northern hemisphere where most field research has been conducted, and on the warmseason perennial coastal bermudagrass for which an extensive database exists. Three primary factors have been identified in the model: (1) an energy driving function, (2) a partition function between lightgathering (leaf) and structural (stem) plant components, and (3) an aging function. These functions are then combined to form a linear differential equation. Integration leads to an analytical solution. A linear relationship is established between biomass production and a growth quantifier for a fixed harvest interval. The theory is further used to describe forage quality (nitrogen concentration and digestible fraction) between leaves and stems of the plants. The theory can be applied to annuals (such as corn) and well as perennials. Crop response to various applied elements (such as nitrogen, phosphorus, potassium, calcium, and magnesium) can be described. The theory contains five parameters: two for the Gaussian energy function, two for the linear partition function, and one for the exponential aging function. Acknowledgement: The authors thank Amy G Buhler, Engineering Librarian, Marston Science Library, University of Florida, for assistance with preparation of this memoir. Overman and Scholtz A Simplified Crop Growth Model A Simplified Theory of Biomass Production by Photosynthesis Allen R. Overman and Richard V. Scholtz III Introduction Photosynthesis is the biochemical processes by which green plants use incident radiant energy to fix C02 from the atmosphere and H from the splitting of the water molecule to form CHO the major content of plant biomass. Mineral elements (such as N, P, K, Ca, Mg, etc) are derived from the rhizosphere (root zone). Readers interested in details of photosynthesis at the molecular and cellular levels are referred to the excellent book by Oliver Morton [1]. The details are extremely complicated. In this article we seek to make simplifying assumptions which lead to a field scale theory relating the rate of accumulation of plant biomass with calendar time, dY/dt, (Mg ha1 wk1) to calendar time, t, (wk). A broader view of progress of science in the 20th century, including the role of physics in photosynthesis, is provided by Gerard Piel [2]. In his classic book The Ascent of Man, Jacob Bronowski [3] traces human history, including cultural and biological evolution. He documents the development of the agricultural revolution from hunter/gatherer to farmer/husbandman that has made modem agriculture possible and which produces the food and fiber upon which humanity now depends. Along with the expansion of technology, agricultural field research has experienced rapid development, beginning with the famous work at Rothamsted, England in about 1850. Today a very large database exists from various locations around the world. It is this database which we will draw upon as the empirical foundation for the present theory. In order to develop a rigorous theory of biomass production by photosynthesis, a sound mathematical framework is required. For this purpose we draw upon two fundamental principles of science as stated by Davies and Gribbin [4, p. 44]: Principle #1: It is possible to know un/et1hing of how nature works without knowing everything about how nature works. Without this principle there would be no science and no understanding. Principle #2: In physics a linear system is one in which a collection of causes leads to a corresponding collection of effects. For a given system it can be shown that this correspondence is unique, and the principle works in both directions. In the sequel this will be referred as the correspondence principle. Since the invention of the calculus by Isaac Newton, it has been common practice to make simplifying assumptions which lead to linear differential equations (ordinary or partial) with analytic solutions. Examples include mechanics (equations of motion), thermodynamics (diffusion of thermal energy), chemical dynamics (diffusion of molecular species), electrical phenomenon, magnetic phenomenon, and even to quantum mechanics (both matrix mechanics and wave mechanics). A similar strategy is followed in the present work. Simplifying assumptions are made in the search to develop a mathematical theory between the time rate of accumulation of biomass, dY/dt, and calendar time, t. The analysis is focused on the northern hemisphere where the greatest collection of field data has been reported. In addition the analysis focuses on field studies with the warmseason perennial coastal bermudagrass [Cynodon dactylon (L.) Pers.]. Numerous studies have been conducted for fixed harvest interval, At. Measurements of biomass accumulation as related to harvest interval have been reported, from which deductions can be Overman and Scholtz A Simplified Crop Growth Model made about effects. The correspondence principle can then be used to make inference about the causes involved. This leads finally to a linear differential equation. Theory Development The first step in this process is to identify key components which contribute to biomass production with calendar time as measured by data from field studies. These factors are then combined into a linear differential equation. The differential equation is then integrated to an analytic solution. Again data are for the warmseason perennial coastal bermudagrass in the northern hemisphere and harvested on a fixed time interval. Energy Driving Function The first step along these lines was taken by Overman [5] in response to requests by environmental regulators to estimate biomass and plant nutrient accumulation with calendar time for a water reclamation/reuse project in Florida. The analysis drew upon a field study at Watkinsville, GA with coastal bermudagrass harvested on a fixed time interval [6]. The experiment consisted of a 2x2 factorial of two harvest intervals (4 wk, 6 wk) and two irrigation treatments (irrigated, nonirrigated). The distribution of biomass with calendar time was shown to follow a Gaussian function described by F = exp (1) where F is the fraction of total biomass at calendar time t (referenced to Jan. 1), /U is time to mean of biomass distribution (referenced to Jan. 1), and i2uo is the time spread of the biomass distribution. It was shown that the distributions were independent of irrigation treatment and harvest interval and followed the equation F = exp . (2) 8.13 ) with exponential values in wk. Details of the analysis are described in Overman and Scholtz [7, Section 3.2] These results raised the interesting question as to the origin of the Gaussian distribution? It is known that incident solar radiation in the northern hemisphere rises from a minimum in January to a maximum in July and decreases again to a minimum in December. Overman and Scholtz [7, Table 1.6] analyzed solar radiation data for Rothamsted, England [8] and showed that the distribution followed the Gaussian distribution F = exp 25.02 (3) 14.8 ) From this analysis it seems logical to assume an energy driving function, E(t), which follows a Gaussian distribution to good approximation Overman and Scholtz A Simplified Crop Growth Model E(t) = exp _ t ) (4) where t is calendar time referenced to Jan. 1, p/ is time to mean of the solar energy distribution referenced to Jan. 1, and 2o is the time spread of the solar energy distribution. All units are in weeks. Partition Function for Biomass The second step in the analysis is to identify an intrinsic growth function that identifies how plants respond to the input of solar energy. Fortunately field experiments have been conducted with coastal bermudagrass at Tifton, GA by Prine and Burton [9]. The factorial experiment consisted of five nitrogen levels (N= 0, 112, 336, 672, and 1008 kg ha1), six harvest intervals (At= 1, 2, 3, 4, 6, and 8 wk), and two years (1953 and 1954). Total biomass yield (Yt, Mg ha1) and total plant nitrogen uptake (Nut, kg ha)) for the entire season (24 wk) were reported. System response is illustrated in Figure 1 with biomass data for N= 672 kg ha1 and for both years. As it turned out 1953 exhibited ideal rainfall (labeled 'wet') and 1954 exhibited the worst drought in 30 years (labeled 'dry'). It appears from Figure 1 that the data points follow straight lines for At < 6 wk. The regression lines are described by 1953: Y, = a +/3At = 11.38 + 2.73At r =0.9950 (5) 1954: Y, =a + 3At = 4.41+1.61At r =0.9836 (6) with correlation coefficients of r = 0.9950 and 0.9836 for 1953 and 1954, respectively. The effect of water availability is clearly evident in these equations. Equations (5) and (6) reveal an additional factor beyond the energy driving function identified above. From the correspondence principle for linear systems, we are led to assume a partition function of the type P(t t,= a+b(t t,) (7) where t is calendar time for the growth interval, wk; t, is the reference time for the growth interval, wk; and a is the intercept parameter, Mg ha1 wk' and b is the slope parameter, Mg ha1 2 wk2. Overman and Wilkinson [10] examined data from the literature on partitioning of dry matter between leaf and stem for coastal bermudagrass and concluded that the first term in Eq. (7) corresponds to the rate of growth of the lightgathering component (leaf) while the second term corresponds to the rate of growth of the structural component (stem). Equation (7) means that reference time t, is reset for each growth interval. It follows logically that Eq. (7) must have a limited time domain of application. Otherwise artificial radiant energy could be supplied to the plants to provide unlimited biomass accumulation. This simply does not happen! It follows that there must be an additional limiting factor in the system. Overman and Scholtz A Simplified Crop Growth Model Aging Function for Biomass The third step in the analysis is to identify an additional factor which imposes a further limit on plant growth. Burton et al. [11] expanded the range of harvest intervals to include At= 3, 4, 5, 6, 8, 12, and 24 wk. Applied nitrogen was N = 672 kg ha1. Seasonal total biomass yield (Ye) and seasonal total plant nitrogen uptake (Nut) were reported as given in Table 1. Plant nitrogen concentration is defined as Nc = Nt/Yt. In order to see the pattern in the response more clearly, results are also shown in Figure 2. The graph of Yt vs. At exhibits a rise, passing through a peak, then following a steady decline. This seems like the place for the timehonored method of intuition as defined by Roger Newton [12, p. 60]. Expanding on the linear partition function from the previous section, we assume a linearexponential function Y, = (a, + f,At).exp( yAt) (8) where a, ,f,, and y are to be evaluated from the data points. It remains to be determined if Eq. (8) exhibits the correct characteristics to describe the response curve. In other words, an operational procedure is required to evaluate the coefficients of Eq. (8). Now if the value of y were known, then we could define a standardized yield Y,* as Y,* = Y, exp(+ y At) = a, +/8y At (9) which leads to a linear equation in At. The procedure is to try values of y which leads to the optimum correlation for Eq. (9). This process leads to an estimate of y = 0.077 wk 1 with the corresponding values in column 5 of Table 1. Linear regression of Y,* vs. At then leads to Y,* = Y, exp(+ 0.077 At) = ay + fy At = 8.91+ 3.49 At r = 0.9995 (10) with a correlation coefficient of r = 0.9995. This result is shown in Figure 3, where the line is drawn from Eq. (10). It follows that the lower curve in Figure 2 is drawn from Y, = (8.91+3.49At).exp( 0.077At) (11) The next step is to define standardized plant nitrogen uptake N,,* as N* = N,, exp(+ yAt)= a,, + /,,At (12) With y = 0.077 wk' 1 as for biomass response, this leads to values in column 6 of Table 1. Linear regression of N,,,* vs. At then leads to Nt*= N,,t .exp(+ 0.077 At) = a,,+, At = 422 +33.OAt r = 0.9780 (13) Overman and Scholtz A Simplified Crop Growth Model with a correlation coefficient of r = 0.9780. This result is shown in Figure 3, where the line is drawn from Eq. (13). It follows that the middle curve in Figure 2 is drawn from N., = (422 + 33.OAt) exp( 0.077 At) (14) Plant nitrogen concentration Nc is then defined by N N, a, +,/nAt 422+33.0 At Yt ay +',yAt 8.91+3.49At The upper curve in Figure 2 is drawn from Eq. (15). The linearexponential function appears to describe the response curves rather well. The question now occurs as to the meaning of the exponential term in Eq. (8). Assuming that the system follows linear behavior, we invoke the correspondence principle to write an aging function, A(t t ) = exp[c(t t)] (16) It should be noted that the 'aging function' is not derived from biochemical or genetic considerations, but represents an operational definition of a decline in capacity of the system to generate new biomass as the plants age. Equation (16) resets for each new value of reference time t,. Linear Differential Equation At this point three independent functions have been identified for the system: a linear partition function P(t t,), an aging function A(t t,), and an energy driving function E(t). The question now is how to combine these functions to form a linear differential equation? Since the functions are considered independent, it seems reasonable to invoke the principle of joint probability and write the equation in product form dY = constant P(t t) A(t t,) E(t) dt 2 (17) = constant. [a + b(t t)] exp[ c(t t)] exp U The procedure of joint probability is the same as that used by James Clerk Maxwell in developing the velocity distribution law in the kinetic theory of gases [13, p. 159]. He first writes the function for one dimension in space. Then the treatment is extended to three dimensions by invoking the principle of joint probability and writing the overall function as the product of the three separate functions. Overman and Scholtz Overman and Scholtz A Simplified Crop Growth Model Equation (17) forms a linear first order differential equation in the time variable t. However, it contains two reference times: t, and /j. Since the partition function and the aging function reset for each growth interval, Equation (17) must be viewed as piecewise continuous in the time interval t t,, and integration must proceed accordingly. The interested reader is referred to Overman and Scholtz [7, Section 4.3] for details of the integration process. The solution for the increase in biomass accumulation for the i th growth interval, AY becomes the simple linear relationship AY = A AQ, (18) where A is the yield factor, Mg ha1 and AQ, is the gi em ith quantifier defined by AQ, (= ( kx, )[erf x erf x, > [exp( x2) exp( x )] exp(aFcx,) (19) The dimensionless partition coefficient, k, in Eq. (19) is defined by k = 2b / a (20) and the dimensionless time variable x is defined by t oc (21) a2o 2 It follows immediately that x, is defined by x t'U oc (22) The error function, erfx, in Eq. (19) is defined by 2 x erf x = exp( u2 )du (23) where u is the variable of integration for the Gaussian function exp( u). The cumulative sum of biomass for n harvests, Yn, is given by ,, = AY = A AQ, = A.Q, (24) 1=1 1=1 A Simplified Crop Growth Model Total biomass yield for the entire season, Y,, is the sum over all harvests and is related to total growth quantifier for the season, Qt, by Yt =A.Q ( A Mathematical Theorem 25) A mathematical theorem is now presented which provides a rigorous connection between Eq. (17) for the linear first order differential equation for each growth interval and Eq. (8) for linear exponential dependence of seasonal total biomass yield, Yt, on a fixed harvest interval, At. Details of the proof are given by Overman and Scholtz [7, Section 4.3]. The theorem establishes that the cumulative sum of biomass production for n harvests is given by Y A' 2 + (Atexp Y, = A' \2 +At exp (26) where A' is defined by A' = constant. 2o. a (27) In Eq. (26) the term in curly brackets confirms the Gaussian distribution of the energy driving function, which approaches 1 in the limit of large t. This means that total biomass yield for the season, Yt, is given by Y = A'.Q;' (28) where seasonal total growth quantifier, Q,' is defined by Q( =2+ A epk A Q' = 2+At exp C At) 2)J Finally, Eq. (28) can be written in regression form C At = (a +ytAt)exp( 7At) This completes the proof of the theorem. It should be noted that = c / 2 is required. The theorem applies for fixed harvest interval At, wk. (29) (30) Overman and Scholtz Y, = 2A'+ kA At exp 1~~~ Vc . At. [I + erf  2 J 2 Ii2T Overman and Scholtz A Simplified Crop Growth Model Application of the Theory to Forage Quality Forage quality is measured primarily by two characteristics: nitrogen concentration (protein content) and digestibility of the biomass by ruminant animals. For perennial grasses quality relates to length of time between harvests since age of plants influences the balance between plant components (leaves vs. stems). In this section data from two studies with perennial grasses are used to evaluate forage quality. Study ith Coastal Bermudagrass Equation (15) can be used to estimate plant nitrogen concentration of the two plant components. In the limit as At > 0, nitrogen concentration of the lightgathering (leaf) component, NcL is estimated from NcL = = 47.4 g kg (31) 8.91 In the limit as At > very large, nitrogen concentration of the structural (stem) component, Ncs, is estimated from Ncs =3.0 9.5 g kg (32) 3.49 Clearly the nitrogen (crude protein) quality of the leaf fraction is considerably higher than of the stem fraction. The study by Burton et al. [11] at Tifton, GA included data on digestible dry matter as measured by the in vitro method [14]. Seasonal total digestible biomass (Dr) along with total biomass (Yt) as related to harvest interval (At) are listed in Table 2. Response of Dr vs. At is assumed to follow a linearexponential function (similar to Eq. (8)) D, = (ad +dAt) exp( At) (33) where ad, fd, and y are to be evaluated from the data points. Standardized digestible dry matter (D,*) is defined by D* = D, exp(+ yAt) = ad +/,d At (34) Again we choose y = 0.077 wk 1 with the corresponding values in column 5 of Table 2. Linear regression of D,* vs. At then leads to D =Dt exp(+ 0.077At)= ad +d At =9.93+1.29At r = 0.9929 (35) A Simplified Crop Growth Model with a correlation coefficient of r = 0.9929. This result leads to the linerexponential equation D, = (9.93+1.29 At) exp( 0.077 At) (36) It follows immediately that digestible fraction,fd, is described by Dt 9.93+1.29At fdY 8.91+3.49At Y8.91 + 3.49At (37) Now Eq. (37) can be used to estimate digestibility of the two plant components. In the limit as At > 0, the digestible fraction of the lightgathering (leaf) fraction, fj, is estimated from (38) 9.93 f = 1.11 8.91 Of course the lightgathering component should not exceed 1.00. The value of 1.11 represents uncertainty in the intercept values in Eq. (37). In fact, it can be shown at the 95% confidence level that ay 8.91+1.41 Mg ha1 and that ad = 9.93 2.52 Mg ha1. Since these values clearly overlap, it seems reasonable to assume that the value of the lightgathering component is approximately 1.00. (39) fdL 1.00 In the limit as At > verylarge, the digestible fraction of the structural (stem) fraction, fds, is estimated from 1.29 fL = 0.37 3.49 (40) Now we see that as At increases, the plants shift from dominance by lightgathering to structural component of the plant with a corresponding decline in forage quality. It can be shown from calculus that harvest intervals to achieve maximum total plant biomass and maximum digestible biomass can be estimated from, respectively, 1 Zay fly 1 a, Atpd ad f=d 1 8.91 10.4 wk =10.4 wk 0.077 3.49 1 9.93 5.3 wk 0.0775.3 wk 0.077 1.29 (41) (42) Overman and Scholtz Overman and Scholtz A Simplified Crop Growth Model Study 11 ih Perennial Peanut A study was conducted by Beltranena et al. [15] with the warmseason legume perennial peanut (Arachis glabrata Benth cv 'Florigraze') on Arredondo loamy fine sand (loamy, siliceous, semiactive, hyperthermic Grossarenic Paleudult). Plants were sampled on fixed harvest intervals of At= 2, 4, 6, 8, 10, and 12 wk. The growing season is considered to be 24 wk. Measurements were made of seasonal total biomass (Ye), seasonal total plant nitrogen uptake (Nrt), and seasonal total digestible biomass (Dr). Results are listed in Table 3. All data are for a clipping height of 3.8 cm. As with the case of coastal bermudagrass, the linearexponential model is assumed to apply. Standardized biomass yield (Y,*), standardized plant nitrogen uptake (N,,*), and standardized digestible biomass yield (D,*) are calculated from Y,* =Y,exp(0.077At) = ayy+ At =4.84 + 2.12At r = 0.9938 (43) Nt' = N,, exp(0.077At) = a, + ,,At = 272 + 33.6At r = 0.9916 (44) D = Dt exp(0.077At) = d + fdAt = 4.97 +1.00At r = 0.9791 (45) Equations (43) through (45) lead to the linearexponential equations Y, = (ay +yAt).exp( At)= (4.84+2.12At)exp( 0.077At) (46) N., = (ac +,8. exp( yAt))= (272 + 33.6At) exp( 0.077At) (47) Equations (46) and (47) lead immediately to N 272+33.6At S Y 4.84+2.12At Equation (48) can be used to estimate nitrogen concentration of each plant component. Nitrogen concentration of the leaf fraction, NcL is estimated from 272 NL = 56.2 g kg1 (49) 4.84 whereas nitrogen concentration of the stem fraction, Ncs, is estimated from 33.6 Ncs = 2.= 15.8 gkg1 (50) 2.12 Again, the leaf fraction contains higher nitrogen concentration than does the stem fraction. A Simplified Crop Growth Model Dependence of seasonal total digestible biomass is described by the linearexponential equation D, =(ad + dAt) exp( yAt)= (4.97 + 1.00At) exp( 0.077At) (51) Equations (46) and (51) can be combined to obtain dependence of digestible fractionfd on harvest interval At Dt 4.97 +1.OOAt fdY 4.84+2.12At Y. 4.84 + 2.12At (52) Digestible fraction for the two plant components leaves and stems, fdL and fds, can be estimated from 4.97 f = =1.03 1.00 (53) 4.84 1.00 fds .= =0.47 (54) 2.12 Again the leaf fraction exhibits higher digestibility than the stem fraction. It can be shown from calculus that the maximum value (peak) of the linearexponential curves corresponds to a peak harvest interval, Atp, for total biomass, total plant nitrogen uptake, and total digestible biomass given by At 1 ay 1 =0 Atpn 1 a At pd ad 1 4.84 4.84 10.7 wk 0.077 2.12 1 272 =4.9 wk 0.077 33.6 1 4.97 8.0 wk 0.0778.0 wk 0.077 1.00 Summary and Conclusions The simplified theory of biomass production by photosynthesis is described by Eqs (18) through (23). Since Eq (19) contains the error function, erfx, this seems like the appropriate place to present the detailed discussion by Abramowitz and Stegun [16, chp 7], including a table of values. A few key characteristics should be noted: (0) = 0, j(+00c) (55) (56) (57) Overman and Scholtz A Simplified Crop Growth Model f( x)=f(+x), f( )= Af(+ )= 1 The theory contains five parameters: two for the energy driving function (j/, V2cr) and three for plant characteristics (k, c, A). Examination of data for the northern hemisphere and for the warmseason perennial coastal bermudagrass lead to the estimates listed in Table 4. References 1. Morton 0 (2007) Eating the Sun: How Plants Power the Planet. London: Harper Collins. 475 p. 2. Piel G (2001) The Age of Science: What Scientists Learned in the Twentieth Century. New York: Basic Books. 459 p. 3. Bronowski J (1973) The Ascent of Man. Boston: Little, Brown & Co. 448 p. 4. Davies P and Gribbin J (1992) The Matter Myth: Dramatic Discoveries That Have Challenged Our Understanding of Physical Reality. New York: Simon & Schuster. 320 p. 5. Overman AR (1984) Estimating crop growth rate with land treatment. J. Env. Eng. Div., American Society of Civil Engineers 110:10091012. 6. Mays DA, Wilkinson SR, and Cole CV (1980) Phosphorus nutrition of forages. In: The Role of Phosphorus in Agriculture. Khasawneh FE and Kamprath EJ (eds). Madison, WI: American Society of Agronomy. pp. 805840. 7. Overman AR and Scholtz RV (2002) Mathematical Models of Crop Growth and Yield. New York: Taylor & Francis. 328 p. 8. Russell EJ (1950) Soil Conditions and Plant Growth 8th ed. London: Longmans, Green & Co. 635 p. 9. Prine GM and Burton GW (1956) The effect of nitrogen rate and clipping frequency upon the yield, protein content, and certain morphological characteristics of coastal bermudagrass [Cynodon dactylon (L.) Pers.]. Agronomy J. 48:296301. 10. Overman AR and Wilkinson SR (1989) Partitioning of dry matter between leaf and stem in coastal bermudagrass. Agricultural Systems 30:3547. 11. Burton GW, Jackson JE, and Hart RH (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. 12. Newton, R (1997) The Truth of Science: Physical Theories and Reality. Cambridge, MA: Harvard University Press. 260 p. 13. Longair MS (1984) Theoretical Concepts in Physics. New York: Cambridge University Press. 366 p. 14. Moore JE and Dunham GD (1971) Procedure for the twostage in vitro organic matter digestion of forages (revised). Nutrition Laboratory, Department of Animal Science. University of Florida. Gainesville, FL 10 p. 15. Beltranena R, Breman J and Prine GM (1981) Yield and quality of Florigraze rhizome peanut (Arachis glabrata Bent.) as affected by cutting height and frequency. Proc. Soil and Crop Science Society of Florida 40:153156. 16. Abramowitz M and Stegun IA (1965) Handbook of Mathematical Functions. New York: Dover Publications. 1046 p. Overman and Scholtz A Simplified Crop Growth Model Table 1. Response of seasonal total biomass yield (Yt), seasonal total plant nitrogen uptake (Nut), and plant nitrogen concentration (NA) to harvest interval (At) at N= 672 kg ha1 for coastal bermudagrass at Tifton, GA.1 At Yt Nt Nc Y, N wk Mg ha1 kg ha1 g kg1 Mg ha1 kg ha1 3 15.2 438 28.8 19.1 552 4 16.2 415 25.6 22.0 565 5 17.8 417 23.4 26.2 613 6 19.9 411 20.6 31.6 652 8 19.9 340 17.1 36.8 630 12 20.1 289 14.4 50.6 728 24 14.6 198 13.6 92.7 1257 1 Data adapted from Burton et al. [ 11]. Table 2. Response of seasonal total biomass yield (Yt), seasonal digestible biomass (Dt), and digestible fraction (fd) to harvest interval (At) at N= 672 kg ha1 for coastal bermudagrass at Tifton, GA.1 At Y, Dt fd D* wk Mg ha1 Mg ha1 Mg ha1 3 15.2 9.91 0.652 12.5 4 16.2 10.3 0.637 14.0 5 17.8   6 19.9 11.9 0.597 18.9 8 19.9 11.3 0.566 20.9 12 20.1 10.6 0.525 26.7 24 14.6 6.31 0.432 40.0 1 Data adapted from Burton et al. [ 11]. Overman and Scholtz A Simplified Crop Growth Model Table 3. Response of seasonal total biomass yield (Yt), seasonal total plant nitrogen uptake (Nut), plant nitrogen concentration (Nc), seasonal digestible biomass (Dt), digestible fraction (fd), standardized total biomass yield (Y,*), standardized total plant nitrogen uptake (Nt*), and standardized total digestible biomass (D,*) to harvest interval (At) for perennial peanut at Gainesville, FL.1 At Yt N,, Nc Dt fd Y Nu* D* wk Mg ha1 kg ha1 g kg1 Mg ha1 Mg ha1 kg ha1 Mg ha1 2 8.0 280 35.0 5.7 0.71 9.3 327 6.6 4 9.0 300 33.3 6.3 0.70 12.2 408 8.6 6 11.4 310 27.2 7.4 0.65 18.1 492 11.7 8 12.4 300 24.2 7.6 0.62 23.0 555 14.1 10 11.6 270 23.3 6.5 0.56 25.1 583 14.0 12 12.0 270 22.5 6.7 0.56 30.2 680 16.9 1 Data adapted from Beltranena et al. [14]. Table 4. Summary of Parameter Values. Parameter Definition Value Units Time to mean of 26.0 wk solar energy distribution1 ,F2cr Time spread of 8.00 wk solar energy distribution k Partition constant 5 none c Aging coefficient 0.150 wk1 A Yield factor varies Mg ha1 1For northern hemisphere, referenced to Jan. 1. Overman and Scholtz A Simplified Crop Growth Model (U 0) E" U) Eo (0 2 4 6 8 1 Harvest Interval, wk Figure 1. Response of total biomass yield (Yt) to harvest interval (At) at N = 672 kg ha1 for coastal bermudagrass grown at Tifton, GA. Data adapted from Prine and Burton [9]. Lines drawn from Eqs. (5) and (6). 1953 1954 Overman and Scholtz A Simplified Crop Growth Model 0 I I Ill 111)1 / 0 6 12 18 24 Harvest Interval, wks Figure 2. Response of total biomass yield (Yt), total plant nitrogen (Nut), and plant nitrogen concentration (Nc) to harvest interval (At) at N 672 kg ha1 for coastal bermudagrass grown at Tifton, GA. Data adapted from Burton et al. [11]. Curves drawn from Eqs. (11), (14), and (15). "C c 400 0 200 ^:: z 0 CU c 2o co 10 E 0 00 0 Overman and Scholtz A Simplified Crop Growth Model 1600 1200 800 400 0 120 80 40 0 Harvest Interval, wk Figure 3. Response of standardized biomass yield (Y,*) and standardized plant nitrogen uptake (Nt*) to harvest interval (At) at N = 672 kg ha1 for coastal bermudagrass grown at Tifton, GA. Lines drawn from Eqs. (10) and (13). J '* I I I I I I ' Overman and Scholtz 
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A Simplified Crop Growth Model Total biomass yield for the entire season, Yt, is the sum over all harvests and is related to total growth quantifier for the season, Qt, by Y,= AQ, A Mathematical Theorem (25) A mathematical theorem is now presented which provides a rigorous connection between Eq. (17) for the linear first order differential equation for each growth interval and Eq. (8) for linear exponential dependence of seasonal total biomass yield, Y,, on a fixed harvest interval, At. Details of the proof are given by Overman and Scholtz [7, Section 4.3]. The theorem establishes that the cumulative sum of biomass production for n harvests is given by Y,, = A 2 + At .exp where A' is defined by A' = constant* a\ 52 2 2At) + erfJ 2 J 2_ F,2ar. (26) (27) In Eq. (26) the term in curly brackets confirms the Gaussian distribution of the energy driving function, which approaches 1 in the limit of large t. This means that total biomass yield for the season, Yt, is given by Y = A'.Q'; (28) where seasonal total growth quantifier, Q, is defined by Q; 2+ At .exp At) Finally, Eq. (28) can be written in regression form (29) K, 2A'+ $JkAf Atjexp c At) = Cy + 8y~t exp (),At (30) This completes the proof of the theorem. It should be noted that y = c/2 is required. The theorem applies for fixed harvest interval At, wk. Overman and Scholtz A MEMOIR ON A Simplified Theory ofBiomass Production by Photosynthesis Allen R. Overman and Richard V. Scholtz III Agricultural and Biological Engineering University of Florida Copyright 2010 Allen R. Overman A Simplified Crop Growth Model Dependence of seasonal total digestible biomass is described by the linearexponential equation D, = (r, + 8,At expy At 4t.97+1.00At7'exp(0.077At (51) Equations (46) and (51) can be combined to obtain dependence of digestible fractionfd on harvest interval At f Dt 4.97 +1.00At Y, 4.84 + 2.12At (52) Digestible fraction for the two plant components leaves and stems,fad andfds, can be estimated from 4.97 fdL = 1.03 = 1.00 (53) 4.84 1.00 fds = = 0.47 (54) 2.12 Again the leaf fraction exhibits higher digestibility than the stem fraction. It can be shown from calculus that the maximum value (peak) of the linearexponential curves corresponds to a peak harvest interval, Atp, for total biomass, total plant nitrogen uptake, and total digestible biomass given by 1 ay ;/fl 1 4.84 = 10.7 wk 0.077 2.12 1 a 1 272 Atp = 27 = 4.9 wk y fl, 0.077 33.6 1 At  pd a, 1 ad 0. Pd 0.077 (55) (56) (57) 4.97 8.0 wk 1.00 Summary and Conclusions The simplified theory of biomass production by photosynthesis is described by Eqs (18) through (23). Since Eq (19) contains the error function, erfx, this seems like the appropriate place to present the detailed discussion by Abramowitz and Stegun [16, chp 7], including a table of values. A few key characteristics should be noted: (0) = 0, J(+ ) = 1 Overman and Scholtz A Simplified Crop Growth Model References 1. Morton O (2007) Eating the Sun: How Plants Power the Planet. London: Harper Collins. 475 p. 2. Piel G (2001) The Age of Science: What Scientists Learned in the Twentieth Century. New York: Basic Books. 459 p. 3. Bronowski J (1973) The Ascent of Man. Boston: Little, Brown & Co. 448 p. 4. Davies P and Gribbin J (1992) The Matter Myth: Dramatic Discoveries That Have Challenged Our Understanding of Physical Reality. New York: Simon & Schuster. 320 p. 5. Overman AR (1984) Estimating crop growth rate with land treatment. J. Env. Eng. Div., American Society of Civil Engineers 110:10091012. 6. Mays DA, Wilkinson SR, and Cole CV (1980) Phosphorus nutrition of forages. In: The Role of Phosphorus in Agriculture. Khasawneh FE and Kamprath EJ (eds). Madison, WI: American Society of Agronomy. pp. 805840. 7. Overman AR and Scholtz RV (2002) Mathematical Models of Crop Growth and Yield. New York: Taylor & Francis. 328 p. 8. Russell EJ (1950) Soil Conditions and Plant Growth 8th ed. London: Longmans, Green & Co. 635 p. 9. Prine GM and Burton GW (1956) The effect of nitrogen rate and clipping frequency upon the yield, protein content, and certain morphological characteristics of coastal bermudagrass [Cynodon dactylon (L.) Pers.]. Agronomy J. 48:296301. 10. Overman AR and Wilkinson SR (1989) Partitioning of dry matter between leaf and stem in coastal bermudagrass. Agricultural Systems 30:3547. 11. Burton GW, Jackson JE, and Hart RH (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. 12. Newton, R (1997) The Truth of Science: Physical Theories and Reality. Cambridge, MA: Harvard University Press. 260 p. 13. Longair MS (1984) Theoretical Concepts in Physics. New York: Cambridge University Press. 366 p. 14. Moore JE and Dunham GD (1971) Procedure for the twostage in vitro organic matter digestion of forages (revised). Nutrition Laboratory, Department of Animal Science. University of Florida. Gainesville, FL 10 p. 15. Beltranena R, Breman J and Prine GM (1981) Yield and quality of Florigraze rhizome peanut (Arachis glabrata Bent.) as affected by cutting height and frequency. Proc. Soil and Crop Science Society of Florida 40:153156. 16. Abramowitz M and Stegun IA (1965) Handbook of Mathematical Functions. New York: Dover Publications. 1046 p. Overman and Scholtz A Simplified Crop Growth Model Aging Function for Biomass The third step in the analysis is to identify an additional factor which imposes a further limit on plant growth. Burton et al. [11] expanded the range of harvest intervals to include At = 3, 4, 5, 6, 8, 12, and 24 wk. Applied nitrogen was N = 672 kg ha1. Seasonal total biomass yield (Y,) and seasonal total plant nitrogen uptake (Nt) were reported as given in Table 1. Plant nitrogen concentration is defined as N = Nt/Yt. In order to see the pattern in the response more clearly, results are also shown in Figure 2. The graph of Yt vs. At exhibits a rise, passing through a peak, then following a steady decline. This seems like the place for the timehonored method of intuition as defined by Roger Newton [12, p. 60]. Expanding on the linear partition function from the previous section, we assume a linearexponential function Y, = (y+ fl At exp Ati (8) where ay, fy, and y are to be evaluated from the data points. It remains to be determined if Eq. (8) exhibits the correct characteristics to describe the response curve. In other words, an operational procedure is required to evaluate the coefficients of Eq. (8). Now if the value of y were known, then we could define a standardized yield Y,* as Y* = Y, exp 4 / At = ay + l At (9) which leads to a linear equation in At. The procedure is to try values of y which leads to the optimum correlation for Eq. (9). This process leads to an estimate of y = 0.077 wk 1 with the corresponding values in column 5 of Table 1. Linear regression of Y,* vs. At then leads to Y, = Y, exp 0.077 At ay + At = 8.91+3.49At r = 0.9995 (10) with a correlation coefficient of r = 0.9995. This result is shown in Figure 3, where the line is drawn from Eq. (10). It follows that the lower curve in Figure 2 is drawn from Y, = 4.91+ 3.49At exp 0.077 At (11) The next step is to define standardized plant nitrogen uptake Nu,* as N, = N, exp ( ) At '= an + P, t (12) With 7 = 0.077wk 1 as for biomass response, this leads to values in column 6 of Table 1. Linear regression of N,,* vs. At then leads to N,* =N, exp4 0.077At = a, + At = 422+33.0At r = 0.9780 (13) Overman and Scholtz A Simplified Crop Growth Model Study 0i ith Perennial Peanut A study was conducted by Beltranena et al. [15] with the warmseason legume perennial peanut (Arachis glabrata Benth cv 'Florigraze') on Arredondo loamy fine sand (loamy, siliceous, semiactive, hyperthermic Grossarenic Paleudult). Plants were sampled on fixed harvest intervals of At= 2, 4, 6, 8, 10, and 12 wk. The growing season is considered to be 24 wk. Measurements were made of seasonal total biomass (Y,), seasonal total plant nitrogen uptake (Nt), and seasonal total digestible biomass (Di). Results are listed in Table 3. All data are for a clipping height of 3.8 cm. As with the case of coastal bermudagrass, the linearexponential model is assumed to apply. Standardized biomass yield (Y,*), standardized plant nitrogen uptake (N,,*), and standardized digestible biomass yield (D,*) are calculated from Y, = Y, exp1 .077At = + PAt = 4.84+2.12At r = 0.9938 (43) N,,* = N, exp .077At= a, + fAt = 272 + 33.6At r = 0.9916 (44) D,* = D, exp ).077At= ad + JdAt = 4.97 + 1.00At r = 0.9791 (45) Equations (43) through (45) lead to the linearexponential equations Y, = y + flAt exp ( At 4.84 + 2.12At exp 0.077At (46) N,, = (y + f, exp ( At = 72 + 33.6At exp(0.077At (47) Equations (46) and (47) lead immediately to N, 272+33.6At N (48) Y, 4.84+2.12At Equation (48) can be used to estimate nitrogen concentration of each plant component. Nitrogen concentration of the leaf fraction, NcL, is estimated from 272 NC = 2 = 56.2 g kg1 (49) 4.84 whereas nitrogen concentration of the stem fraction, Ncs, is estimated from 33.6 NCS 33.6 15.8 gkg (50) 2.12 Again, the leaf fraction contains higher nitrogen concentration than does the stem fraction. Overman and Scholtz A Simplified Crop Growth Model E ex)p 2] (4) where t is calendar time referenced to Jan. 1, p is time to mean of the solar energy distribution referenced to Jan. 1, and J2o is the time spread of the solar energy distribution. All units are in weeks. Partition Function for Biomass The second step in the analysis is to identify an intrinsic growth function that identifies how plants respond to the input of solar energy. Fortunately field experiments have been conducted with coastal bermudagrass at Tifton, GA by Prine and Burton [9]. The factorial experiment consisted of five nitrogen levels (N= 0, 112, 336, 672, and 1008 kg ha1), six harvest intervals (At= 1, 2, 3, 4, 6, and 8 wk), and two years (1953 and 1954). Total biomass yield (Yt, Mg ha') and total plant nitrogen uptake (Nt, kg ha') for the entire season (24 wk) were reported. System response is illustrated in Figure 1 with biomass data for N = 672 kg ha and for both years. As it turned out 1953 exhibited ideal rainfall (labeled 'wet') and 1954 exhibited the worst drought in 30 years (labeled 'dry'). It appears from Figure 1 that the data points follow straight lines for At < 6 wk. The regression lines are described by 1953: Y, = a + jAt = 11.38 + 2.73At r = 0.9950 (5) 1954: Y,= a + At = 4.41+1.61At r = 0.9836 (6) with correlation coefficients ofr = 0.9950 and 0.9836 for 1953 and 1954, respectively. The effect of water availability is clearly evident in these equations. Equations (5) and (6) reveal an additional factor beyond the energy driving function identified above. From the correspondence principle for linear systems, we are led to assume a partition function of the type P(t =a+b(t, (7) where t is calendar time for the growth interval, wk; t, is the reference time for the growth interval, wk; and a is the intercept parameter, Mg ha1 wk' and b is the slope parameter, Mg ha1 wk2. Overman and Wilkinson [10] examined data from the literature on partitioning of dry matter between leaf and stem for coastal bermudagrass and concluded that the first term in Eq. (7) corresponds to the rate of growth of the lightgathering component (leaf) while the second term corresponds to the rate of growth of the structural component (stem). Equation (7) means that reference time t, is reset for each growth interval. It follows logically that Eq. (7) must have a limited time domain of application. Otherwise artificial radiant energy could be supplied to the plants to provide unlimited biomass accumulation. This simply does not happen! It follows that there must be an additional limiting factor in the system. Overman and Scholtz PAGE 1 A MEMOIR ON A Simplified Theory of Biomass Production by Photosynthesis Allen R. Overman and Richard V. Scholtz III Agricultural and Biological Engineering University of Florida Copyright 2010 Allen R. Overman PAGE 2 Overman and Scholtz A Simplified Crop Growth Model 1 Key words : Plant growth, mathematical model, photosynthesis This memoir is focused on a simplified theory of biomass production by photosynthesis. It describes accumulation of biomass with calendar time. The theory is structured on a rigorous mathematical framework and a sound empirical foundation using data from the literature. Particular focus in on the northern hemisphere where most field research has been conducted, and on the warmseason perennial coastal bermudagr ass for which an extensive database exists. Three primary factors have been identified in the model: (1) an energy driving function, (2) a partition function between lightgat hering (leaf) and structural (stem) plant components, and (3) an aging function. These functions are then co mbined to form a linear differential equation. Integration leads to an analytical solution. A lin ear relationship is esta blished between biomass production and a growth quantifier for a fixed harv est interval. The theory is further used to describe forage quality (nitrogen concentration a nd digestible fraction) between leaves and stems of the plants. The theory can be applied to annua ls (such as corn) and we ll as perennials. Crop response to various applied elements (such as nitrogen, phosphorus, potassium, calcium, and magnesium) can be described. The theory contains five parameters: two for the Gaussian energy function, two for the linear part ition function, and one for th e exponential aging function. Acknowledgement : The authors thank Amy G Buhler, E ngineering Librarian, Marston Science Library, University of Florida, for assist ance with preparation of this memoir. PAGE 3 Overman and Scholtz A Simplified Crop Growth Model 2 A Simplified Theory of Biomass Production by Photosynthesis Allen R. Overman and Richard V. Scholtz III Introduction Photosynthesis is the biochemical processes by which green plants use incident radiant energy to fix CO2 from the atmosphere and H from the spl itting of the water molecule to form CHO the major content of plant biomass. Mineral elements (such as N, P, K, Ca, Mg, etc) are derived from the rhizosphere (root zone). Readers interested in details of photosynthesis at the molecular and cellular levels are referred to the excellent book by Oliver Morton [1]. The details are extremely complicated. In this article we seek to make simplifying assumptions which lead to a field scale theory relating the rate of accu mulation of plant biomass with calendar time, dY/dt (Mg ha1 wk1) to calendar time, t (wk). A broader view of pr ogress of science in the 20th century, including the role of physic s in photosynthesis, is provided by Gerard Piel [2]. In his classic book The Ascent of Man Jacob Bronowski [3] traces huma n history, including cultural and biological evolution. He documents the de velopment of the agricultural revolution from hunter/gatherer to farmer/husbandman that has made modern agriculture possible and which produces the food and fiber upon which humanity now depends. Along with the expansion of technology, agricultural field resear ch has experienced rapid deve lopment, beginning with the famous work at Rothamsted, England in about 1850. Today a very large database exists from various locations around the world. It is this da tabase which we will draw upon as the empirical foundation for the present theory. In order to develop a rigorous theory of biomass production by photosynthesis, a sound mathematical framework is required. For this purpose we draw upon two fundamental principles of science as stated by Davies and Gribbin [ 4, p. 44]: Principle #1: It is possible to know something of how nature works without knowing everything about how nature works. Without this principle there would be no science and no understanding. Principle #2: In physics a linear system is one in which a collection of causes l eads to a corresponding collection of effects. For a given system it can be shown that this correspon dence is unique, and the principle works in both directions. In the sequel this will be referred as the correspondence principle Since the invention of the calculus by Isaac Newton, it has been comm on practice to make simplifying assumptions which lead to linear differential equations (ordinar y or partial) with anal ytic solutions. Examples include mechanics (equations of motion), thermodynamics (diffusion of thermal energy), chemical dynamics (diffusion of molecula r species), electrical phenomenon, magnetic phenomenon, and even to quantum mechanics (bot h matrix mechanics and wave mechanics). A similar strategy is followed in the present work. Simplifying assumptions are made in the search to develop a mathematical theory betwee n the time rate of accumulation of biomass, dY/dt and calendar time, t The analysis is focused on the northern hemisphere where the greatest collection of field data has been reported. In additi on the analysis focuses on field studies with the warmseason pe rennial coastal bermudagrass [ Cynodon dactylon (L.) Pers.]. Numerous studies have been conduc ted for fixed harvest interval,t Measurements of biomass accumulation as related to harvest interval have been reported, from which deductions can be PAGE 4 Overman and Scholtz A Simplified Crop Growth Model 3 made about effects. The correspondence principle can then be used to make inference about the causes involved. This leads finally to a linear differential equation. Theory Development The first step in this process is to identif y key components which contribute to biomass production with calendar time as measured by data from field studies. These factors are then combined into a linear differential equation. The differential equation is then integrated to an analytic solution. Again data are for the warm season perennial coastal bermudagrass in the northern hemisphere and harvested on a fixed time interval. Energy Driving Function The first step along these lines was taken by Overman [5] in response to requests by environmental regulators to estimate biomass a nd plant nutrient accumulation with calendar time for a water reclamation/reuse project in Flor ida. The analysis dr ew upon a field study at Watkinsville, GA with coastal bermudagrass ha rvested on a fixed time interval [6]. The experiment consisted of a 2x2 factorial of two ha rvest intervals (4 wk, 6 wk) and two irrigation treatments (irrigated, nonirrigated) The distribution of biomass w ith calendar time was shown to follow a Gaussian function described by 22 exp t F (1) where F is the fraction of total biomass at calendar time t (referenced to Jan. 1), is time to mean of biomass distribution (referenced to Jan. 1), and 2 is the time spread of the biomass distribution. It was shown that the distributions were independe nt of irrigation treatment and harvest interval and followed the equation 213 8 8 27 exp t F (2) with exponential values in wk. Details of the an alysis are described in Overman and Scholtz [7, Section 3.2] These result s raised the interesting question as to the origin of the Gaussian distribution? It is known that in cident solar radiation in the no rthern hemisphere rises from a minimum in January to a maximum in July a nd decreases again to a minimum in December. Overman and Scholtz [7, Table 1.6] analyzed sola r radiation data for Rothamsted, England [8] and showed that the distribution fo llowed the Gaussian distribution 28 14 0 25 exp t F (3) From this analysis it seems logical to assume an energy driving function, E ( t ), which follows a Gaussian distribution to good approximation PAGE 5 Overman and Scholtz A Simplified Crop Growth Model 4 22 exp t t E (4) where t is calendar time referenced to Jan. 1, is time to mean of the solar energy distribution referenced to Jan. 1, and 2 is the time spread of the solar en ergy distribution. All units are in weeks. Partition Function for Biomass The second step in the analysis is to identify an intrinsic growth func tion that identifies how plants respond to the input of solar energy. Fo rtunately field experime nts have been conducted with coastal bermudagrass at Tifton, GA by Prin e and Burton [9]. The factorial experiment consisted of five nitrogen levels ( N = 0, 112, 336, 672, and 1008 kg ha1), six harvest intervals (t = 1, 2, 3, 4, 6, and 8 wk), and two year s (1953 and 1954). Total biomass yield ( Yt Mg ha1) and total plant nitrogen uptake ( Nut, kg ha1) for the entire season (24 wk) were reported. System response is illustrated in Fi gure 1 with biomass data for N = 672 kg ha1 and for both years. As it turned out 1953 exhibited ideal rainfall (labeled wet) and 1954 exhibited the worst drought in 30 years (labeled dry). It appe ars from Figure 1 that the data points follow straight lines for wk 6 t The regression lines are described by 1953: t t Yt 73 2 38 11 r = 0.9950 (5) 1954: t t Yt 61 1 41 4 r = 0.9836 (6) with correlation coefficients of r = 0.9950 and 0.9836 for 1953 and 1954, respectively. The effect of water availability is clearly evident in these equations. Equations (5) and (6) reveal an additional factor beyond the ener gy driving function identified above. From the correspondence principle for linear systems, we are led to assume a partition function of the type i it t b a t t P (7) where t is calendar time for the growth interval, wk; ti is the reference time for the growth interval, wk; and a is the intercept parameter, Mg ha1 wk1 and b is the slope parameter, Mg ha1 wk2. Overman and Wilkinson [10] examined data from the literature on partitioning of dry matter between leaf and stem for coastal bermudag rass and concluded that the first term in Eq. (7) corresponds to the rate of growth of the lightgathering co mponent (leaf) while the second term corresponds to the rate of growth of the structural component (stem). Equation (7) means that reference time ti is reset for each growth interval. It follows logically that Eq. (7) must have a limited time domain of application. Otherwise artificial radiant energy coul d be supplied to the plants to provide unlimited biomass accumulation. This simply does not happen! It follo ws that there must be an additional limiting factor in the system. PAGE 6 Overman and Scholtz A Simplified Crop Growth Model 5 Aging Function for Biomass The third step in the analysis is to identify an additional factor which imposes a further limit on plant growth. Burton et al. [11] expanded th e range of harvest intervals to include t = 3, 4, 5, 6, 8, 12, and 24 wk. Applied nitrogen was N = 672 kg ha1. Seasonal total biomass yield ( Yt) and seasonal total plant nitrogen uptake ( Nut) were reported as given in Table 1. Plant nitrogen concentration is defined as Nc = Nut/Yt In order to see the pattern in the response more clearly, results are also shown in Figure 2. The graph of Yt vs.t exhibits a rise, passing through a peak, then following a steady decline. This seems like the place for the timehonored method of intuition as defined by Roger Newton [ 12, p. 60]. Expanding on the li near partition function from the previous section, we a ssume a linearexponential function t t Yy y t exp (8) where and ,y y are to be evaluated from the data point s. It remains to be determined if Eq. (8) exhibits the correct characteristics to desc ribe the response curve. In other words, an operational procedure is required to evaluate the coefficients of Eq. (8). Now if the value of were known, then we could define a standardized yield tY as t t Y Yy y t t exp (9) which leads to a linear equation in t The procedure is to try values of which leads to the optimum correlation for Eq. (9). This process leads to an estimate of 1wk 077 0 with the corresponding values in column 5 of Table 1. Linear regression of tY vs. t then leads to t t t Y Yy y t t 49 3 91 8 077 0 exp r = 0.9995 (10) with a correlation coefficient of r = 0.9995. This result is shown in Figure 3, where the line is drawn from Eq. (10). It follows that th e lower curve in Figure 2 is drawn from t t Yt 077 0 exp 49 3 91 8 (11) The next step is to define standardized plant nitrogen uptake utN as t t N Nn n ut ut exp (12) With 1wk 077 0 as for biomass response, this leads to values in column 6 of Table 1. Linear regression of utN vs. t then leads to t t t N Nn n ut ut 0 33 422 077 0 exp r = 0.9780 (13) PAGE 7 Overman and Scholtz A Simplified Crop Growth Model 6 with a correlation coefficient of r = 0.9780. This result is shown in Figure 3, where the line is drawn from Eq. (13). It follows that the middle curve in Figure 2 is drawn from t t Nut 077 0 exp 0 33 422 (14) Plant nitrogen concentration Nc is then defined by t t t t Y N Ny y n n t ut c 49 3 91 8 0 33 422 (15) The upper curve in Figure 2 is drawn from Eq (15). The linearexponentia l function appears to describe the response curves rather well. The question now occurs as to th e meaning of the expone ntial term in Eq. (8 ). Assuming that the system follows linear behavior, we invoke th e correspondence principl e to write an aging function, ] exp[i it t c t t A (16) It should be noted that the a ging function is not derived from biochemical or genetic considerations, but represents an operational defin ition of a decline in capacity of the system to generate new biomass as the plan ts age. Equation (16) resets for each new value of reference time ti. Linear Differential Equation At this point three independent functions have been identified for the system: a linear partition function it t P an aging function it t A and an energy driving function t E. The question now is how to combine these functions to form a linear differential equati on? Since the functions are considered independent, it seems reasonable to invoke the principle of joint probability and write the equation in product form 22 exp exp constant constant t t t c t t b a t E t t A t t P dt dYi i i i (17) The procedure of joint probability is the sa me as that used by James Clerk Maxwell in developing the velocity distributi on law in the kinetic theory of gases [13, p. 159]. He first writes the function for one dimension in space. Then th e treatment is extended to three dimensions by invoking the principle of joint probability and wr iting the overall function as the product of the three separate functions. PAGE 8 Overman and Scholtz A Simplified Crop Growth Model 7 Equation (17) forms a linear first order di fferential equation in the time variable t However, it contains two reference times: ti and Since the partition function and the aging function reset for each growth interval, Equation (17) must be viewed as piecewise continuous in the time interval it t and integration must proceed accordingly. The interested read er is referred to Overman and Scholtz [7, Section 4.3] for details of the integration process. The solution for the increase in biomass accumulation for the i th growth interval, iY becomes the simple linear relationship i iQ A Y (18) where A is the yield factor, Mg ha1 and iQ is the growth quantifier defined by i i i i icx x x k x x kx Q2 exp exp exp erf erf 12 2 (19) The dimensionless partition coefficient, k in Eq. (19) is defined by a b k / 2 (20) and the dimensionless time variable x is defined by 2 2 2 c t x (21) It follows immediately that xi is defined by 2 2 2 c t xi i (22) The error function, erf x in Eq. (19) is defined by xdu u x0 2exp 2 erf (23) where u is the variable of integra tion for the Gaussian function 2exp u The cumulative sum of biomass for n harvests, Yn, is given by n i n i n i i nQ A Q A Y Y1 1 (24) PAGE 9 Overman and Scholtz A Simplified Crop Growth Model 8 Total biomass yield for the entire season, Yt, is the sum over all harvests and is related to total growth quantifier for the season, Qt, by t tQ A Y (25) A Mathematical Theorem A mathematical theorem is now presented wh ich provides a rigorous connection between Eq. (17) for the linear first order differential equati on for each growth interval and Eq. (8) for linearexponential dependence of seasonal total biomass yield, Yt, on a fixed ha rvest interval, t Details of the proof are given by Overman and Sc holtz [7, Section 4.3]. The theorem establishes that the cumulative sum of biomass production for n harvests is given by 2 erf 1 2 1 2 exp 2 2 t t c t k A Yn (26) where A is defined by 2 2 constant a A (27) In Eq. (26) the term in curly brackets confirms the Gaussian distributi on of the energy driving function, which approaches 1 in the limit of large t This means that total biomass yield for the season, Yt is given by t tQ A Y (28) where seasonal total growth quantifier, tQ is defined by t c t k Qt2 exp 2 2 (29) Finally, Eq. (28) can be wr itten in regression form t t t c t A k A Yy y t exp 2 exp 2 2 (30) This completes the proof of the theorem. It should be noted that 2 / c is required. The theorem applies for fixed harvest intervalt wk. PAGE 10 Overman and Scholtz A Simplified Crop Growth Model 9 Application of the Theory to Forage Quality Forage quality is measured primarily by two characteristics: nitrogen concentration (protein content) and digestibility of the biomass by ru minant animals. For perennial grasses quality relates to length of time between harvests sin ce age of plants influences the balance between plant components (leaves vs. stems). In this secti on data from two studies with perennial grasses are used to evaluate forage quality. Study with Coastal Bermudagrass Equation (15) can be used to estimate plant nitr ogen concentration of the two plant components. In the limit as 0 t, nitrogen concentration of the lightgathering (leaf) component, NcL is estimated from 4 47 91 8 422 cLN g kg1 (31) In the limit as large very t, nitrogen concentration of the structural (stem) component, NcS is estimated from 5 9 49 3 0 33 cSN g kg1 (32) Clearly the nitrogen (crude protei n) quality of the leaf fraction is considerably higher than of the stem fraction. The study by Burton et al. [11] at Tifton, GA included data on dige stible dry matter as measured by the in vitro method [14]. Seasonal total digestible biomass ( Dt) along with total biomass ( Yt) as related to harvest interval (t ) are listed in Table 2. Response of Dt vs. t is assumed to follow a linearexponential func tion (similar to Eq. (8)) t t Dd d t exp (33) where and ,d d are to be evaluated from the data poi nts. Standardized digestible dry matter (tD) is defined by t t D Dd d t t exp (34) Again we choose 1wk 077 0 with the corresponding values in column 5 of Table 2. Linear regression of tD vs. t then leads to t t t D Dd d t t 29 1 93 9 077 0 exp r = 0.9929 (35) PAGE 11 Overman and Scholtz A Simplified Crop Growth Model 10 with a correlation coefficient of r = 0.9929. This result leads to the linerexponential equation t t Dt 077 0 exp 29 1 93 9 (36) It follows immediately that digestible fraction, fd, is described by t t Y D ft t d 49 3 91 8 29 1 93 9 (37) Now Eq. (37) can be used to estimate digestibil ity of the two plant components. In the limit as 0 t, the digestible fraction of the lightgathering (leaf) fraction, dLf is estimated from 11 1 91 8 93 9 dLf (38) Of course the lightgathering component should not exceed 1.00. The value of 1.11 represents uncertainty in the intercept values in Eq. (37) In fact, it can be shown at the 95% confidence level that 41 1 91 8 y Mg ha1 and that 52 2 93 9 d Mg ha1. Since these values clearly overlap, it seems reasonable to assume that th e value of the lightgathering component is approximately 1.00. 00 1 dLf (39) In the limit as large very t, the digestible fraction of the structural (stem) fraction, dSf is estimated from 37 0 49 3 29 1 dLf (40) Now we see that as t increases, the plants shift from domi nance by lightgathering to structural component of the plant with a corr esponding decline in forage quality. It can be shown from calculus that harvest inte rvals to achieve maximum total plant biomass and maximum digestible biomass can be estimated from, respectively, 4 10 49 3 91 8 077 0 1 1 y y pyt wk (41) 3 5 29 1 93 9 077 0 1 1 d d pdt wk (42) PAGE 12 Overman and Scholtz A Simplified Crop Growth Model 11 Study with Perennial Peanut A study was conducted by Beltranena et al. [15] with the warmseason legu me perennial peanut ( Arachis glabrata Benth cv Florigraze) on Arredondo lo amy fine sand (loamy, siliceous, semiactive, hyperthermic Grossarenic Paleudult). Pl ants were sampled on fixed harvest intervals of t = 2, 4, 6, 8, 10, and 12 wk. The growing seas on is considered to be 24 wk. Measurements were made of seasonal total biomass ( Yt), seasonal total plant nitrogen uptake ( Nut), and seasonal total digestible biomass ( Dt). Results are listed in Table 3. All data are for a clipping height of 3.8 cm. As with the case of coastal bermudagr ass, the linearexponential model is assumed to apply. Standardized biomass yield (tY), standardized plant nitrogen uptake (utN), and standardized digestible biomass yield (tD) are calculated from t t t Y Yy y t t 12 2 84 4 077 0 exp r = 0.9938 (43) t t t N Nn n ut ut 6 33 272 077 0 exp r = 0.9916 (44) t t t D Dd d t t 00 1 97 4 077 0 exp r = 0.9791 (45) Equations (43) through (45) lead to the linearexpone ntial equations t t t t Yy y t 077 0 exp 12 2 84 4 exp (46) t t t Nn n ut 077 0 exp 6 33 272 exp (47) Equations (46) and (47) lead immediately to t t Y N Nt ut c 12 2 84 4 6 33 272 (48) Equation (48) can be used to estimate nitrogen concentration of each plant component. Nitrogen concentration of the leaf fraction, NcL is estimated from 2 56 84 4 272 cLN g kg1 (49) whereas nitrogen concentrati on of the stem fraction, NcS is estimated from 8 15 12 2 6 33 cSN g kg1 (50) Again, the leaf fraction contains higher nitr ogen concentration than does the stem fraction. PAGE 13 Overman and Scholtz A Simplified Crop Growth Model 12 Dependence of seasonal total digestible bioma ss is described by the linearexponential equation t t t t Dd d t 077 0 exp 00 1 97 4 exp (51) Equations (46) and (51) can be combined to obtain dependence of digestible fraction fd on harvest interval t t t Y D ft t d 12 2 84 4 00 1 97 4 (52) Digestible fraction for the two plant components leaves and stems, fdL and fdS can be estimated from 00 1 03 1 84 4 97 4 dLf (53) 47 0 12 2 00 1 dSf (54) Again the leaf fraction exhibits higher digestibility than the stem fraction. It can be shown from calculus that the maximu m value (peak) of the linearexponential curves corresponds to a peak harvest interval,pt for total biomass, total plant nitrogen uptake, and total digestible biomass given by 7 10 12 2 84 4 077 0 1 1 y y pyt wk (55) 9 4 6 33 272 077 0 1 1 n n pnt wk (56) 0 8 00 1 97 4 077 0 1 1 d d pdt wk (57) Summary and Conclusions The simplified theory of biomass production by p hotosynthesis is described by Eqs (18) through (23). Since Eq (19) contai ns the error function, erf x this seems like the appropriate place to present the detailed discussion by Abramowitz and Stegun [16, chp 7], including a table of values. A few key characteristics should be noted: f (0) = 0, f ( ) = 1 PAGE 14 Overman and Scholtz A Simplified Crop Growth Model 13 f ( x ) = f ( x ), f ( ) = f ( ) = 1 The theory contains five parameters : two for the energy driving function ( 2 ,) and three for plant characteristics (k, c, A). Examination of data for the northern hemisphere and for the warmseason perennial coastal bermudagrass lead to the estimates listed in Table 4. References 1. Morton O (2007) Eating the S un: How Plants Power the Plan et. London: Harper Collins. 475 p. 2. Piel G (2001) The Age of Scien ce: What Scientists Learned in the Twentieth Century. New York: Basic Books. 459 p. 3. Bronowski J (1973) The Ascent of Man. Boston: Little, Brown & Co. 448 p. 4. Davies P and Gribbin J (1992) The Matter Myth: Dramatic Discoveries That Have Challenged Our Understanding of Physical R eality. New York: Simon & Schuster. 320 p. 5. Overman AR (1984) Estimating crop growth rate w ith land treatment. J. Env. Eng. Div., American Society of Civil Engineers 110:10091012. 6. Mays DA, Wilkinson SR, and Cole CV (1980) Phosphorus nutrition of forages. In: The Role of Phosphorus in Agriculture. Khasawneh FE and Kamprath EJ (eds). Madison, WI: American Society of Agronomy. pp. 805840. 7. Overman AR and Scholtz RV (2002) Mathemat ical Models of Crop Gr owth and Yield. New York: Taylor & Francis. 328 p. 8. Russell EJ (1950) Soil Conditions and Plan t Growth 8th ed. London: Longmans, Green & Co. 635 p. 9. Prine GM and Burton GW (1956) The effect of nitrogen rate and clipping frequency upon the yield, protein content, and certain morphological characteri stics of coastal bermudagrass [Cynodon dactylon (L.) Pers.]. Agronomy J. 48:296301. 10. Overman AR and Wilkinson SR (1989) Partitioning of dry matte r between leaf and stem in coastal bermudagrass. Agricultural Systems 30:3547. 11. Burton GW, Jackson JE, and Hart RH (1963) Effects of cutting frequency and nitrogen on yield, in vitro digestibility, and protein, fiber and carotene co ntent of coastal bermudagrass. Agronomy J. 55:500502. 12. Newton, R (1997) The Truth of Science: P hysical Theories and Reality. Cambridge, MA: Harvard University Press. 260 p. 13. Longair MS (1984) Theoreti cal Concepts in Physics. Ne w York: Cambridge University Press. 366 p. 14. Moore JE and Dunham GD (1971) Procedure for the twostage in vitro organic matter digestion of forages (revised). Nutrition La boratory, Department of Animal Science. University of Florida. Gainesville, FL 10 p. 15. Beltranena R, Breman J and Prine GM (1981) Yield and quality of Florigraze rhizome peanut (Arachis glabrata Bent.) as affected by cutting hei ght and frequency. Proc. Soil and Crop Science Society of Florida 40:153156. 16. Abramowitz M and Stegun IA (1965) Handbook of Mathematical Functions. New York: Dover Publications. 1046 p. PAGE 15 Overman and Scholtz A Simplified Crop Growth Model 14 Table 1. Response of seasonal total biomass yield (Yt), seasonal total plant nitrogen uptake (Nut), and plant nitrogen concentration (Nc) to harvest interval (t ) at N = 672 kg ha1 for coastal bermudagrass at Tifton, GA.1 t Yt Nut Nc tY utN wk Mg ha1 kg ha1 g kg1 Mg ha1 kg ha1 3 15.2 438 28.8 19.1 552 4 16.2 415 25.6 22.0 565 5 17.8 417 23.4 26.2 613 6 19.9 411 20.6 31.6 652 8 19.9 340 17.1 36.8 630 12 20.1 289 14.4 50.6 728 24 14.6 198 13.6 92.7 1257 1 Data adapted from Burton et al. [11]. Table 2. Response of seasonal total biomass yield (Yt), seasonal digestible biomass (Dt), and digestible fraction (fd) to harvest interval (t ) at N = 672 kg ha1 for coastal bermudagrass at Tifton, GA.1 t Yt Dt fd tD wk Mg ha1 Mg ha1 Mg ha1 3 15.2 9.91 0.652 12.5 4 16.2 10.3 0.637 14.0 5 17.8 6 19.9 11.9 0.597 18.9 8 19.9 11.3 0.566 20.9 12 20.1 10.6 0.525 26.7 24 14.6 6.31 0.432 40.0 1 Data adapted from Burton et al. [11]. PAGE 16 Overman and Scholtz A Simplified Crop Growth Model 15 Table 3. Response of seasonal total biomass yield (Yt), seasonal total plant nitrogen uptake (Nut), plant nitrogen concentration (Nc), seasonal digestible biomass (Dt), digestible fraction (fd), standardized tota l biomass yield (tY), standardized total plant nitrogen uptake (utN), and standardized total digestible biomass (tD) to harvest interval (t ) for perennial peanut at Gainesville, FL.1 t Yt utN Nc Dt fd tY utN tD wk Mg ha1 kg ha1 g kg1 Mg ha1 Mg ha1 kg ha1 Mg ha1 2 8.0 280 35.0 5.7 0.71 9.3 327 6.6 4 9.0 300 33.3 6.3 0.70 12.2 408 8.6 6 11.4 310 27.2 7.4 0.65 18.1 492 11.7 8 12.4 300 24.2 7.6 0.62 23.0 555 14.1 10 11.6 270 23.3 6.5 0.56 25.1 583 14.0 12 12.0 270 22.5 6.7 0.56 30.2 680 16.9 1 Data adapted from Beltranena et al. [14]. Table 4. Summary of Parameter Values. Parameter Definition Value Units Time to mean of 26.0 wk solar energy distribution1 2 Time spread of 8.00 wk solar energy distribution k Partition constant 5 none c Aging coefficient 0.150 wk1 A Yield factor varies Mg ha1 1For northern hemisphere, referenced to Jan. 1. PAGE 17 Overman and Scholtz A Simplified Crop Growth Model 16 Figure 1. Response of total biomass yield (Yt) to harvest interval (t ) at N = 672 kg ha1 for coastal bermudagrass grown at Tifton, GA. Data adapted from Prine and Burton [9]. Lines drawn from Eqs. (5) and (6). PAGE 18 Overman and Scholtz A Simplified Crop Growth Model 17 Figure 2. Response of total biomass yield (Yt), total plant nitrogen (Nut), and plant nitrogen concentration (Nc) to harvest interval (t ) at N = 672 kg ha1 for coastal bermudagrass grown at Tifton, GA. Data adapted from Bu rton et al. [11]. Curves drawn fr om Eqs. (11), (14), and (15). PAGE 19 Overman and Scholtz A Simplified Crop Growth Model 18 Figure 3. Response of sta ndardized biomass yield ( tY) and standardized plant nitrogen uptake ( utN) to harvest interval (t ) at N = 672 kg ha1 for coastal bermudagrass grown at Tifton, GA. Lines drawn from Eqs. (10) and (13). Overman and Scholtz A Simplified Crop Growth Model 1600 1200 z 800 N 0 c 400 cl) 0 "V a, S120 E I o m 80 N co 40 lc 0 0 6 12 18 24 30 36 Harvest Interval, wk Figure 3. Response of standardized biomass yield ( Y,) and standardized plant nitrogen uptake (N,,*) to harvest interval (At) at N= 672 kg ha'1 for coastal bermudagrass grown at Tifton, GA. Lines drawn from Eqs. (10) and (13). A Simplified Crop Growth Model with a correlation coefficient of r = 0.9929. This result leads to the linerexponential equation D, = 1.93 +1.29At exp 0.077 At (36) It follows immediately that digestible fraction,fd, is described by D, 9.93+1.29At fd t (37) Y, 8.91 + 3.49At Now Eq. (37) can be used to estimate digestibility of the two plant components. In the limit as At 0, the digestible fraction of the lightgathering (leaf) fraction, fd, is estimated from 9.93 fd 1.11 (38) 8.91 Of course the lightgathering component should not exceed 1.00. The value of 1.11 represents uncertainty in the intercept values in Eq. (37). In fact, it can be shown at the 95% confidence level that a, = 8.91 1.41 Mg ha1 and that ad = 9.93 + 2.52 Mg ha1. Since these values clearly overlap, it seems reasonable to assume that the value of the lightgathering component is approximately 1.00. fdL 1.00 (39) In the limit as At > very large the digestible fraction of the structural (stem) fraction, fds, is estimated from 1.29 fdL = 0.37 (40) 3.49 Now we see that as At increases, the plants shift from dominance by lightgathering to structural component of the plant with a corresponding decline in forage quality. It can be shown from calculus that harvest intervals to achieve maximum total plant biomass and maximum digestible biomass can be estimated from, respectively, 1 a 1 8.91 AtP 10.4 wk (41) y Py 0.077 3.49 1 a 1 9.93 At pd a 5.3 wk (42) 'y pd 0.077 1.29 Overman and Scholtz A Simplified Crop Growth Model A Simplified Theory of Biomass Production by Photosynthesis Allen R. Overman and Richard V. Scholtz III Introduction Photosynthesis is the biochemical processes by which green plants use incident radiant energy to fix C02 from the atmosphere and H from the splitting of the water molecule to form CHO the major content of plant biomass. Mineral elements (such as N, P, K, Ca, Mg, etc) are derived from the rhizosphere (root zone). Readers interested in details of photosynthesis at the molecular and cellular levels are referred to the excellent book by Oliver Morton [1]. The details are extremely complicated. In this article we seek to make simplifying assumptions which lead to a field scale theory relating the rate of accumulation of plant biomass with calendar time, dY/dt, (Mg ha1 wk') to calendar time, t, (wk). A broader view of progress of science in the 20th century, including the role of physics in photosynthesis, is provided by Gerard Piel [2]. In his classic book The Ascent ofMan, Jacob Bronowski [3] traces human history, including cultural and biological evolution. He documents the development of the agricultural revolution from hunter/gatherer to farmer/husbandman that has made modern agriculture possible and which produces the food and fiber upon which humanity now depends. Along with the expansion of technology, agricultural field research has experienced rapid development, beginning with the famous work at Rothamsted, England in about 1850. Today a very large database exists from various locations around the world. It is this database which we will draw upon as the empirical foundation for the present theory. In order to develop a rigorous theory of biomass production by photosynthesis, a sound mathematical framework is required. For this purpose we draw upon two fundamental principles of science as stated by Davies and Gribbin [4, p. 44]: Principle #1: It is possible to know uill,, hing of how nature works without knowing everything about how nature works. Without this principle there would be no science and no understanding. Principle #2: In physics a linear system is one in which a collection of causes leads to a corresponding collection of effects. For a given system it can be shown that this correspondence is unique, and the principle works in both directions. In the sequel this will be referred as the correspondence principle. Since the invention of the calculus by Isaac Newton, it has been common practice to make simplifying assumptions which lead to linear differential equations (ordinary or partial) with analytic solutions. Examples include mechanics (equations of motion), thermodynamics (diffusion of thermal energy), chemical dynamics (diffusion of molecular species), electrical phenomenon, magnetic phenomenon, and even to quantum mechanics (both matrix mechanics and wave mechanics). A similar strategy is followed in the present work. Simplifying assumptions are made in the search to develop a mathematical theory between the time rate of accumulation of biomass, dY/dt, and calendar time, t. The analysis is focused on the northern hemisphere where the greatest collection of field data has been reported. In addition the analysis focuses on field studies with the warmseason perennial coastal bermudagrass [Cynodon dactylon (L.) Pers.]. Numerous studies have been conducted for fixed harvest interval, At. Measurements of biomass accumulation as related to harvest interval have been reported, from which deductions can be Overman and Scholtz A Simplified Crop Growth Model Application of the Theory to Forage Quality Forage quality is measured primarily by two characteristics: nitrogen concentration (protein content) and digestibility of the biomass by ruminant animals. For perennial grasses quality relates to length of time between harvests since age of plants influences the balance between plant components (leaves vs. stems). In this section data from two studies with perennial grasses are used to evaluate forage quality. Study 1 i/th Coastal Bermudagrass Equation (15) can be used to estimate plant nitrogen concentration of the two plant components. In the limit as At > 0, nitrogen concentration of the lightgathering (leaf) component, NcL, is estimated from 422 N =22= 47.4 g kg' (31) 8.91 In the limit as At very large nitrogen concentration of the structural (stem) component, Ncs, is estimated from 33.0 Nc 33.= 9.5 g kg (32) 3.49 Clearly the nitrogen (crude protein) quality of the leaf fraction is considerably higher than of the stem fraction. The study by Burton et al. [11] at Tifton, GA included data on digestible dry matter as measured by the in vitro method [14]. Seasonal total digestible biomass (Dt) along with total biomass (Y,) as related to harvest interval (At) are listed in Table 2. Response of D, vs. At is assumed to follow a linearexponential function (similar to Eq. (8)) D, = + dAt exp yAt (33) where a /d,, and y are to be evaluated from the data points. Standardized digestible dry matter (D,*) is defined by D,* = D, exp 4 YAt= a, + /, At (34) Again we choose y = 0.077 wk 1 with the corresponding values in column 5 of Table 2. Linear regression of D,* vs. At then leads to D,* = D, exp 4 0.077 At = ad + ,d At = 9.93 + 1.29At r = 0.9929 (35) Overman and Scholtz A Simplified Crop Growth Model Table 3. Response of seasonal total biomass yield (Yt), seasonal total plant nitrogen uptake (Nt), plant nitrogen concentration (Nc), seasonal digestible biomass (Dt), digestible fraction (fd), standardized total biomass yield (Y, ), standardized total plant nitrogen uptake (N,,*), and standardized total digestible biomass (D,*) to harvest interval (At) for perennial peanut at Gainesville, FL.1 At Y, N,, Nc Dt fd Y, Nut D,* wk Mg ha kg ha1 g kg' Mg ha1 Mg ha1 kg ha1 Mg ha1 2 8.0 280 35.0 5.7 0.71 9.3 327 6.6 4 9.0 300 33.3 6.3 0.70 12.2 408 8.6 6 11.4 310 27.2 7.4 0.65 18.1 492 11.7 8 12.4 300 24.2 7.6 0.62 23.0 555 14.1 10 11.6 270 23.3 6.5 0.56 25.1 583 14.0 12 12.0 270 22.5 6.7 0.56 30.2 680 16.9 1 Data adapted from Beltranena et al. [14]. Table 4. Summary of Parameter Values. Parameter Definition Value Units u Time to mean of 26.0 wk solar energy distribution1 ,2o Time spread of 8.00 wk solar energy distribution k Partition constant 5 none c Aging coefficient 0.150 wk1 A Yield factor varies Mg ha1 'For northern hemisphere, referenced to Jan. 1. Overman and Scholtz A Simplified Crop Growth Model Table 1. Response of seasonal total biomass yield (Yt), seasonal total plant nitrogen uptake (Nt), and plant nitrogen concentration (Nc) to harvest interval (At) at N= 672 kg ha1 for coastal bermudagrass at Tifton, GA.' At Yt N, No Y, N, wk Mg ha1 kg ha1 g kg' Mg ha1 kg ha1 3 15.2 438 28.8 19.1 552 4 16.2 415 25.6 22.0 565 5 17.8 417 23.4 26.2 613 6 19.9 411 20.6 31.6 652 8 19.9 340 17.1 36.8 630 12 20.1 289 14.4 50.6 728 24 14.6 198 13.6 92.7 1257 1 Data adapted from Burton et al. [11]. Table 2. Response of seasonal total biomass yield (Yt), seasonal digestible biomass (Dt), and digestible fraction (fd) to harvest interval (At) at N= 672 kg ha1 for coastal bermudagrass at Tifton, GA.' At Yt Dt fd D, wk Mg ha1 Mg ha1 Mg ha1 3 15.2 9.91 0.652 12.5 4 16.2 10.3 0.637 14.0 5 17.8  6 19.9 11.9 0.597 18.9 8 19.9 11.3 0.566 20.9 12 20.1 10.6 0.525 26.7 24 14.6 6.31 0.432 40.0 1 Data adapted from Burton et al. [11]. Overman and Scholtz A Simplified Crop Growth Model with a correlation coefficient of r = 0.9780. This result is shown in Figure 3, where the line is drawn from Eq. (13). It follows that the middle curve in Figure 2 is drawn from N,, = 422 + 33.0At exp 0.077 At (14) Plant nitrogen concentration N, is then defined by N,, a, + aPnAt 422 +33.0 At N = (15) Y, a, + fyAt 8.91+3.49At The upper curve in Figure 2 is drawn from Eq. (15). The linearexponential function appears to describe the response curves rather well. The question now occurs as to the meaning of the exponential term in Eq. (8). Assuming that the system follows linear behavior, we invoke the correspondence principle to write an aging function, A(t, exp[c( t, (16) It should be noted that the 'aging function' is not derived from biochemical or genetic considerations, but represents an operational definition of a decline in capacity of the system to generate new biomass as the plants age. Equation (16) resets for each new value of reference time t,. Linear Differential Equation At this point three independent functions have been identified for the system: a linear partition function P( t _, an aging function A ( t, and an energy driving function E (. The question now is how to combine these functions to form a linear differential equation? Since the functions are considered independent, it seems reasonable to invoke the principle of joint probability and write the equation in product form dY = constant P(t A(t tE( dt 2 _(17) = constant* I +b(t t exp c texp (17) The procedure of joint probability is the same as that used by James Clerk Maxwell in developing the velocity distribution law in the kinetic theory of gases [13, p. 159]. He first writes the function for one dimension in space. Then the treatment is extended to three dimensions by invoking the principle of joint probability and writing the overall function as the product of the three separate functions. Overman and Scholtz PAGE 1 A MEMOIR ON A Simplified Theory of Biomass Production by Photosynthesis Allen R. Overman and Richard V. Scholtz III Agricultural and Biological Engineering University of Florida Copyright 2010 Allen R. Overman PAGE 2 Overman and Scholtz A Simplified Crop Growth Model 1 Key words : Pl ant growth, mathematical model, photosynthesis This memoir is focused on a simplified theory of biomass production by photosynthesis. It describes accumulation of biomass with calendar time. The theory is structured on a rigorous mathematical framework a nd a sound empirical foundation using data from the literature. Particular focus in on the northern hemisphere where most field research has been conducted, and on the warm season perennial coastal bermudagrass for which an extensive database exists. Three primary factors have been identified in the model: (1) an energy driving function, (2) a partition function between light gathering (leaf) and structural (stem) plant components, and (3) an aging function. These functions are then combined to form a linea r differential equation. Integration leads to an analytical solution. A linear relationship is established between biomass production and a growth quantifier for a fixed harvest interval. The theory is further used to describe forage quality (nitrogen conc entration and digestible fraction) between leaves and stems of the plants. The theory can be applied to annuals (such as corn) and well as perennials. Crop response to various applied elements (such as nitrogen, phosphorus, potassium, calcium, and magnesiu m) can be described. The theory contains five parameters: two for the Gaussian energy function, two for the linear partition function, and one for the exponential aging function. Acknowledgement : The authors thank Amy G Buhler, Engineering Librarian, Ma rston Science Library, University of Florida, for assistance with preparation of this memoir. PAGE 3 Overman and Scholtz A Simplified Crop Growth Model 2 A Simplified Theory of Biomass Production by Photosynthesis Allen R. Overman and Richard V. Scholtz III Introduction Photosynthesis is the biochemical proces ses by which green plants use incident radiant energy to fix CO2 from the atmosphere and H from the splitting of the water molecule to form C HO the major content of plant biomass. Mineral elements (such as N, P, K, Ca, Mg, etc) are derived from the rhizosp here (root zone). Readers interested in details of photosynthesis at the molecular and cellular level s are referred to the excellent book by Oliver Morton [1]. The details are extremely complicated. In this article we seek to make simplifying assumptions w hich lead to a field scale theory relating the rate of accumulation of plant biomass with calendar time, dY/dt (Mg ha 1 wk 1 ) to calendar time, t (wk). A broader view of progress of science in the 20th century, including the role of physics in photosynth esis, is provided by Gerard Piel [2]. In his classic book The Ascent of Man Jac ob Bronowski [3] traces human history, including cultural and biological evolution. He documents the development of the agricultural revolution from hunter/gatherer to farmer/h usband man that has made modern agriculture possible and which produces the food and fiber upon which humanity now depends. Along with the expansion of technology, agricultural field research has experienced rapid development, beginning with the famous work at Rothamsted, England i n about 1850. Today a very large database exists from various locations around the world. It is this database which we will draw upon as the empirical foundation for the present theory. In order to develop a rigorous theory of bio mass production by photosynthe sis, a sound mathematical frame work is required. For this purpose we draw upon two fundamental principles of science as stated by Davies and Gribbin [4 p. 44 ]: Principle #1: It is possible to know something of how nature work s without knowing everything about how nature works. Without this principle there would be no science and no understanding. Principle #2: In physics a linear system is one in which a collection of causes leads to a corresponding collection of effects. For a given system it can be shown that this c orrespondence is unique, and the principle works in both directions. In the sequel this will be referred as the correspondence principle Since the invention of the calculus by Isa ac Newton, it has been common prac tice to make simplifying assumptions which lead to linear differential equations (ordi nary or partial) with analytic solutions. Examples include mechanics (equations of motion ), thermodynamics (diffusion of thermal energy), chemical dynamics (diffusion of molecular species), electrical phenomenon, magnetic phenomenon, and even to quantum mechanics (both matrix mechanics and wave mechanics). A similar strategy is followed in the present work. Sim plifying assumptions are made in the search to develop a math ematical theory between the time rate of accumulation of biomass, dY/dt and calendar time, t The analysis is focused on the northern hemi sphere where the greatest collection of f ield data has been reported In addition the analysis focuses on field studi es with the warm season perennial coastal bermu dagrass [ Cynodon dactylon (L.) Pers.] Numerous studies have been conducted for fixed harvest interval, Measurements of biomass accumulation as related to harvest interval have been rep orted, from which deductions can be PAGE 4 Overman and Scholtz A Simplified Crop Growth Model 3 made about effects. The correspondence principle can the n be used to make inference about the causes involved. This leads finally to a linear differential equation. Theory Development The first step in this process is to identify ke y components which contribute to biomass production with calendar time as measured by data from field studies. These factors are then combined into a linear differential equation. The differential equation is then integrated to an analytic s olution. Again data are for the warm season perennial coastal bermudagrass in the northern hemisphere and harvested on a fixed time interval. Energy Driving Function The first step along these lines was taken by Overman [5] in response to requests by envi ronmental regulators to estimate biomass and plant nutrient accumulation with calendar time for a water reclamation/reuse project in Florida. The analysis drew upon a field study at Watkinsville, GA with coastal bermudagrass harvested on a fixed time inter val [6]. The experiment consisted of a 2x2 factorial of two harvest interval s (4 wk, 6 wk) and two irrigation treatments (irrigated, nonirrigated). The distribution of biomass with calend ar time was shown to follow a G a u ssian function described by (1) where F is the fraction of total biomass at calendar time t (referenced to Jan. 1), is time to mean of biomass distribution (referenced to Jan. 1), and is the time spread of the biomass distribution. It was shown that the distributions were independent of irrigation treatment and harvest interval and followed the equation (2) with exponential values in wk. Details of the analysis are described in Overman and Scho ltz [7, Section 3.2] These results raised the interesting question as to the origin of the Gaussian distribution? It is known that incident solar radiation in the northern hemisphere rises from a minimum in January to a maximum in July and decreases again to a minimum in December. Overman and Scholt z [7, Table 1.6] analyzed solar radiation data for Rothamsted, England [8] and showed that the distribution followed the Gaussian distribution (3) From this analys is it seems logical to assume an energy driving function, E ( t ), which follows a Gaussian distribution to good approximation PAGE 5 Overman and Scholtz A Simplified Crop Growth Model 4 (4) where t is calendar time referenced to Jan. 1, is time to mean of the solar energy distribution refer enced to Jan. 1, and is the time spread of the solar energy distribution. All units are in weeks. Partition Function for Biomass The second step in the analysis is to identify an intrinsic growth function that identifies how plant s respond to the input of solar energy. Fortunately field experiments have been conducted with coastal bermudagrass at Tifton, GA by Prine and Burton [9]. The factorial experiment consisted of five nitrogen levels ( N = 0, 112, 336, 672, and 1008 kg ha 1 ), six harvest intervals ( = 1, 2, 3, 4, 6, and 8 wk), and two years (1953 and 1954). Total biomass yield ( Y t Mg ha 1 ) and total plant nitrogen uptake ( N ut kg ha 1 ) for the entire season (24 wk) were reported System response is illus trated in Figure 1 with biomass data for N = 672 kg ha 1 and for both years. As it ) and 1954 exhibited the worst drought in It appears from Figure 1 that the data points foll ow straight lines for The regression lines are described by 1953: r = 0.9950 (5) 1954: r = 0.9836 (6) with correlation coefficients of r = 0.9950 and 0. 9836 for 1953 and 1954, respectively. The effect of water availability is clearly evident in these equations. Equations (5) and (6) reveal an additional factor beyond the energy driving function identified above. From the correspondence principle for linea r systems, we are led to assume a partition function of the type (7) where t is calendar time for the growth interval, wk; t i is the reference time for the growth interval, wk; and a is the intercept parameter, Mg ha 1 wk 1 and b is the slope parameter, Mg ha 1 wk 2 Overman and Wilkinson [10] examined data from the literature on partitioning of dry matter between leaf and stem for coastal bermudagrass and concluded that the first term in Eq. (7) corresponds to the rate of growth of the light gathering component (leaf) while the second term corresponds to the rate of growth of the s tructural component (stem). Equation (7) means that reference time t i is reset for each growth interval. It follows logically that Eq. (7) must have a limited time domain of application. Otherwise artificial radiant energy could be supplied to the plants to provide unlimited biomass accumulation. This simply does not happen! It follows that there must be an addition al limiting factor in the system. PAGE 6 Overman and Scholtz A Simplified Crop Growth Model 5 Agi ng Function for Biomass The third step in the analysis is to identify an addition al factor which imposes a further limit on plant growth. Burton et al. [11] expanded the range of harvest intervals to include = 3, 4, 5, 6, 8, 12, and 24 wk. Applied nitrogen was N = 672 kg ha 1 Seasonal total biomass yield ( Y t ) and seasonal total plant nitrogen uptake ( N ut ) were reported as given in Table 1. Plant nitrogen concentration is defined as N c = N ut /Y t In order to see the pattern in the re sponse more clearly, results are also shown in Figure 2. The graph of Y t vs. exhibits a rise, passing through a peak, then following a steady decline. This seems like the place for the time honored method of intuition as defined by Roger Newton [12, p. 60]. Expanding on the linear partition function from the previous section, we assume a linear exponential function (8) where are to be evaluated from the data points. It remains to be d etermined if Eq. (8) exhibits the correct characteristics to describe the response curve. In other words, an operational procedure is required to evaluate the coefficients of Eq. (8). Now if the value of were known, then we could de fine a standardized yield as (9) which leads to a linear equation in The procedure is to try values of which leads to the optimum correlation for Eq. (9). T his process leads to an estimate of with the corresponding values in column 5 of Table 1. Linear regression of vs. then leads to r = 0.9995 (10) wit h a correlation coefficient of r = 0.9995. This result is shown in Figure 3, where the line is drawn from Eq. (10). It follows that the lower curve in Figure 2 is drawn from (11) The next step is to define standardized plant nitr ogen uptake as (12) With as for biomass response, this leads to values in column 6 of Table 1. Linear regression of vs. then leads to r = 0.9780 (13) PAGE 7 Overman and Scholtz A Simplified Crop Growth Model 6 with a correlation coefficient of r = 0.9780. This result is shown in Figure 3, where the line is drawn from Eq. (13). It follows that the middle curve in Figure 2 is drawn from (14) Plant nitrogen concentration N c is then defined by (15) The upper curve in Figure 2 is drawn from Eq. (15). The linear exponential function appears to describe the response curves rather well. The question now occurs as to the meaning of the exponential term in Eq. (8). Assuming that the system follows linear behavior, we invoke the correspondence principle to write an aging function, (16) om biochemical or genetic co nsiderations, but represents an operational definition of a decline in capacity of the system to generate new biomass as the plants age. Equation (16) resets for each new value of reference time t i Linear Differential Equatio n At this point three independent functions have been identified for the system: a linear partition function an aging function and an energy driving function The question now is ho w to combine these functions to form a linear differential equation? Since the functions are considered independent, it seems reasonable to invoke the principle of joint probability and write the equation in product form (17) The procedure of joint probability is the same as that used by James Clerk Maxwell in developing the velocity distribution law in the kinetic theory of gases [13, p. 159]. He first writes the function for one dimension in space. Then the treatment is extended to three dimensions by invoking the principle of joint probability and writing the overall function as the product of the three separate functions. PAGE 8 Overman and Scholtz A Simplified Crop Growth Model 7 Equation (17) forms a linear first order differential equation in the time variable t However, it contain s two reference times: t i and Since the partition function and the aging function reset for each growth interval, Equation (17) must be viewed as piecewise continuous in the time interval and integration mus t proceed accordingly. The interested reader is referred to Overman and Scholtz [7, Section 4.3] for details of the integration process. The solution for the increase in biomass accumulation for the i th growth interval, becomes the simple linear relationship (18) where A is the yield factor, Mg ha 1 and is the growth quantifier defined by (19) The dimensionless partition coefficient k in Eq. (19) is define d by (20) and the dimensionless time variable x is defined by (21) It follows immediately that x i is defined by (22) The error function, erf x in Eq. (19) is defined by (23) where u is the variable of integration for the Gaussian function The cumulative sum of biomass for n harvests Y n is given by (24) PAGE 9 Overman and Scholtz A Simplified Crop Growth Model 8 Total biomass yield for the entire season, Y t i s the sum over all harvests and is related to total growth quantifier for the season, Q t by (25) A Mathematical Theorem A mathematical theorem is now presented which provides a rigorous connection between Eq. (17) for the linear first ord er differential equation for each growth interval and Eq. (8) for linear exponential dependence of seasonal total biomass yield, Y t on a fixe d harvest interval, Details of the proof are given by Overman and Scholtz [7, S ection 4.3]. The theorem establishes that the cumulative sum of biomass production for n harvests is given by (26) where is defined by (27) In Eq. (26) the term in curly brackets c onfirms the Gaussian distribution of the energy driving function, which approaches 1 in the limit of large t This means that total biomass yield for the season, Y t is given by (28) where seasonal total growth quantifier, is defined by (29) Finally, Eq. (28) can be written in regression form (30) This completes the proof of the theorem. It should be noted that is required. The the orem app lies for fixed harvest interval wk. PAGE 10 Overman and Scholtz A Simplified Crop Growth Model 9 Application of the Theory to Forage Quality Forage quality is measure d primarily by two characteristics: nitrogen concentration (protein content) and digestibility of the biomass by ru minant animals. For perennial grasses quality relates to length of time between harvests since age of plants influences the balance between plant components (leaves vs. stems). In this section data from two studies with perennial grasses are used to evalua te forage quality. Study with Coastal Bermudagrass Equation (15) can be used to estimate plant nitrogen concentration of the two plant components. In the limit as nitrogen concentration of the light gathering (leaf) component N cL is estimated from g kg 1 (31) In the limit as nitroge n concentration of the structural (stem ) component, N cS is estimated from g kg 1 (32) Clearly the nitrogen (crude prote in) quality of the leaf fraction is considerably higher than of the stem fraction. The study by Burton et al. [11] at Tifton, GA included data on digestible dry matter as measured by the in vitro method [14] Seasonal total digestible biomass ( D t ) along w ith total biomass ( Y t ) as related to harvest interval ( ) are listed in Table 2. Response of D t vs. is assume d to follow a linear exponential function (similar to Eq. (8)) (33 ) where are to be evaluated from the data points. Standardized digestible dry matter ( ) is defined by (34 ) Again we choose with the corresponding values in column 5 of Table 2. Linear regression of vs. then leads to r = 0.9929 (35 ) PAGE 11 Overman and Scholtz A Simplified Crop Growth Model 10 with a correlation coefficient of r = 0.9929. This result leads to the liner exponential equation (36 ) It follows immediately that digestible fraction f d is described by (37 ) Now Eq. (37 ) can be used to estimate digestibility of the two plant components. In the limit as the digestible fraction of the light gathering (leaf) fraction, is estimated from (38) Of course the light gathering component should not exceed 1.00. The value of 1.11 represents uncertainty in the intercept values in Eq. (37 ). In fact, it can be shown at the 95% confidence level that Mg ha 1 and that Mg ha 1 Since these values clearly overlap, it seems reasonable to assume that the value of the light gathering component is approximately 1.00. (39) In the limit as the digestible fraction of the structural (stem ) fraction, is estimated from (40 ) Now we see that as increases, the plants shift from dominance by light gathering to structural component of the plant with a correspon ding decline in forage quality. It can be shown from calculus that harvest interval s to achieve maximum total plant biomass and maximum digestible biomass can be estimated from, respectively wk (41 ) wk (42 ) PAGE 12 Overman and Scholtz A Simplified Crop Growth Model 11 Study with Perennial Peanut A study was co nducted by Beltranena et al. [15 ] with the warm season legume perennial pean ut ( Arachis glabrata on Arredondo loamy fine sand (loamy, siliceous, semiactive, hyperthermic Grossarenic Paleudult). Plants were sampled on fixed harvest intervals of = 2, 4, 6, 8, 10, and 12 wk. The growing s eason is considered to be 24 wk. Measurements were made of seasonal total biomass ( Y t ) seasonal total plant nitrogen uptake ( N ut ), and seasonal total digestible biomass ( D t ). Results are listed in Table 3. All data are for a clipping height of 3.8 cm. As with the case of coastal bermudagrass, the linear exponential model is assumed to apply. Standardized biomass yield ( ) standardized plant nitrogen uptake ( ), and standardized digestible biomass yield ( ) are calculated from r = 0.9938 (43 ) r = 0.9916 (44 ) r = 0.9791 (45 ) Equation s (43) through (45 ) lead to the linear exponential equations (46 ) (47 ) Equations (46) and (47 ) lead immediately to (48 ) Equation (48 ) can be used to estimate nitrogen concentration of each plant component. Nitrogen concentration of the leaf fract ion, N cL is estimated from g kg 1 (49 ) whereas nitrogen concentration of the stem fraction, N cS is estimated from g kg 1 (50 ) Again, the leaf fr action contains higher nitrogen concentration than does the stem fraction. PAGE 13 Overman and Scholtz A Simplified Crop Growth Model 12 Dependence of seasonal total digestible biomass is described by the linear exponential equation (51 ) Equations (46) and (51) can be combined to obtain dependence of digestible fraction f d on harvest interval (52) Digestible fraction for the two plant components leaves and stems, f dL and f dS can be estimated from (53) (54) Again the le af fraction exhibits high er digestibility than the stem fraction. It can be shown from calculus that the maximum value (peak) of the linear exponential curves corresponds to a peak harvest interval, for total biomass, total plant nitrogen uptake, and total digestible biomass given by wk (55 ) wk (56) wk (57 ) Summary and Conclusions The simplified theory of biomass production by photosynthesis is described by Eqs (18) through (23). Since Eq (19) contains the error function, erf x this seems like the appropriate place to present the detailed discussion by Abramowitz and Stegun [16, chp 7], including a table of values. A few key characteristics should be noted: f (0) = 0, f ( ) = 1 PAGE 14 Overman and Scholtz A Simplified Crop Growth Model 13 f ( ) = f ( ), f ( f ( The theory contains five parameters: two for the energy driving function ( ) and three for plant characteristics ( k, c, A ). Examination of data for the northern hemisphere and for the warm season perennial coastal bermudagrass lead to the estimates listed in Table 4. PAGE 15 Overman and Scholtz A Simplified Crop Growth Model 14 References 1. Morton O (2007) Eating the Sun: How Plants Power the Planet. London: Harper Collins. 475 p. 2. Piel G (20 01) The Age of Science: What Scientists Learned in the Twentieth Century. New York: Basic Books. 459 p. 3. Bron owski J (1973 ) The Ascent of Man. Boston: Little, Brown & Co. 448 p. 4. Davies P and Gribbin J (1992) The Matter Myth: Dramatic Discoveries That Have Challenged Our Understanding of Physical Reality. New York: Simon & Schuster. 320 p. 5. Overman AR (1984) Estimating crop growth rate with land treatment. J. Env. Eng. Div., American Society of Civil Engineers 110: 1009 1012. 6. Mays DA, Wilkinson SR, and Cole CV (1980) Phosphorus nutrition of forages. In: The Role of Phosphorus in Agriculture. Khasawneh FE and Kamprath EJ (eds). Madison, WI: American Society of Agronomy. pp. 805 840. 7. Overman AR and Scholtz RV (2002) Mathematical Models of Crop Growt h and Yield. New York: Taylor & Francis. 328 p. 8. Russell EJ (1950) Soil Conditions and Plant Growth 8th ed. London : Longmans, Green & Co. 635 p. 9. Prine GM and Burton GW (1956) The effect of nitrogen rate and clipping frequency upon the yield, protein c ontent, and certain morphological characteristics of coastal bermudagrass [ Cynodon dactylon ( L .) Pers.]. Agronomy J. 48:296 301. 10. Overman AR and Wilkinson SR (1989) Partitioning of dry matter between leaf and stem in coastal bermudagrass. Agricultural S ystems 30:35 47. 11. Burton GW, Jackson JE, and Hart RH (1963) Effects of cutting frequency and nitrogen on yield, in vitro digestibility, and protein, fiber and carotene content of coastal bermudagrass. Agronomy J. 55:500 502. 12. Newton, R (1997) The Tru th of Science: Physical Theories and Reality. Cambridge, MA: Harvard University Press. 260 p. 13. Longair MS (1984) Theoretical Concepts in Physics. New York: Cambridge University Press. 366 p. 14. Moor e JE and Dunham GD (1971) Proc edure for the two stage in vitro organic matter digestion of forages (revised). Nutrition Laboratory, Department of Animal Science. University of Florida. Gainesville, FL 10 p. 15 Belt ranena R, Breman J and Prine GM (1981) Yield and quality of Florigraze rhizome peanut ( Arachis glabrata Bent.) as affected by cutting height and frequency. Proc. Soil and Crop Science Society of Florida 40:153 156. 16. Abramowitz M and Stegun IA (1965) Handbook of Mathematical Functions. New York: Dover Publications. 1046 p. PAGE 16 Overman and Scholtz A Simplified Crop Growth Model 15 Table 1. Response of seasonal total biomass yield ( Y t ), seasonal total plant nitrogen uptake ( N ut ) and plant nitrogen concentration ( N c ) to harvest interval ( ) at N = 672 kg ha 1 for coastal bermudagrass at Tifton, GA. 1 Y t N ut N c w k Mg ha 1 kg ha 1 g kg 1 Mg ha 1 kg ha 1 3 15.2 438 28.8 19.1 552 4 16.2 415 25.6 22.0 565 5 17.8 41 7 23.4 26.2 613 6 19.9 411 20.6 31.6 652 8 19.9 340 17.1 36.8 630 12 20.1 289 14.4 50.6 728 24 14.6 198 13.6 92.7 1257 1 Data adapted from Burton et al. [11]. Table 2. Response of seasonal total biomass yield ( Y t ), seasonal digestible biomass ( D t ), and digestible fraction ( f d ) to harvest interval ( ) at N = 672 kg ha 1 for coastal bermudagrass at Tifton, GA. 1 Y t D t f d wk Mg ha 1 Mg ha 1 Mg ha 1 3 15.2 9.91 0.652 12.5 4 16.2 10.3 0.637 14.0 5 17.8 6 19.9 11.9 0.597 18.9 8 19.9 11.3 0.566 20.9 12 20.1 10.6 0.525 26.7 24 14.6 6.31 0.432 40.0 1 Data adapted from Burton et al. [11]. PAGE 17 Overman and Scholtz A Simplified Crop Growth Model 16 Table 3. Response of seasonal total biomas s yield ( Y t ), seasonal total plant nitrogen uptake ( N ut ), plant nitrogen concentration ( N c ), seasonal digestible biomass ( D t ), digestible fraction ( f d ), standardized total biomass yield ( ), standardized total plant nitrogen uptake ( ), and standardized total digestible biomass ( ) to harvest interval ( ) for perennial peanut at Gainesville, FL 1 Y t N c D t f d wk Mg ha 1 kg ha 1 g kg 1 Mg ha 1 Mg ha 1 kg ha 1 Mg ha 1 2 8.0 280 35.0 5.7 0.71 9.3 327 6.6 4 9.0 300 33.3 6.3 0.70 12.2 408 8.6 6 11.4 310 27.2 7.4 0.65 18.1 492 11.7 8 12.4 30 0 24.2 7.6 0.62 23.0 555 14.1 10 11.6 270 23.3 6.5 0.56 25.1 583 14.0 12 12.0 270 22.5 6.7 0.56 30.2 680 16.9 1 Data adapted from Beltranena et al. [14]. Table 4. Summary of Parameter Values. Parameter Definition Value Units Time to mean of 26.0 wk solar energy distribution 1 Time spread of 8.00 wk solar energy distribution k Partition constant 5 none c Aging coefficient 0.150 wk 1 A Yield factor varies Mg ha 1 1 For northern h emisphere, referenced to Jan. 1. PAGE 18 Overman and Scholtz A Simplified Crop Growth Model 17 Figure 1. Response of total biomass yield ( Y t ) to harvest interval ( ) at N = 672 kg ha 1 for coastal bermudagrass grown at Tifton, GA. Data adapted from Prine and Burton [9]. Lines drawn fr om Eqs. (5) and (6). PAGE 19 Overman and Scholtz A Simplified Crop Growth Model 18 Figure 2. Response of total biomass yield ( Y t ), total plant nitrogen ( N ut ), and plant nitrogen concentration ( N c ) to harvest interval ( ) at N = 672 kg ha 1 for coastal bermudagrass grown at Tifton, GA. Data adapted from Burton et al. [11]. Curves drawn from Eqs. (11), (14), and (15). PAGE 20 Overman and Scholtz A Simplified Crop Growth Model 19 Figure 3. Response of standardized biomass yield ( ) and standardized plant nitrogen uptake ( ) to harvest interval ( ) at N = 672 kg ha 1 for coastal bermudagrass grown at Tifton, GA. Lines drawn from Eqs. (10) and (13). A Simplified Crop Growth Model 0 II E). 0 6 12 18 24 Harvest Interval, wks Figure 2. Response of total biomass yield (Y,), total plant nitrogen (Nt), and plant nitrogen concentration (Nc) to harvest interval (At) at N= 672 kg ha1 for coastal bermudagrass grown at Tifton, GA. Data adapted from Burton et al. [11]. Curves drawn from Eqs. (11), (14), and (15). " 400 (U c200 z Overman and Scholtz Overman and Scholtz A Simplified Crop Growth Model f(x)=fx(+x), f(o) = f(+0) 1 The theory contains five parameters: two for the energy driving function (p/, F2cr) and three for plant characteristics (k, c, A). Examination of data for the northern hemisphere and for the warmseason perennial coastal bermudagrass lead to the estimates listed in Table 4. A Simplified Crop Growth Model made about effects. The correspondence principle can then be used to make inference about the causes involved. This leads finally to a linear differential equation. Theory Development The first step in this process is to identify key components which contribute to biomass production with calendar time as measured by data from field studies. These factors are then combined into a linear differential equation. The differential equation is then integrated to an analytic solution. Again data are for the warmseason perennial coastal bermudagrass in the northern hemisphere and harvested on a fixed time interval. Energy Driving Function The first step along these lines was taken by Overman [5] in response to requests by environmental regulators to estimate biomass and plant nutrient accumulation with calendar time for a water reclamation/reuse project in Florida. The analysis drew upon a field study at Watkinsville, GA with coastal bermudagrass harvested on a fixed time interval [6]. The experiment consisted of a 2x2 factorial of two harvest intervals (4 wk, 6 wk) and two irrigation treatments (irrigated, nonirrigated). The distribution of biomass with calendar time was shown to follow a Gaussian function described by F=exp 22 (1) where F is the fraction of total biomass at calendar time t (referenced to Jan. 1), p is time to mean of biomass distribution (referenced to Jan. 1), and v2o is the time spread of the biomass distribution. It was shown that the distributions were independent of irrigation treatment and harvest interval and followed the equation F = exp t 27.8 )J (2) 8.13 with exponential values in wk. Details of the analysis are described in Overman and Scholtz [7, Section 3.2] These results raised the interesting question as to the origin of the Gaussian distribution? It is known that incident solar radiation in the northern hemisphere rises from a minimum in January to a maximum in July and decreases again to a minimum in December. Overman and Scholtz [7, Table 1.6] analyzed solar radiation data for Rothamsted, England [8] and showed that the distribution followed the Gaussian distribution F =exp t 25.0 ) (3) 14.8 From this analysis it seems logical to assume an energy driving function, E(t), which follows a Gaussian distribution to good approximation Overman and Scholtz A Simplified Crop Growth Model Key words: Plant growth, mathematical model, photosynthesis This memoir is focused on a simplified theory of biomass production by photosynthesis. It describes accumulation of biomass with calendar time. The theory is structured on a rigorous mathematical framework and a sound empirical foundation using data from the literature. Particular focus in on the northern hemisphere where most field research has been conducted, and on the warmseason perennial coastal bermudagrass for which an extensive database exists. Three primary factors have been identified in the model: (1) an energy driving function, (2) a partition function between lightgathering (leaf) and structural (stem) plant components, and (3) an aging function. These functions are then combined to form a linear differential equation. Integration leads to an analytical solution. A linear relationship is established between biomass production and a growth quantifier for a fixed harvest interval. The theory is further used to describe forage quality (nitrogen concentration and digestible fraction) between leaves and stems of the plants. The theory can be applied to annuals (such as corn) and well as perennials. Crop response to various applied elements (such as nitrogen, phosphorus, potassium, calcium, and magnesium) can be described. The theory contains five parameters: two for the Gaussian energy function, two for the linear partition function, and one for the exponential aging function. Acknowledgement: The authors thank Amy G Buhler, Engineering Librarian, Marston Science Library, University of Florida, for assistance with preparation of this memoir. Overman and Scholtz A Simplified Crop Growth Model Equation (17) forms a linear first order differential equation in the time variable t. However, it contains two reference times: t, and p. Since the partition function and the aging function reset for each growth interval, Equation (17) must be viewed as piecewise continuous in the time interval t t,, and integration must proceed accordingly. The interested reader is referred to Overman and Scholtz [7, Section 4.3] for details of the integration process. The solution for the increase in biomass accumulation for the i th growth interval, AY, becomes the simple linear relationship AY, = AAQ, (18) where A is the yield factor, Mg ha' and AQ, is the gi ,n ih quantifier defined by AQ ( kx, rf x erf x, _ k x ( x2 exp ( x2 exp 2(cx (19) The dimensionless partition coefficient, k, in Eq. (19) is defined by k = 2r b/a (20) and the dimensionless time variable x is defined by t p 2crc x= + 2 (21) V2o 2 It follows immediately that x, is defined by x +' P 0C (22) The error function, erfx, in Eq. (19) is defined by 2 x erf x = ,exp (u2 du (23) where u is the variable of integration for the Gaussian function exp (u2 The cumulative sum of biomass for n harvests, Y,, is given by Y, = AY, A AQ, = AQ, (24) 1=1 1=1 Overman and Scholtz A Simplified Crop Growth Model 50 40  S 20  _o 0 10 1954 0 I I II 0 2 4 6 8 10 Harvest Interval, wk Figure 1. Response of total biomass yield (Y,) to harvest interval (At) at N= 672 kg ha'1 for coastal bermudagrass grown at Tifton, GA. Data adapted from Prine and Burton [9]. Lines drawn from Eqs. (5) and (6). Overman and Scholtz 