A MEMOIR ONufdcimages.uflib.ufl.edu/AA/00/00/15/73/00001/A... · A broader view of progress of science in the 20th century, including the role of physics in photosynthesis, is provided

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

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 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 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.


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 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 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 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 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

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 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 warm-season perennial coastal bermudagrass [Cynodon dactylon (L.) Pers.].

Numerous studies have been conducted for fixed harvest interval, t . Measurements of biomass

accumulation as related to harvest interval have been reported, from which deductions can be


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 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 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 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

2

2exp

tF (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 independent of irrigation treatment and

harvest interval and followed the equation

2

13.8

8.27exp

tF (2)

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

2

8.14

0.25exp

tF (3)

From this analysis it seems logical to assume an energy driving function, E(t), which follows a

Gaussian distribution to good approximation


4

2

2exp

ttE (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 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 ha-1

), six harvest intervals

( t = 1, 2, 3, 4, 6, and 8 wk), and two years (1953 and 1954). Total biomass yield (Yt , Mg ha-1

)

and total plant nitrogen uptake (Nut, kg ha-1

) for the entire season (24 wk) were reported. System

response is illustrated in Figure 1 with biomass data for N = 672 kg ha-1

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

wk6t . The regression lines are described by

1953: ttYt 73.238.11 r = 0.9950 (5)

1954: ttYt 61.141.4 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 energy driving function identified above. From the correspondence

principle for linear systems, we are led to assume a partition function of the type

ii ttbattP (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 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 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 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.


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 the range of harvest intervals to include t = 3, 4, 5, 6,

8, 12, and 24 wk. Applied nitrogen was N = 672 kg ha-1

. 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 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

ttY yyt exp (8)

where and,, yy 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

were known, then we could define a standardized yield tY as

ttYY yytt 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 1wk077.0 with the

corresponding values in column 5 of Table 1. Linear regression of tY vs. t then leads to

tttYY yytt 49.391.8077.0exp 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

ttYt 077.0exp49.391.8 (11)

The next step is to define standardized plant nitrogen uptake utN as

ttNN nnutut exp (12)

With 1wk077.0 as for biomass response, this leads to values in column 6 of Table 1. Linear

regression of utN vs. t then leads to

tttNN nnutut 0.33422077.0exp r = 0.9780 (13)


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

ttNut 077.0exp0.33422 (14)

Plant nitrogen concentration Nc is then defined by

t

t

t

t

Y

NN

yy

nn

t

ut

c49.391.8

0.33422 (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,

]exp[ ii ttcttA (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 ti.

Linear Differential Equation

At this point three independent functions have been identified for the system: a linear partition

function ittP , an aging function ittA , and an energy driving function tE . 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

2

2expexpconstant

constant

tttcttba

tEttAttPdt

dY

ii

ii

(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.


7

Equation (17) forms a linear first order differential 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 itt , 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, iY becomes the simple linear

relationship

ii QAY (18)

where A is the yield factor, Mg ha-1

and iQ is the growth quantifier defined by

iiiii cxxxk

xxkxQ 2expexpexpπ

erferf1 22 (19)

The dimensionless partition coefficient, k, in Eq. (19) is defined by

abk /2 (20)

and the dimensionless time variable x is defined by

2

2

2

ctx (21)

It follows immediately that xi is defined by

2

2

2

ctx i

i (22)

The error function, erf x, in Eq. (19) is defined by

x

duux0

2expπ

2erf (23)

where u is the variable of integration for the Gaussian function 2exp u .

The cumulative sum of biomass for n harvests, Yn, is given by

n

i

ni

n

i

in QAQAYY11

(24)


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

tt QAY (25)

A Mathematical Theorem

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, t .

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

2erf1

2

1

2exp

22

tt

ct

kAYn

(26)

where A is defined by

2

π2constant aA (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

tt QAY (28)

where seasonal total growth quantifier, tQ is defined by

tc

tk

Qt2

exp2

2 (29)

Finally, Eq. (28) can be written in regression form

tttc

tAk

AY yyt exp2

exp2

2 (30)

This completes the proof of the theorem. It should be noted that 2/c is required. The

theorem applies for fixed harvest interval t , wk.


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 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 with Coastal Bermudagrass

Equation (15) can be used to estimate plant nitrogen concentration of the two plant components.

In the limit as 0t , nitrogen concentration of the light-gathering (leaf) component, NcL , is

estimated from

4.4791.8

422cLN g kg

-1 (31)

In the limit as largeveryt , nitrogen concentration of the structural (stem) component, NcS ,

is estimated from

5.949.3

0.33cSN g kg

-1 (32)

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 (Yt)

as related to harvest interval ( t ) are listed in Table 2. Response of Dt vs. t is assumed to

follow a linear-exponential function (similar to Eq. (8))

ttD ddt exp (33)

where and,, dd are to be evaluated from the data points. Standardized digestible dry matter

( tD ) is defined by

ttDD ddtt exp (34)

Again we choose 1wk077.0 with the corresponding values in column 5 of Table 2. Linear

regression of tD vs. t then leads to

tttDD ddtt 29.193.9077.0exp r = 0.9929 (35)


10

with a correlation coefficient of r = 0.9929. This result leads to the liner-exponential equation

ttDt 077.0exp29.193.9 (36)

It follows immediately that digestible fraction, fd, is described by

t

t

Y

Df

t

t

d49.391.8

29.193.9 (37)

Now Eq. (37) can be used to estimate digestibility of the two plant components. In the limit as

0t , the digestible fraction of the light-gathering (leaf) fraction, dLf , is estimated from

11.191.8

93.9dLf (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 41.191.8y Mg ha-1

and that 52.293.9d 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.

00.1dLf (39)

In the limit as largeveryt , the digestible fraction of the structural (stem) fraction, dSf , is

estimated from

37.049.3

29.1dLf (40)

Now we see that as t increases, the plants shift from dominance by light-gathering 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,

4.1049.3

91.8

077.0

11

y

y

pyt wk (41)

3.529.1

93.9

077.0

11

d

d

pdt wk (42)


11

Study with Perennial Peanut

A study was conducted by Beltranena et al. [15] with the warm-season 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 t = 2, 4, 6, 8, 10, and 12 wk. The growing season 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 bermudagrass, the linear-exponential model is assumed to

apply. Standardized biomass yield (tY ), standardized plant nitrogen uptake ( utN ), and

standardized digestible biomass yield ( tD ) are calculated from

tttYY yytt 12.284.4077.0exp r = 0.9938 (43)

tttNN nnutut 6.33272077.0exp r = 0.9916 (44)

tttDD ddtt 00.197.4077.0exp r = 0.9791 (45)

Equations (43) through (45) lead to the linear-exponential equations

ttttY yyt 077.0exp12.284.4exp (46)

tttN nnut 077.0exp6.33272exp (47)

Equations (46) and (47) lead immediately to

t

t

Y

NN

t

ut

c12.284.4

6.33272 (48)

Equation (48) can be used to estimate nitrogen concentration of each plant component. Nitrogen

concentration of the leaf fraction, NcL , is estimated from

2.5684.4

272cLN g kg

-1 (49)

whereas nitrogen concentration of the stem fraction, NcS , is estimated from

8.1512.2

6.33cSN g kg

-1 (50)

Again, the leaf fraction contains higher nitrogen concentration than does the stem fraction.


12

Dependence of seasonal total digestible biomass is described by the linear-exponential equation

ttttD ddt 077.0exp00.197.4exp (51)

Equations (46) and (51) can be combined to obtain dependence of digestible fraction fd on

harvest interval t

t

t

Y

Df

t

t

d12.284.4

00.197.4 (52)

Digestible fraction for the two plant components leaves and stems, fdL and fdS , can be estimated

from

00.103.184.4

97.4dLf (53)

47.012.2

00.1dSf (54)

Again the leaf fraction exhibits higher 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, pt , for total biomass, total plant nitrogen uptake, and

total digestible biomass given by

7.1012.2

84.4

077.0

11

y

y

pyt wk (55)

9.46.33

272

077.0

11

n

n

pnt wk (56)

0.800.1

97.4

077.0

11

d

d

pdt 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


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

warm-season perennial coastal bermudagrass lead to the estimates listed in Table 4.


14

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 110:1009-1012.

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 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. 48:296-301.

10. Overman AR and Wilkinson SR (1989) Partitioning of dry matter between leaf and stem in

coastal bermudagrass. Agricultural Systems 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 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 two-stage 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 40:153-156.

16. Abramowitz M and Stegun IA (1965) Handbook of Mathematical Functions. New York:

Dover Publications. 1046 p.


15

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 ha-1

for coastal

bermudagrass at Tifton, GA.1

t Yt Nut Nc tY utN

wk 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 17.8 417 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 ha-1

for coastal bermudagrass at

Tifton, GA.1

t Yt Dt fd tD

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].


16

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 (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 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 12.4 300 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 wk-1

A Yield factor varies Mg ha-1

1For northern hemisphere, referenced to Jan. 1.


17

Figure 1. Response of total biomass yield (Yt) to harvest interval ( t ) 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).


18

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 ha-1

for coastal bermudagrass grown at

Tifton, GA. Data adapted from Burton et al. [11]. Curves drawn from Eqs. (11), (14), and (15).


19

Figure 3. Response of standardized biomass yield ( tY ) and standardized plant nitrogen uptake

( utN ) to harvest interval ( t ) at N = 672 kg ha-1

for coastal bermudagrass grown at Tifton, GA.

Lines drawn from Eqs. (10) and (13).

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