A MEMOIR ON. Allen R. Overman and Richard V. Scholtz III. Agricultural and Biological Engineering. University of Florida

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1 A MEMOIR O A Smplfed Theory of Bomass Producon by Phoosynhess Allen R. Overman and Rchard V. Scholz III Agrculural and Bologcal Engneerng Unversy of Florda Copyrgh 010 Allen R. Overman

2 Key words: Plan growh, mahemacal model, phoosynhess Ths memor s focused on a smplfed heory of bomass producon by phoosynhess. I descrbes accumulaon of bomass wh calendar me. The heory s srucured on a rgorous mahemacal framework and a sound emprcal foundaon usng daa from he leraure. Parcular focus n on he norhern hemsphere where mos feld research has been conduced, and on he warm-season perennal coasal bermudagrass for whch an exensve daabase exss. Three prmary facors have been denfed n he model: (1) an energy drvng funcon, () a paron funcon beween lgh-gaherng (leaf) and srucural (sem) plan componens, and (3) an agng funcon. These funcons are hen combned o form a lnear dfferenal equaon. Inegraon leads o an analycal soluon. A lnear relaonshp s esablshed beween bomass producon and a growh quanfer for a fxed harves nerval. The heory s furher used o descrbe forage qualy (nrogen concenraon and dgesble fracon) beween leaves and sems of he plans. The heory can be appled o annuals (such as corn) and well as perennals. Crop response o varous appled elemens (such as nrogen, phosphorus, poassum, calcum, and magnesum) can be descrbed. The heory conans fve parameers: wo for he Gaussan energy funcon, wo for he lnear paron funcon, and one for he exponenal agng funcon. Acknowledgemen: The auhors hank Amy G Buhler, Engneerng Lbraran, Marson Scence Lbrary, Unversy of Florda, for asssance wh preparaon of hs memor. 1

3 A Smplfed Theory of Bomass Producon by Phoosynhess Allen R. Overman and Rchard V. Scholz III Inroducon Phoosynhess s he bochemcal processes by whch green plans use ncden radan energy o fx CO from he amosphere and H from he splng of he waer molecule o form CHO he major conen of plan bomass. Mneral elemens (such as, P, K, Ca, Mg, ec) are derved from he rhzosphere (roo zone). Readers neresed n deals of phoosynhess a he molecular and cellular levels are referred o he excellen book by Olver Moron [1]. The deals are exremely complcaed. In hs arcle we seek o make smplfyng assumpons whch lead o a feld scale heory relang he rae of accumulaon of plan bomass wh calendar me, dy/d, (Mg ha -1 wk -1 ) o calendar me,, (wk). A broader vew of progress of scence n he 0h cenury, ncludng he role of physcs n phoosynhess, s provded by Gerard Pel []. In hs classc book The Ascen of Man, Jacob Bronowsk [3] races human hsory, ncludng culural and bologcal evoluon. He documens he developmen of he agrculural revoluon from huner/gaherer o farmer/husbandman ha has made modern agrculure possble and whch produces he food and fber upon whch humany now depends. Along wh he expanson of echnology, agrculural feld research has experenced rapd developmen, begnnng wh he famous work a Rohamsed, England n abou Today a very large daabase exss from varous locaons around he world. I s hs daabase whch we wll draw upon as he emprcal foundaon for he presen heory. In order o develop a rgorous heory of bomass producon by phoosynhess, a sound mahemacal framework s requred. For hs purpose we draw upon wo fundamenal prncples of scence as saed by Daves and Grbbn [4, p. 44]: Prncple #1: I s possble o know somehng of how naure works whou knowng everyhng abou how naure works. Whou hs prncple here would be no scence and no undersandng. Prncple #: In physcs a lnear sysem s one n whch a collecon of causes leads o a correspondng collecon of effecs. For a gven sysem can be shown ha hs correspondence s unque, and he prncple works n boh drecons. In he sequel hs wll be referred as he correspondence prncple. Snce he nvenon of he calculus by Isaac ewon, has been common pracce o make smplfyng assumpons whch lead o lnear dfferenal equaons (ordnary or paral) wh analyc soluons. Examples nclude mechancs (equaons of moon), hermodynamcs (dffuson of hermal energy), chemcal dynamcs (dffuson of molecular speces), elecrcal phenomenon, magnec phenomenon, and even o quanum mechancs (boh marx mechancs and wave mechancs). A smlar sraegy s followed n he presen work. Smplfyng assumpons are made n he search o develop a mahemacal heory beween he me rae of accumulaon of bomass, dy/d, and calendar me,. The analyss s focused on he norhern hemsphere where he greaes collecon of feld daa has been repored. In addon he analyss focuses on feld sudes wh he warm-season perennal coasal bermudagrass [Cynodon dacylon (L.) Pers.]. umerous sudes have been conduced for fxed harves nerval,. Measuremens of bomass accumulaon as relaed o harves nerval have been repored, from whch deducons can be

4 made abou effecs. The correspondence prncple can hen be used o make nference abou he causes nvolved. Ths leads fnally o a lnear dfferenal equaon. Theory Developmen The frs sep n hs process s o denfy key componens whch conrbue o bomass producon wh calendar me as measured by daa from feld sudes. These facors are hen combned no a lnear dfferenal equaon. The dfferenal equaon s hen negraed o an analyc soluon. Agan daa are for he warm-season perennal coasal bermudagrass n he norhern hemsphere and harvesed on a fxed me nerval. Energy Drvng Funcon The frs sep along hese lnes was aken by Overman [5] n response o requess by envronmenal regulaors o esmae bomass and plan nuren accumulaon wh calendar me for a waer reclamaon/reuse projec n Florda. The analyss drew upon a feld sudy a Waknsvlle, GA wh coasal bermudagrass harvesed on a fxed me nerval [6]. The expermen conssed of a x facoral of wo harves nervals (4 wk, 6 wk) and wo rrgaon reamens (rrgaed, nonrrgaed). The dsrbuon of bomass wh calendar me was shown o follow a Gaussan funcon descrbed by F exp (1) where F s he fracon of oal bomass a calendar me (referenced o Jan. 1), s me o mean of bomass dsrbuon (referenced o Jan. 1), and s he me spread of he bomass dsrbuon. I was shown ha he dsrbuons were ndependen of rrgaon reamen and harves nerval and followed he equaon 7.8 F exp () 8.13 wh exponenal values n wk. Deals of he analyss are descrbed n [7, Secon 3.] These resuls rased he neresng queson as o he orgn of he Gaussan dsrbuon? I s known ha ncden solar radaon n he norhern hemsphere rses from a mnmum n January o a maxmum n July and decreases agan o a mnmum n December. [7, Table 1.6] analyzed solar radaon daa for Rohamsed, England [8] and showed ha he dsrbuon followed he Gaussan dsrbuon 5.0 F exp (3) 14.8 From hs analyss seems logcal o assume an energy drvng funcon, E(), whch follows a Gaussan dsrbuon o good approxmaon 3

5 E exp (4) where s calendar me referenced o Jan. 1, s me o mean of he solar energy dsrbuon referenced o Jan. 1, and s he me spread of he solar energy dsrbuon. All uns are n weeks. Paron Funcon for Bomass The second sep n he analyss s o denfy an nrnsc growh funcon ha denfes how plans respond o he npu of solar energy. Forunaely feld expermens have been conduced wh coasal bermudagrass a Tfon, GA by Prne and Buron [9]. The facoral expermen conssed of fve nrogen levels ( = 0, 11, 336, 67, and 1008 kg ha -1 ), sx harves nervals ( = 1,, 3, 4, 6, and 8 wk), and wo years (1953 and 1954). Toal bomass yeld (Y, Mg ha -1 ) and oal plan nrogen upake ( u, kg ha -1 ) for he enre season (4 wk) were repored. Sysem response s llusraed n Fgure 1 wh bomass daa for = 67 kg ha -1 and for boh years. As urned ou 1953 exhbed deal ranfall (labeled we ) and 1954 exhbed he wors drough n 30 years (labeled dry ). I appears from Fgure 1 ha he daa pons follow sragh lnes for 6 wk. The regresson lnes are descrbed by 1953: Y r = (5) 1954: Y r = (6) wh correlaon coeffcens of r = and for 1953 and 1954, respecvely. The effec of waer avalably s clearly evden n hese equaons. Equaons (5) and (6) reveal an addonal facor beyond he energy drvng funcon denfed above. From he correspondence prncple for lnear sysems, we are led o assume a paron funcon of he ype P a b (7) where s calendar me for he growh nerval, wk; s he reference me for he growh nerval, wk; and a s he nercep parameer, Mg ha -1 wk -1 and b s he slope parameer, Mg ha -1 wk -. Overman and Wlknson [10] examned daa from he leraure on paronng of dry maer beween leaf and sem for coasal bermudagrass and concluded ha he frs erm n Eq. (7) corresponds o he rae of growh of he lgh-gaherng componen (leaf) whle he second erm corresponds o he rae of growh of he srucural componen (sem). Equaon (7) means ha reference me s rese for each growh nerval. I follows logcally ha Eq. (7) mus have a lmed me doman of applcaon. Oherwse arfcal radan energy could be suppled o he plans o provde unlmed bomass accumulaon. Ths smply does no happen! I follows ha here mus be an addonal lmng facor n he sysem. 4

6 Agng Funcon for Bomass The hrd sep n he analyss s o denfy an addonal facor whch mposes a furher lm on plan growh. Buron e al. [11] expanded he range of harves nervals o nclude = 3, 4, 5, 6, 8, 1, and 4 wk. Appled nrogen was = 67 kg ha -1. Seasonal oal bomass yeld (Y ) and seasonal oal plan nrogen upake ( u ) were repored as gven n Table 1. Plan nrogen concenraon s defned as c = u /Y. In order o see he paern n he response more clearly, resuls are also shown n Fgure. The graph of Y vs. exhbs a rse, passng hrough a peak, hen followng a seady declne. Ths seems lke he place for he me-honored mehod of nuon as defned by Roger ewon [1, p. 60]. Expandng on he lnear paron funcon from he prevous secon, we assume a lnear-exponenal funcon Y y y exp (8) where y, y, and are o be evaluaed from he daa pons. I remans o be deermned f Eq. (8) exhbs he correc characerscs o descrbe he response curve. In oher words, an operaonal procedure s requred o evaluae he coeffcens of Eq. (8). ow f he value of were known, hen we could defne a sandardzed yeld Y as Y Y exp y y (9) whch leads o a lnear equaon n. The procedure s o ry values of whch leads o he opmum correlaon for Eq. (9). Ths process leads o an esmae of correspondng values n column 5 of Table 1. Lnear regresson of Y vs wk wh he hen leads o Y Y exp y y r = (10) wh a correlaon coeffcen of r = Ths resul s shown n Fgure 3, where he lne s drawn from Eq. (10). I follows ha he lower curve n Fgure s drawn from Y exp (11) The nex sep s o defne sandardzed plan nrogen upake u as u u exp n n (1) Wh regresson of wk as for bomass response, hs leads o values n column 6 of Table 1. Lnear u vs. hen leads o u u exp n n r = (13)

7 wh a correlaon coeffcen of r = Ths resul s shown n Fgure 3, where he lne s drawn from Eq. (13). I follows ha he mddle curve n Fgure s drawn from u exp (14) Plan nrogen concenraon c s hen defned by c Y u n y n y (15) The upper curve n Fgure s drawn from Eq. (15). The lnear-exponenal funcon appears o descrbe he response curves raher well. The queson now occurs as o he meanng of he exponenal erm n Eq. (8). Assumng ha he sysem follows lnear behavor, we nvoke he correspondence prncple o wre an agng funcon, A exp[ c ] (16) I should be noed ha he agng funcon s no derved from bochemcal or genec consderaons, bu represens an operaonal defnon of a declne n capacy of he sysem o generae new bomass as he plans age. Equaon (16) reses for each new value of reference me. Lnear Dfferenal Equaon A hs pon hree ndependen funcons have been denfed for he sysem: a lnear paron funcon P, an agng funcon A, and an energy drvng funcon E. The queson now s how o combne hese funcons o form a lnear dfferenal equaon? Snce he funcons are consdered ndependen, seems reasonable o nvoke he prncple of jon probably and wre he equaon n produc form dy d consan consan P a b A exp E c exp (17) The procedure of jon probably s he same as ha used by James Clerk Maxwell n developng he velocy dsrbuon law n he knec heory of gases [13, p. 159]. He frs wres he funcon for one dmenson n space. Then he reamen s exended o hree dmensons by nvokng he prncple of jon probably and wrng he overall funcon as he produc of he hree separae funcons. 6

8 Equaon (17) forms a lnear frs order dfferenal equaon n he me varable. However, conans wo reference mes: and. Snce he paron funcon and he agng funcon rese for each growh nerval, Equaon (17) mus be vewed as pecewse connuous n he me nerval, and negraon mus proceed accordngly. The neresed reader s referred o [7, Secon 4.3] for deals of he negraon process. The soluon for he ncrease n bomass accumulaon for he h growh nerval, Y becomes he smple lnear relaonshp Y A Q (18) where A s he yeld facor, Mg ha -1 and Q s he growh quanfer defned by k Q 1 kx erf x erf x exp x exp x exp π cx (19) The dmensonless paron coeffcen, k, n Eq. (19) s defned by k b / a (0) and he dmensonless me varable x s defned by c x (1) I follows mmedaely ha x s defned by c x () The error funcon, erf x, n Eq. (19) s defned by x erf x exp u du (3) π 0 where u s he varable of negraon for he Gaussan funcon The cumulave sum of bomass for n harvess, Y n, s gven by exp u. Y n n 1 Y A n 1 Q A Q n (4) 7

9 Toal bomass yeld for he enre season, Y, s he sum over all harvess and s relaed o oal growh quanfer for he season, Q, by Y A Q (5) A Mahemacal Theorem A mahemacal heorem s now presened whch provdes a rgorous connecon beween Eq. (17) for he lnear frs order dfferenal equaon for each growh nerval and Eq. (8) for lnearexponenal dependence of seasonal oal bomass yeld, Y, on a fxed harves nerval,. Deals of he proof are gven by [7, Secon 4.3]. The heorem esablshes ha he cumulave sum of bomass producon for n harvess s gven by k c 1 Y n A exp 1 erf (6) where A s defned by π A consan a (7) In Eq. (6) he erm n curly brackes confrms he Gaussan dsrbuon of he energy drvng funcon, whch approaches 1 n he lm of large. Ths means ha oal bomass yeld for he season, Y, s gven by Y A Q (8) where seasonal oal growh quanfer, Q s defned by k c Q exp (9) Fnally, Eq. (8) can be wren n regresson form Y ka c A exp y y exp (30) Ths complees he proof of he heorem. I should be noed ha c / s requred. The heorem apples for fxed harves nerval, wk. 8

10 Applcaon of he Theory o Forage Qualy Forage qualy s measured prmarly by wo characerscs: nrogen concenraon (proen conen) and dgesbly of he bomass by rumnan anmals. For perennal grasses qualy relaes o lengh of me beween harvess snce age of plans nfluences he balance beween plan componens (leaves vs. sems). In hs secon daa from wo sudes wh perennal grasses are used o evaluae forage qualy. Sudy wh Coasal Bermudagrass Equaon (15) can be used o esmae plan nrogen concenraon of he wo plan componens. In he lm as 0, nrogen concenraon of he lgh-gaherng (leaf) componen, cl, s esmaed from 4 cl 47.4 g kg -1 (31) 8.91 In he lm as very large, nrogen concenraon of he srucural (sem) componen, cs, s esmaed from 33.0 cs 9.5 g kg -1 (3) 3.49 Clearly he nrogen (crude proen) qualy of he leaf fracon s consderably hgher han of he sem fracon. The sudy by Buron e al. [11] a Tfon, GA ncluded daa on dgesble dry maer as measured by he n vro mehod [14]. Seasonal oal dgesble bomass (D ) along wh oal bomass (Y ) as relaed o harves nerval ( ) are lsed n Table. Response of D vs. s assumed o follow a lnear-exponenal funcon (smlar o Eq. (8)) D d d exp (33) where d, d, and are o be evaluaed from he daa pons. Sandardzed dgesble dry maer ( D ) s defned by D D exp d d (34) Agan we choose regresson of D vs wk wh he correspondng values n column 5 of Table. Lnear hen leads o D D exp d d r = (35) 9

11 wh a correlaon coeffcen of r = Ths resul leads o he lner-exponenal equaon D exp (36) I follows mmedaely ha dgesble fracon, f d, s descrbed by f d D Y (37) ow Eq. (37) can be used o esmae dgesbly of he wo plan componens. In he lm as 0, he dgesble fracon of he lgh-gaherng (leaf) fracon, f dl, s esmaed from 9.93 f dl 1.11 (38) 8.91 Of course he lgh-gaherng componen should no exceed The value of 1.11 represens uncerany n he nercep values n Eq. (37). In fac, can be shown a he 95% confdence level ha y Mg ha -1 and ha d Mg ha -1. Snce hese values clearly overlap, seems reasonable o assume ha he value of he lgh-gaherng componen s approxmaely f 1.00 (39) dl In he lm as esmaed from very large, he dgesble fracon of he srucural (sem) fracon, f ds, s 1.9 f dl 0.37 (40) 3.49 ow we see ha as ncreases, he plans shf from domnance by lgh-gaherng o srucural componen of he plan wh a correspondng declne n forage qualy. I can be shown from calculus ha harves nervals o acheve maxmum oal plan bomass and maxmum dgesble bomass can be esmaed from, respecvely, 1 y py 10.4 wk (41) y 1 d pd 5.3 wk (4) d 10

12 Sudy wh Perennal Peanu A sudy was conduced by Belranena e al. [15] wh he warm-season legume perennal peanu (Arachs glabraa Benh cv Florgraze ) on Arredondo loamy fne sand (loamy, slceous, semacve, hyperhermc Grossarenc Paleudul). Plans were sampled on fxed harves nervals of =, 4, 6, 8, 10, and 1 wk. The growng season s consdered o be 4 wk. Measuremens were made of seasonal oal bomass (Y ), seasonal oal plan nrogen upake ( u ), and seasonal oal dgesble bomass (D ). Resuls are lsed n Table 3. All daa are for a clppng hegh of 3.8 cm. As wh he case of coasal bermudagrass, he lnear-exponenal model s assumed o apply. Sandardzed bomass yeld ( Y ), sandardzed plan nrogen upake ( sandardzed dgesble bomass yeld ( D ) are calculaed from u ), and Y Y exp y y r = (43) u u exp n n r = (44) D D exp d d r = (45) Equaons (43) hrough (45) lead o he lnear-exponenal equaons Y y y exp exp (46) u n n exp exp (47) Equaons (46) and (47) lead mmedaely o c Y u (48) Equaon (48) can be used o esmae nrogen concenraon of each plan componen. rogen concenraon of he leaf fracon, cl, s esmaed from 7 cl 56. g kg -1 (49) 4.84 whereas nrogen concenraon of he sem fracon, cs, s esmaed from 33.6 cs 15.8 g kg -1 (50).1 Agan, he leaf fracon conans hgher nrogen concenraon han does he sem fracon. 11

13 Dependence of seasonal oal dgesble bomass s descrbed by he lnear-exponenal equaon D d d exp exp (51) Equaons (46) and (51) can be combned o oban dependence of dgesble fracon f d on harves nerval f d D Y (5) Dgesble fracon for he wo plan componens leaves and sems, f dl and f ds, can be esmaed from 4.97 f dl (53) f ds 0.47 (54).1 Agan he leaf fracon exhbs hgher dgesbly han he sem fracon. I can be shown from calculus ha he maxmum value (peak) of he lnear-exponenal curves corresponds o a peak harves nerval,, for oal bomass, oal plan nrogen upake, and oal dgesble bomass gven by p 1 y py 10.7 wk (55) y 1 n 1 7 pn 4.9 wk (56) n 1 d pd 8.0 wk (57) d Summary and Conclusons The smplfed heory of bomass producon by phoosynhess s descrbed by Eqs (18) hrough (3). Snce Eq (19) conans he error funcon, erf x, hs seems lke he approprae place o presen he dealed dscusson by Abramowz and Segun [16, chp 7], ncludng a able of values. A few key characerscs should be noed: f(0) = 0, f( ) = 1 1

14 f( x ) = f( x ), f( ) = f( ) = 1 The heory conans fve parameers: wo for he energy drvng funcon (, ) and hree for plan characerscs (k, c, A). Examnaon of daa for he norhern hemsphere and for he warm-season perennal coasal bermudagrass lead o he esmaes lsed n Table 4. 13

15 References 1. Moron O (007) Eang he Sun: How Plans Power he Plane. London: Harper Collns. 475 p.. Pel G (001) The Age of Scence: Wha Scenss Learned n he Tweneh Cenury. ew York: Basc Books. 459 p. 3. Bronowsk J (1973) The Ascen of Man. Boson: Lle, Brown & Co. 448 p. 4. Daves P and Grbbn J (199) The Maer Myh: Dramac Dscoveres Tha Have Challenged Our Undersandng of Physcal Realy. ew York: Smon & Schuser. 30 p. 5. Overman AR (1984) Esmang crop growh rae wh land reamen. J. Env. Eng. Dv., Amercan Socey of Cvl Engneers 110: Mays DA, Wlknson SR, and Cole CV (1980) Phosphorus nuron of forages. In: The Role of Phosphorus n Agrculure. Khasawneh FE and Kamprah EJ (eds). Madson, WI: Amercan Socey of Agronomy. pp Overman AR and Scholz RV (00) Mahemacal Models of Crop Growh and Yeld. ew York: Taylor & Francs. 38 p. 8. Russell EJ (1950) Sol Condons and Plan Growh 8h ed. London: Longmans, Green & Co. 635 p. 9. Prne GM and Buron GW (1956) The effec of nrogen rae and clppng frequency upon he yeld, proen conen, and ceran morphologcal characerscs of coasal bermudagrass [Cynodon dacylon (L.) Pers.]. Agronomy J. 48: Overman AR and Wlknson SR (1989) Paronng of dry maer beween leaf and sem n coasal bermudagrass. Agrculural Sysems 30: Buron GW, Jackson JE, and Har RH (1963) Effecs of cung frequency and nrogen on yeld, n vro dgesbly, and proen, fber and caroene conen of coasal bermudagrass. Agronomy J. 55: ewon, R (1997) The Truh of Scence: Physcal Theores and Realy. Cambrdge, MA: Harvard Unversy Press. 60 p. 13. Longar MS (1984) Theorecal Conceps n Physcs. ew York: Cambrdge Unversy Press. 366 p. 14. Moore JE and Dunham GD (1971) Procedure for he wo-sage n vro organc maer dgeson of forages (revsed). uron Laboraory, Deparmen of Anmal Scence. Unversy of Florda. Ganesvlle, FL 10 p. 15. Belranena R, Breman J and Prne GM (1981) Yeld and qualy of Florgraze rhzome peanu (Arachs glabraa Ben.) as affeced by cung hegh and frequency. Proc. Sol and Crop Scence Socey of Florda 40: Abramowz M and Segun IA (1965) Handbook of Mahemacal Funcons. ew York: Dover Publcaons p. 14

16 Table 1. Response of seasonal oal bomass yeld (Y ), seasonal oal plan nrogen upake ( u ), and plan nrogen concenraon ( c ) o harves nerval ( ) a = 67 kg ha -1 for coasal bermudagrass a Tfon, GA. 1 Y u c Y u wk Mg ha -1 kg ha -1 g kg -1 Mg ha -1 kg ha Daa adaped from Buron e al. [11]. Table. Response of seasonal oal bomass yeld (Y ), seasonal dgesble bomass (D ), and dgesble fracon (f d ) o harves nerval ( ) a = 67 kg ha -1 for coasal bermudagrass a Tfon, GA. 1 Y D f d D wk Mg ha -1 Mg ha -1 Mg ha Daa adaped from Buron e al. [11]. 15

17 Table 3. Response of seasonal oal bomass yeld (Y ), seasonal oal plan nrogen upake ( u ), plan nrogen concenraon ( c ), seasonal dgesble bomass (D ), dgesble fracon (f d ), sandardzed oal bomass yeld ( Y ), sandardzed oal plan nrogen upake ( u ), and sandardzed oal dgesble bomass ( D ) o harves nerval ( Ganesvlle, FL. 1 ) for perennal peanu a Y u c D f d Y u D wk Mg ha -1 kg ha -1 g kg -1 Mg ha -1 Mg ha -1 kg ha -1 Mg ha Daa adaped from Belranena e al. [14]. Table 4. Summary of Parameer Values. Parameer Defnon Value Uns Tme o mean of 6.0 wk solar energy dsrbuon 1 Tme spread of 8.00 wk solar energy dsrbuon k Paron consan 5 none c Agng coeffcen wk -1 A Yeld facor vares Mg ha -1 1 For norhern hemsphere, referenced o Jan

18 Fgure 1. Response of oal bomass yeld (Y ) o harves nerval ( ) a = 67 kg ha -1 for coasal bermudagrass grown a Tfon, GA. Daa adaped from Prne and Buron [9]. Lnes drawn from Eqs. (5) and (6). 17

19 Fgure. Response of oal bomass yeld (Y ), oal plan nrogen ( u ), and plan nrogen concenraon ( c ) o harves nerval ( ) a = 67 kg ha -1 for coasal bermudagrass grown a Tfon, GA. Daa adaped from Buron e al. [11]. Curves drawn from Eqs. (11), (14), and (15). 18

20 Fgure 3. Response of sandardzed bomass yeld ( Y ) and sandardzed plan nrogen upake ( u ) o harves nerval ( ) a = 67 kg ha -1 for coasal bermudagrass grown a Tfon, GA. Lnes drawn from Eqs. (10) and (13). 19