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PAGE 1 A MEMOIR ON A Model of Yield and Phosphorus Uptake in Response to Applied Phosphorus by Potato Allen R. Overman and Richard V. Scholtz III Agricultural and Biological Engineering University of Florida Copyright 2012 Allen R. Overman PAGE 2 Overman and Scholtz Response to Applied P by Potato i Key words : Mathematical model, plant yield, phosphorus uptake, potato This memoir is focused on a model of yiel d and phosphorus uptake in response to applied phosphorus by potato ( Solanum tuberosum L.). Mathematical analysis utilizes the extended logistic model published previously by the auth ors. Data analysis is based on a field study conducted at Hastings, FL, USA and published in the literature. The model describes vegetative yield ( Y ) and phosphorus uptake ( Pu) by potato reasonably well with a common response coefficient ( cp). The results confirm the phase relations ( Y and Pc vs. Pu) for the system. It is further shown that extractable soil phosphorus is related to applied phosphor us through a logistic equation ( Pex vs. P ) with the same response coefficient as for yield and plant P uptake. New phase relations are then established with extractable soil phosphorus ( Pu, Y, and Pc vs. Pex). This analysis resolves the mystery of the basis for th e buffer capacity of the system it resides in the extractable soil phosphorus (the soil component) rath er than in the plant component. In fact it relates to anion exchange between soil particle surfaces and the liquid phase. Acknowledgement : The authors thank Amy G. Buhler, E ngineering Librarian, Marston Science Library, University of Florida, for assi stance with preparation of this memoir. PAGE 3 Overman and Scholtz Response to Applied P by Potato 1 A Model of Yield and Phosphorus Uptake in Response to Applied Phosphorus by Potato Allen R. Overman and Richard V. Scholtz III Introduction The extended logistic model was developed to describe dependence of crop yield and nutrient uptake on applied nutrients [1]. It was later applied to response of corn ( Zea mays L.) to applied nitrogen and water availability [2]. The model has been applied to numerous crops, soils, and environmental conditions [3]. In this memoir the model is used to coupl e plant response to applied phosphorus for potato. Dependence of soil extractable P on applied P is al so discussed. This provides a basis to explain the mystery of the buffer capacity of the system it resides in the soil rather than in the plants. Model Development The extended logistic model of crop response to applied nutrients is f ounded on two postulates: (1) plant nutrient uptak e is related to applied nu trient by a logistic equati on and (2) biomass yield is related to nutrient uptake by a hyperbolic e quation. The first postulate can be written as ) exp( 1 P c b A Pp p p u (1) where P is applied phosphorus, kg ha1; Pu is plant phosphorus uptake, kg ha1; Ap is maximum P uptake at high P kg ha1; bp is intercept parameter for P uptake; and cp is response coefficient to applied P, ha kg1. Note that units on cp are the reciproc al of those on P The second postulate can be written as u p u mP K P Y Y (2) where Y is biomass yield, Mg ha1; Ym is potential maximum yield, Mg ha1; and Kp is response coefficient for P uptake, kg ha1. Note that for Pu = Kp, Y = Ym/2 Substitution of Eq. (1) into Eq. (2) leads to a second logistic equation ) exp( 1 P c b A Yp y y (3) where Ay is maximum yield at high P Mg ha1; and by is intercept parameter for yield. Hyperbolic and logistic pa rameters are related by ) exp( 1 b A Yy m (4) PAGE 4 Overman and Scholtz Response to Applied P by Potato 2 1 ) exp( b A Kp p (5) where b is defined by y pb b b (6) Now plant P concentration is defined by ) exp( 1 ) exp( 1 P c b P c b P Y P Pp p p y cm u c (7) where Pcm = Ap/Ay = maximum plant P concentration at high P g kg1. Of the three variables ( Y, Pu and Pc) only two are independent. The approach here is to model Y and Pu and then to estimate Pc from the definition Pc = Pu/Y In fact, it would be mo re appropriate to call Pc specific phosphorus (mass of plant P per unit of plant biomass) However, the term concentration is so entrenched that traditional convention will be followed in this memoir. In order to facilitate parameter estimation some of these equations can be linearized. Equation (1) can be rearranged to the form p p u p pb P c P A Z 1 ln (8) which requires an estimate of Ap. Equation (2) can be lin earized to the form u m m p u cP Y Y K Y P P 1 (9) which provides an easy test of the hyperbolic phase relation ( Pc vs. Pu) from data. The lower limit on plant P concentration, Pcl at highly depleted soil P is gi ven from the intercept in Eq. (9) by m p clY K P (10) It can be shown from Eqs. (4) and (5) that lowe r and maximum plant P concentrations are related by b P Pcm cl exp (11) PAGE 5 Overman and Scholtz Response to Applied P by Potato 3 Since Pcl < Pcm it follows from Eq. (11) that 0 b, i.e. b must be positive and bp > by These relationships will be illu strated through analysis of data. Data Analysis Data for this analysis are adapted from a field study at Hastings, FL with potato (cv Atlantic) by Rhue et al. [4] on Placid sand (sandy, siliceous hyperthermic Typic Humaquept). Phosphorus was applied at rates of 0, 56, 112, and 168 kg ha1. Applied N and K rates were 156 and 168 kg ha1, respectively. Planting was on 8 February 1978, with treatments being replicated three times. Whole plant (above ground) samples were collected at early bloom stage (10 wk after planting). Plant biomass (dry matter) and plant P were determ ined, as listed in Table 1 and shown in Figure 1. Also listed in Table 1 are values of extractable soil P (MehlichI) measured in 1979. It is apparent from Figure 1 that yield, plant P uptake, and plant P concentration all increase with applied P. From Pu vs. P maximum plant P uptake is estimated visually as Ap = 8.3 kg ha1. It follows from Eq. (8) by linear regression ( Zp vs. P ) that 00 2 0315 0 1 3 8 ln P b P c P Zp p u p r = 0.986 (12) with a correlation coefficient of r = 0.986. Plant P uptake is then estimated from ) 0315 0 80 1 exp( 1 3 8 P Pu (13) where the intercept parameter has b een adjusted to 1.80 to provide better estimates at low P, as shown in Figure 1. Visual inspection of Figure 1 also leads to estimates of Ay = 2.00 Mg ha1 and Pcm = 4.15 g kg1. It remains to estimate by. The phase plots ( Y and Pc vs. Pu) are shown in Figure 2. From Eq (9) regression analysis gives u cP P 369 0 05 1 r = 0.9905 (14) which leads to the hyperbolic relation u uP P Y 85 2 71 2 (15) The line in Figure 2 is drawn from Eq. (14), wh ile the curve is drawn from Eq. (15). This confirms the phase relations. It follows from Eq. (14) that Pcl = 1.05 g kg1. Since Pcm = 4.15 g kg1 from Figure 1, Eq. (11) leads to 43 0 37 1 80 1 37 1 15 4 05 1 ln yb b (16) PAGE 6 Overman and Scholtz Response to Applied P by Potato 4 The logistic equation for yield response to applied P becomes ) 0315 0 43 0 exp( 1 00 2 P Y (17) Response of plant P concentration to applied P is then given by ) 0315 0 80 1 exp( 1 ) 0315 0 43 0 exp( 1 15 4 P P Pc (18) Curves in Figure 1 are drawn from Eqs. (13), (17), and (18). Agreement between the model and data appear quite reasonable. Correlation of extractable soil P, Pex, with applied P is examined next (Figure 3). Following Overman and Scholtz [3], a l ogistic relation is assumed ) exp( 1 P c b A Pp e e ex (19) where Ae is maximum extractable soil P at high P g (Mg)1; and be is intercept parameter for extractable P. From Figure 3 Ae = 36.0 g (Mg)1 is estimated visually. Eq uation (19) can then be linearized to give 20 2 0315 0 1 0 36 ln P b P c P Ze p ex ex r = 0.974 (20) which leads to the estimation equation ) 0315 0 20 2 exp( 1 0 36 P Pex (21) This suggests a possible link betwee n plant P uptake and extractable so il P, as shown in Figure 4. Now if both Pu and Pex follow logistic dependence on P with a common response coefficient cp, then it can be shown that Pu and Pex are coupled through a hyperbolic relation ex e ex um uP K P P P (22) where Pum is potential maximum plant P uptake, kg ha1; and Ke is response coefficient for extractable P, g Mg1. Hyperbolic and logistic parameters are related by PAGE 7 Overman and Scholtz Response to Applied P by Potato 5 2 25 ) 40 0 exp( 1 3 8 ) exp( 1 b A Pp um kg ha1 (23) 2 73 1 ) 40 0 exp( 0 36 1 ) exp( b A Ke e g Mg1 (24) Substitution of these parameters into Eq. (20) leads to ex ex uP P P 2 73 2 25 (25) Following a similar line of reasoning, coupling between Y and Pex should follow the hyperbolic equation ex e ex mP K P Y Y (26) where hyperbolic and logistic parameters are related by 41 2 ) 77 1 exp( 1 00 2 ) exp( 1 b A Yy m Mg ha1 (27) 39 7 1 ) 77 1 exp( 0 36 1 ) exp( b A Ke e kg ha1 (28) Substitution of these parameters into Eq. (26) leads to ex exP P Y 39 7 41 2 (29) It follows from Eqs. (25) and ( 29) that plant P concentration ( Pc) and extractable soil P ( Pex) are related by ex ex ex e ex e m um u cP P P K P K Y P Y P P 2 73 39 7 4 10 (30) Curves in Figure 4 are drawn from Eqs. (25), (2 9), and (30). According to Eq. (30) the lower limit on Pc as Pex 0 is 1.05 g kg1, which is in agreement with Eq. (14) as expected. Correlation of tuber yield with ve getative yield is shown in Figure 5. Since there appears to be a somewhat linear relationship, the line is drawn from PAGE 8 Overman and Scholtz Response to Applied P by Potato 6 Y Y Y Y Y Y Yt t1 10 96 10 9 110 2 (31) where the line has been constrained to pass through (0, 0) on intuitive grounds. Summary and Conclusions The extended logistic model appears to pr ovide reasonable response for the vegetative component of potato to applied P (Figures 1 and 2). It is easily shown that applied P to reach 50% of maximum yield is given by 14 0315 0 43 02 / 1 p yc b P kg ha1 (32) Similarly, applied P required to reach 50% of maximum plant P uptake is given by 57 0315 0 80 12 / 1 p pc b P kg ha1 (33) According to Eq. (17), P = 85 kg ha1 would be required to produce 90% of maximum yield. It should be noted that the intercept parameter by reflects the base level of soil P. As shown by Eq. (11), b is controlled by the ratio of lower and maximum plant P concentrations, which is a characteristic of the plant species. Parameter Ay is influenced by water availability [5] and plant population [6]. The ratio Ap/Ay is controlled by maximum plant P concentration (see Eq. (7)). This analysis supports the idea of logistic depe ndence of extractable soil P on applied P (Figure 3), in agreement with previous results ([4], [7 ]). This leads to hyperbol ic coupling of plant P uptake and yield on soil extractable P (see Figur e 4). Perhaps available soil P relates to exchangeable P described by the LangmuirHinsh elwood model of soil phosphorus kinetics [8]. If phosphorus adsorption is viewed as reversible anion ex change between H2PO4 1 and OH1, then Ae represents the maximum concentration of site s available for anion exchange in the soil. This point clearly deserves further research. A logical basis for the logistic model has been published [9]. The model is mathematically well behaved and straight forward to use for estim ation of production and plant nutrient uptake. The logistic model exhibits the important pr operty of mathematical symmetry. In 1915 the mathematician Emmy Noether proved an important theorem which established that for dynamic systems symmetry relates to a conservation princi ple, meaning that some quantity in the system is conserved [10]. These concepts are now applied to response of potato to applied P. Note that this is a two state system comprised of filled and unfilled capacity. For plant P uptake we have filled capacity ( Pu) + unfilled capacity ( Ap Pu) = total capacity ( Ap) (34) PAGE 9 Overman and Scholtz Response to Applied P by Potato 7 It follows that the conserved quant ity is measured by the parameter Ap. For plant biomass yield we obtain filled capacity ( Y ) + unfilled capacity ( Ay Y ) = total capacity ( Ay) (35) Finally for extractable soil P we have filled capacity ( Pex) + unfilled capacity ( Ae Pex) = total capacity ( Ae) (36) While these results appear simple, they do illu strate the connection between symmetry and a conservation principle, which clarifies coup ling between plant and soil components with a common response coefficient ( cp). A simplified theory of biomass production by photosynthesis with cal endar time has been developed [11]. The theory utilized rigorous principles from mathematics, physics, and chemistry to construct a linear differential equation which was then integrated over time. Field data from the literature formed the empirical f oundation to guide developmen t of the theory. This led to a simple linear relationshi p between biomass accumulation and a growth quantifier. Three basic processes were identified: (1) seasonal distribution of sola r energy described by a Gaussian function, (2) partitioning of biomass into lightg athering and structural components, and (3) an aging function described by an exponential function. Data from field studies with a warmseason perennial grass were used in the analysis. It has been the long term goal of this effort to develop a compre hensive theory which would have broad application to a wide range of crops, soils, and environm ental conditions. Work continues on application to additional crop sp ecies and soils. We have tried to follow the guiding principles outlined by Michael Faraday, viz work, finish, and publish [12]. PAGE 10 Overman and Scholtz Response to Applied P by Potato 8 Table 1. Response of soil extractable phosphorus ( Pex), vegetative biomass yield ( Y ), plant P uptake ( Pu), plant P concentration ( Pc), and tuber yield ( Yt) to applied phosphorus ( P ) for potato at Hastings, FL, USA (1978).1 P Pex Y Pc Pu Yt kg ha1 g Mg1 Mg ha1 g kg1 kg ha1 Mg ha1 0 5 0.55 1.4 0.77 7.3 56 12 1.85 2.6 4.8 18.6 112 24 1.80 3.6 6.5 19.4 168 35 2.00 4.0 8.0 18.8 1Data adapted from Rhue et al. (1981). PAGE 11 Overman and Scholtz Response to Applied P by Potato 9 References 1. Overman AR, Wilkinson SR and Wilson DM (1 994) An extended logistic model of forage grass response to applied n itrogen. Agronomy J. 86:617620. 2. Reck WR and Overman AR (1996) Estimati on of corn response to water and applied nitrogen. J. Plant Nutrition 19:201214. 3. Overman AR and Scholtz RV III (2002) Mathem atical Models of Plant Growth and Yield. New York: Taylor & Francis. 328 p. 4. Rhue RD, Hensel DR, Yuan TL, and Robert son WK (1981) Response of potatoes to soil and fertilizer phosphorus in Northeast Florida. Pr oc. Soil and Crop Science Society of Florida 40:5861. 5. Overman AR, Neff CR, Wilkinson SR, and Ma rtin FG (1990) Water, harvest interval, and applied nitrogen effects on forage yield of bermudagrass and bahiagrass. Agronomy J. 82:10111016. 6. Overman AR and Scholtz RV III (2011) Model of yield response of corn to plant population and absorption of solar energy. PLoS One 6(1):e16117.doi:10.1371/journal.pone.0016117. 7. Overman AR (1995a) Coupling among applied, soil, root, and top components for forage crop production. Commun. in Soil Scie nce and Plant Analysis 26:11791202. 8. Overman AR and Scholtz RV III (1999) LangmuirHinshelwood model of soil phosphorus kinetics. Commun. in Soil Science and Plant Analysis 30:109119. 9. Overman AR (1995b) Rational basis for the l ogistic model for forage grasses. J. Plant Nutrition 18:9951012. 10. Lederman LM and Hill CT (2008) Symmet ry and the Beautiful Universe: New York: Prometheus Books. 363 p. 11. Overman AR (2010) A Memoir on a Si mplified Theory of Biomass Production by Photosynthesis. University of Fl orida. Gainesville, FL. 19 p. http://ufdc.ufl.edu/AA00001573/00001 12. Russell CA (2000) Michael Faraday Physic s and Faith. New York: Oxford University Press. 124 p. PAGE 12 Overman and Scholtz Response to Applied P by Potato 10 Figure 1. Response of vegetative biomass yield ( Y ), plant P uptake ( Pu), and plant P concentration ( Pc) to applied phosphorus ( P ) for potato at Hastings FL, USA (1978). Data adapted from Rhue et al. (1981). Curves drawn from Eqs. (13), (17), and (18). PAGE 13 Overman and Scholtz Response to Applied P by Potato 11 Figure 2. Phase plots between vegetative biomass yield ( Y ) and plant P concentration ( Pc) and plant P uptake ( Pu) for potato at Hastings, FL, USA ( 1978). Data adapted from Rhue et al. (1981). Line drawn from Eq. (14); curve from Eq. (15). PAGE 14 Overman and Scholtz Response to Applied P by Potato 12 Figure 3. Response of soil extractable phosphorus( Pex) to applied phosphorus ( P ) for potato at Hastings, FL, USA (1979). Data adapted from Rhue et al. (1978) Curve drawn from Eq. (21). PAGE 15 Overman and Scholtz Response to Applied P by Potato 13 Figure 4. Phase plots between plant P uptake ( Pu), vegetative biomass yield ( Y ), and plant P concentration ( Pc) with extractable soil P ( Pex) for potato at Hastings FL, USA (1978). Data adapted from Rhue et al. (1978). Curves drawn from Eqs. (25), (29), and (30). PAGE 16 Overman and Scholtz Response to Applied P by Potato 14 Figure 5. Correlation between tuber yield ( Yt) and vegetative biomass yield ( Y ) for potato at Hastings, FL, USA (1978). Data adapted from Rhue et al. (1978) Line drawn from Eq. (31). 