THE interest in modeling agricultural systems is rapidly increasing,

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1 Teaching Crop Physiology with the Soil Water Balance Model N. Z. Jovanovic, J. G. Annandale,* and P. S. Hammes ABSTRACT Soil Water Balance (SWB) is a mechanistic, generic crop irrigation scheduling model, making use of weather, soil, and crop databases to simulate crop growth and the soil water balance. The crop growth unit of the SWB model was presented to undergraduate, fourth year crop production students in the crop physiology course at the University of Pretoria, South Africa. The objectives were to assess the effectiveness of the SWB model as a teaching tool, and to investigate the students perceptions of the SWB classroom demonstration. The crop growth principles were first discussed using a flow diagram of the crop unit. The model was then demonstrated using a data projector, and a homework exercise was assigned and marked. None of the students had a score lower than 50%, which was arbitrarily assumed to be the passing mark. The average score was 69.4% ± 9.0% (the lowest score was 58.3% and the highest 87.5%). The results of the homework were encouraging for both the instructors and students, and the classroom demonstration was well received by the students. The SWB model proved to be a useful tool for providing instruction to crop physiology students. THE interest in modeling agricultural systems is rapidly increasing, particularly since personal computers have become widely affordable. A large number of crop physiology models have been developed for different applications (Hanks and Ritchie, 1991). Most of these models can be adapted and used for teaching purposes. Educational software was developed to teach the effect of soil salinity and varying water conditions on crop yield (Robbins et al., 1995; Khan et al., 1996), or to quantify N balance reactions in the soil plant system (Torbert et al., 1994). In this study, a computer tool for teaching the mechanisms of crop growth and development is presented. Soil Water Balance (SWB) is a user-friendly, mechanistic, real time, generic crop, soil water balance, irrigation scheduling model (Annandale et al., 1999), based on the improved generic crop version of NEWSWB (Campbell and Diaz, 1988). It gives a detailed description of the soil plant atmosphere system, making use of weather, soil, and crop databases. The SWB model calculates the water balance and crop growth with three units, namely weather, soil, and crop. The purpose of the weather unit of SWB is to calculate potential evapotranspiration from available meteorological input data (Allen et al., 1998; Smith, 1992; Smith et al., 1996). Daily Penman-Monteith grass reference evapotranspiration (ETo) and crop potential evapotranspiration (PET) are calculated and used by the soil unit to compute actual crop transpiration (T) and soil evaporation (E). The purpose of the soil unit of SWB is to simulate the dynamics of water movement in the soil profile to determine E Dep. of Plant Production and Soil Science, Univ. of Pretoria, 0001, Pretoria, South Africa. Received 24 June *Corresponding author (annan@ scientia.up.ac.za). Published in J. Nat. Resour. Life Sci. Educ. 29:23 30 (2000). and T. The SWB model has a multilayer soil component and water contents and potentials of the various layers are calculated on a daily time step. Cascading water movement is simulated once canopy water interception and surface runoff have been accounted for. The two components of potential evapotranspiration (potential evaporation and potential transpiration) are separated according to canopy cover (Ritchie, 1972). Actual evaporation proceeds at the potential rate until the water content in the top soil layer reaches the permanent wilting point. Thereafter, it is equal to the product of potential evaporation and the square of the remaining evaporable water down to the air dry water content (Campbell and Diaz, 1988). Actual transpiration is determined as either supply-limited or demand-limited water uptake (Campbell and Norman, 1998). The SWB model assumes that water is taken from soil layers with highest root densities when total soil water potential is uniform throughout the profile. Profile root-weighted soil water potential is calculated from each soil layer s total potential and the fraction of the total root length in that layer (Campbell and Stockle, 1993). Reduction in soil water potential eventually closes stomata and decreases T and therefore dry matter production. In a related paper (Jovanovic and Annandale, 2000), the weather and soil units of SWB were presented to undergraduate students in the soil physics and irrigation courses at the University of Pretoria, South Africa. In that study, SWB proved to be a useful teaching tool. In this work, the SWB model was presented to undergraduate students in the PGW 411 crop physiology course in the Department of Plant Production and Soil Science, University of Pretoria. Particular attention was given to the crop unit of SWB, which simulates the accumulation and partitioning of dry matter as well as phenological development and canopy growth. The aims of this classroom demonstration were to familiarize the students with crop modeling, to strengthen their understanding of some of the theoretical principles explained during the course, and to help them in developing problem solving skills. The objectives of this study were to assess the effectiveness of a computer application as a teaching tool for crop physiology students and to investigate the students perceptions of the SWB classroom demonstration. MATERIALS AND METHODS Description of the Crop Unit of Soil Water Balance The purpose of the crop unit of SWB is to simulate crop growth and development so that evaporation and transpiration can be accurately simulated to ensure reasonable crop water use estimates. A flow diagram of the crop unit of SWB is shown in Fig. 1. The crop unit includes three procedures, namely initialization, planting, and day step calculation. Crop initialization sets initial values of crop parameters. The procedure for crop planting is initiated once a valid planting date has been identified. The crop day step procedure is then performed on a daily basis. Abbreviations: SWB, Soil Water Balance model. J. Nat. Resour. Life Sci. Educ., Vol. 29,

2 Fig. 1. Flow diagram of the crop unit of the Soil Water Balance (SWB) model. 24 J. Nat. Resour. Life Sci. Educ., Vol. 29, 2000

3 Crop development is simulated using thermal time, an approach suggested by Monteith (1977) and shown to work well by Olivier and Annandale (1998). The calculation of growing day degrees (GDD) starts after crop planting. Growing day degrees are accumulated daily and the growing day degrees increment for each day (GDD i ) is calculated as follows: GDD i =(T avg - T b ) 1day [1] where T avg is the daily average air temperature and T b is the base temperature, both in C. When T avg is below the base temperature, GDD i is set to 0. If T avg > T cutoff, then: GDD i =(T cutoff - T b ) 1day [2] where T cutoff is an optimal temperature for crop development in C. The succession of phenological stages is simulated using day degree requirements for emergence (EMDD), completion of vegetative growth or flowering day degrees (FLDD), transition period between vegetative and reproductive growth (TransDD), and maturity day degrees (MTDD). At emergence, aboveground dry matter or top dry matter (TDM) is set to TDM at emergence. For most crops, TDM at emergence is estimated to be equivalent to seed mass per unit area. Root dry matter (RDM) at emergence is estimated as: f r TDM RDM = [3] (1 - f r ) where f r is the fraction of dry matter partitioned to the roots. Leaf area index (LAI) at emergence is calculated as: LAI = SLA TDM [4] where SLA is the specific leaf area in m 2 kg -1. Fractional interception of radiation or canopy cover is used to determine the portion of radiation available for crop transpiration and evaporation from the soil surface. The two parameters calculated are: FI transp =1-exp (-K LAI) [5] FI evap =1-exp [-K (LAI + ylai)] [6] where FI transp is the fraction of radiation intercepted by the canopy and available for photosynthesis and transpiration. The amount of radiation penetrating the canopy and used for evaporation from the soil surface is given by (1 - FI evap ). The crop specific parameter, K, is the canopy solar radiation extinction coefficient, while ylai is the leaf area index of senesced or yellowed leaves. Crop height (Hc) is used in the calculation of PET in the weather unit of SWB. It is assumed to be m until emergence. After emergence, Hc increases linearly with day degrees until the end of the transition period between vegetative and reproductive growth, when it reaches its maximum (Hc max ). The SWB model calculates Hc daily, using the following equation: (GDD -EMDD) (Hc max ) Hc = [7] (FLDD + TransDD -EMDD) After the transition period between the vegetative and reproductive stages has been completed, crop height remains equal to Hc max. The SWB model calculates daily dry matter increments (DM i ) as being either transpiration- or radiation-limited. Transpiration-limited DM i (kg m -2 ) is predicted using the relationship between dry matter accumulation and transpiration (Tanner and Sinclair, 1983): T DM i = DWR [8] VPD where DWR is the dry matter/water ratio in Pa, VPD is the vapor pressure deficit in Pa, and T is in mm or kg m -2. Under conditions of radiation-limited crop growth, DM i is calculated using the equation recommended by Monteith (1977): DM i = E c T f FI transp R s [9] where E c is the radiation use efficiency (kg MJ -1 ), R s is solar radiation (MJ m -2 ) for a particular day, and T f is the temperature factor for radiation-limited crop growth, calculated as follows: T avg - T b T f = [10] T lo - T b where T lo is the temperature of optimum light-limited growth in C. The upper limit of T f is set to 1 when T avg > T lo. Daily dry matter increment is chosen as the lesser of the two DM i values calculated with Eq. [8] and [9]. The SWB model assumes that, after flowering, DM i is first partitioned to reproductive sinks, then to the other plant organs. The calculation of daily harvestable dry matter increment is therefore the first in the series of calculations carried out to determine dry matter partitioning to plant organs. On the day when flowering commences, initial harvestable dry matter (HDM) of the crop is calculated as follows: HDM = Transl SDM [11] where Transl is the factor determining translocation of dry matter from stem to grain, and SDM is the stem dry matter (kg m -2 ). During the flowering stage, the following equation is used to calculate the daily harvestable dry matter increment (HDM i ): HDM i =rpf DM i [12] where rpf is the reproductive partitioning fraction. This is calculated as follows: GDD -FLDD rpf = [13] TransDD J. Nat. Resour. Life Sci. Educ., Vol. 29,

4 The upper limit of rpf is set to 1 (all dry matter produced is partitioned to the reproductive portion). If the crop has not flowered, rpf is set to 0. Once the HDM calculation has been completed, the remaining DM i is first partitioned to roots, then to leaves and the remainder into the stem. Daily dry matter increment for roots (RDM i ) is calculated as follows: RDM i = f r DM i [14] The f r factor is set to 0 once root depth has reached a maximum value. Canopy dry matter daily increment (TDM i ) is then calculated as follows: TDM i =(1-f r )DM i [15] Daily increment of leaf dry matter (LDM i ) is calculated with the following equation: LDM i = f l TDM i [16] where the fraction of top dry matter partitioned into leaves (f l ) is calculated as a function of aboveground dry matter: 1 f l = [17] (1+PART TDM) 2 and PART is the leaf-stem partitioning factor in m 2 kg -1. The daily increment of stem dry matter (SDM i ) is calculated as follows: SDM i = TDM i - LDM i [18] HDM i is finally added to TDM i to include grain dry matter into TDM. Assimilate partitioning is affected by water stress. Water stress is simulated when the calculated daily water stress index (SI) is lower than a crop specific threshold value. Stress index is calculated in the soil unit of SWB as the ratio of actual to potential transpiration. Under conditions of water stress (when SI is smaller than the threshold value), half of LDM i is partitioned into roots and the other half into the stem thereby limiting canopy expansion. If the root system has already reached the maximum depth (f r = 0), LDM i is fully partitioned into the stem. After emergence has taken place, LAI daily increments (LAI i ) are calculated using the following relationship: LAI i = LDM i SLA [19] LAI is then calculated by summing LAI i values, and this represents the green leaf or photosynthetically active canopy, which contributes to transpiration and dry matter production. Leaf senescence is also accounted for in SWB. The age in d C of each individual day s leaf area increment (LAIage i ) is kept track of from the day it was generated. Once the LAI i reaches the maximum age, it is classified as yellow or senesced leaves (ylai i ) and it stops contributing to transpiration and dry matter production. The green LAI value is then reduced by ylai i. Leaf area index of senesced leaves (ylai) is increased by ylai i, so as to estimate shading of the soil for the evaporation calculation in the soil unit. A water stress factor (wsf) is used to simulate premature leaf senescence under water stress conditions. When SI is lower than the threshold value, wsf is calculated as: Fig. 2. Example of crop specific growth parameters for the Soil Water Balance (SWB) model. The parameters are (from top to bottom, and from left to right): canopy radiation extinction coefficient, dry matter/water ratio, radiation use efficiency, base temperature, temperature for optimum lightlimited crop growth, cutoff temperature, day degrees for emergence, flowering, maturity, transition period from vegetative to reproductive growth and leaf senescence, maximum crop height, maximum root depth, fraction of dry matter translocated from stem to grain, canopy storage, leaf water potential at maximum transpiration, maximum transpiration, specific leaf area, leaf-stem partitioning parameter, aboveground dry matter at emergence, fraction of dry matter partitioned to roots, root growth rate, and stress index. 26 J. Nat. Resour. Life Sci. Educ., Vol. 29, 2000

5 wsf = 1/(SI) [20] Aging of leaves is then hastened by multiplying the daily thermal time increment by wsf: LAIage i = wsf GDD i [21] The upper limit of wsf is set to 2, indicating that the aging of leaves under water stress conditions can be at most twice as fast as that under well-watered conditions. Rooting depth is calculated with the following equation: RD = RGR RDM 1/2 [22] where RGR is the root growth rate (m 2 kg -1/2 ). Root depth is used in the calculation of transpiration in the soil unit. The simulation ends when day degrees required for maturity are reached, or when daily weather input data are exhausted. The SWB model can be used to simulate the growth and soil water balance of any crop provided that crop specific growth parameters are known. These parameters can be edited in a window of SWB, and they are shown in Fig. 2 for maize (Zea mays L. PNR 6552 ). The model includes an extensive database of growth parameters for several crops (Annandale et al., 1999). This was compiled using data from both field research and literature. Guidelines for the determination of crop specific growth parameters required by the model were given by Jovanovic et al. (1999). The SWB model is available for use with Windows 95 on an IBM-PC or compatible computer. The minimum requirement is 16 Mb RAM and a CD-ROM drive. The time required to complete a seasonal simulation is 3 to 5 s on a Pentium 166. The program is supplied in executable code on CD, with a quick reference user guide manual making extensive use of graphics. Copies of the program are available through John G. Annandale, Dep. Plant Production and Soil Science, Univ. of Pretoria, 0001 Pretoria, South Africa; address: annan@scientia.up.ac.za. Reproduction and shipping costs are payable. The SWB model, which is used by several irrigation consultants in South Africa, was not specially modified or adapted for the purpose of the classroom presentation. Classroom Demonstration The theoretical background of the crop unit of SWB was discussed in a three-period lecture (each period was 50 min) to a class of seven students. The flow diagram shown in Fig. 1 was presented, and the daily calculations performed by the SWB model were discussed step-by-step. Particular attention was given to the concepts and procedures describing the mechanisms of crop growth and development, as well as to the crop specific growth parameters required as input. A week later, SWB was demonstrated using a data projector in a two-period classroom session. The students were shown how to edit crop input parameters, run the model, see the results in graphic format and print graphs. A simulation was done for irrigated maize grown in Bloemfontein (South Africa) using the crop parameters shown in Fig. 2. The SWB model makes extensive use of graphics and several output variables can be displayed. An example of output graphs is shown in Fig. 3, with measured (symbols) and simulated (solid lines) root depth, leaf area index, aboveground and harvestable dry Fig. 3. Measured (symbols) and simulated (lines) root depth (RD), leaf area index (LAI), aboveground dry matter (TDM, left), and harvestable dry matter production (HDM, right), as well as fractional interception of solar radiation (FI evap ) for irrigated maize (Zea mays L. PNR 6552 ). The parameters of the statistical analysis between measured and simulated data are number of observations (N), coefficient of determination (r 2 ), Willmott s index of agreement (D), root mean square error (RMSE), and mean absolute error (MAE). J. Nat. Resour. Life Sci. Educ., Vol. 29,

6 matter, and fractional interception of solar radiation. Parameters of the statistical analysis between measured and simulated data are shown in the top right corner of each graph. These are number of observations (N), coefficient of determination (r 2 ), Willmott s (1982) index of agreement (D), root mean square error (RMSE), and the mean absolute error (MAE). These statistical parameters were recommended by De Jager (1994) to assess a model s accuracy. Simulations were run again after changing the following crop specific growth parameters: K, DWR, E c, EMDD, FLDD, MTDD, TransDD, maximum leaf age, maximum RD, SLA, and PART. These parameters are of basic importance in describing the mechanisms of crop growth and development, and are very well known to crop modelers. Theoretical principles were then used to explain changes in the simulation results noted in the output graphs. It was interesting for the students to note that the interaction of many parameters caused output results that were different than they would expect from fairly straightforward single equations. For example, Fig. 4 presents output results obtained using a radiation use efficiency of kg MJ -1 as crop input parameter. This value of E c is half that used for the simulation in Fig. 3. Radiation use efficiency is used in SWB to calculate radiation-limited DM i (Eq. [9]). Simulated leaf area index and dry matter production were much lower than half the values simulated in Fig. 3. This indicates that by decreasing E c under irrigation (water nonlimiting), solar radiation is the dominant limiting resource for crop production. In addition, a lower production of leaf dry matter gives a smaller leaf area (Eq. [19]), which in turn intercepts less solar radiation that can be used by the canopy for the production of dry matter. A homework exercise was assigned. The questions are summarized in Table 1. For Question 1, the task was to obtain the lowest possible total MAE for the simulation (Fig. 3), by starting with a poor fit and changing the values of crop specific growth parameters. The tasks of the other homework questions were to interpret output results and logically explain processes. The students were asked to use the study material from previous course work, and encouraged to use the model to answer the questions. In addition, study material with a detailed description of the crop unit of SWB was distributed in written format with references. Recommended literature included a series of papers published in Agronomy Journal, where several scientists gave a general review on the use, importance, advantages, and disadvantages of crop modeling (Baker, 1996; Boote et al., 1996; Monteith, 1996; Passioura, 1996; Sinclair and Seligman, 1996). Copies of the model were made available on CD s and in the students computer room. The students were given 2 wk to complete the homework and hand in the answers. A questionnaire was also distributed to determine whether the materials and methods used in this classroom demonstration were useful in improving their understanding of the topics covered in the course. The authors were also very open to any suggestions or criticism. The questions for the students evaluation are summarized in Table 2. Fig. 4. Measured (symbols) and simulated (lines) root depth (RD), leaf area index (LAI), aboveground dry matter (TDM, left), and harvestable dry matter production (HDM, right), as well as fractional interception of solar radiation (FI evap ) for irrigated maize (Zea mays L. PNR 6552 ). The parameters of the statistical analysis between measured and simulated data are number of observations (N), coefficient of determination (r 2 ), Willmott s index of agreement (D), root mean square error (RMSE), and mean absolute error (MAE). The radiation use efficiency was kg MJ -1 (half the value used for the simulation in Fig. 3). 28 J. Nat. Resour. Life Sci. Educ., Vol. 29, 2000

7 Table 1. Homework exercise, points assigned, and students score. Questions Points Score and SD, % 1. Use irrigated maize (Zea mays L. PNR 6552 ) planted on 14 Dec at Bloemfontein (South Africa) for your seasonal simulation with the Soil Water Balance (SWB) model. Change the crop specific growth parameters listed below, until the simulations of leaf area index, root depth, aboveground dry matter, and harvestable dry matter, as well as fractional interception of radiation (canopy cover) fit measured data points best. The mean absolute error (MAE) between measured and simulated data should be a minimum. Bonus points (+10%) will be assigned to the student with the lowest total MAE for the simulations. Print the output graphs to show your results, and give the values of the crop specific growth parameters that you used ± 11.7 Change the values of the following crop specific growth parameters (permissible range shown in brackets): i) Canopy radiation extinction coefficient ( ). ii) Dry matter/water ratio (2 9 Pa). iii) Radiation use efficiency ( kg MJ -1 ). iv) Growing day degrees for emergence (0 100 d C), completion of vegetative growth ( d C), maturity ( d C), transition period between vegetative and reproductive growth ( d C), and leaf senescence ( d C). v) Maximum root depth ( m). vi) Specific leaf area (5 20 m 2 kg -1 ). vii) Leaf-stem partitioning parameter (0.1 2 m 2 kg -1 ). After you have run several simulations, answer the following questions: 2. Discuss thermal time requirements for the completion of phenological stages ± How does the canopy radiation extinction coefficient affect crop growth? ± What is the difference between transpiration-limited and radiation-limited dry matter production? ± How does canopy cover affect the production of dry matter? 2 50 ± How does assimilate partitioning occur after flowering? ± How does assimilate partitioning change by changing the leaf-stem partitioning parameter? ± How does the transition period from vegetative to reproductive growth affect the final yield? ± Explain the concepts of leaf area index and specific leaf area ± How does leaf age affect the production of dry matter? ± How does the maximum root depth affect crop growth? ± 44.0 RESULTS The homework exercise was marked and results statistically processed. The results are shown in Table 1. The sensitivity analysis (Question 1) helped the students to get more familiar with the mechanisms of crop growth and the values to expect for each crop parameter (permissible ranges were given in Table 1), as well as with computer modeling. This was the most successful part of the homework exercise judging from the score. The lowest total MAE for the simulation was 17%, and the student who achieved this had 10% bonus points added to his score for this exercise. The homework questions did not generally present a problem for the students, except Questions 3 and 10, which were both related to canopy structure and leaf senescence. For Question 3, the students were expected to discuss the estimation of FI transp and FI evap for the partitioning of energy into potential soil evaporation and potential transpiration (Eq. [5] and [6]). None of the students explained the partitioning of energy in the presence of senesced leaves. In response to Question 10, the students did not correctly describe the physiological reaction of enhanced leaf senescence under conditions of water stress (Eq. [20] and [21]). The students score was high for Questions 2 and 4. This was encouraging, as the students were expected to discuss two processes that are of basic importance in modeling crop growth and development, as well as the soil water balance. The students score was variable but satisfactory for the other homework questions. Statistical analyses of the homework results gave an average score of 69.4% and a standard deviation of 9% (the lowest score was 58.3% and the highest 87.5%). None of the students had a score lower than 50%, which was arbitrarily assumed to be the pass mark. According to the results of the questionnaire, the theory lecture followed by the classroom demonstration of SWB was well received by the students (Table 2), and the use of computers as instruction tools should be appreciated. The students Table 2. Students perceptions questionnaire. Question Answer 1. The theoretical background information was useful. 4.8 ± The style of presentation, through the step-by-step discussion of the flow diagram, was appropriate. 4.5 ± The study material supplied was adequate. 4.3 ± The SWB (Soil Water Balance) model was easy to use. 4±0 5. The graphics of SWB added to your understanding of the theory. 4.3 ± Your impression of the potential of computers for instruction is positive. 4.8 ± How many hours did you spend on the SWB exercise? 4.3 ± Any suggestions to improve the practical on modeling? -- 1 = Strongly disagree; 2 = disagree; 3 = indecisive; 4 = agree; 5 = strongly agree. agreed that the SWB model is user-friendly enough, and that the demonstration added to their understanding of the theory. The number of hours the students spent working on the exercise indicated that the homework load was not excessive. It was disappointing that few suggestions were made for the improvement of the SWB classroom demonstration. The students suggested a classroom discussion after returning the homework, and that more time should be spent on modeling in the crop physiology course. CONCLUSIONS The SWB computer model is a useful tool for providing instruction to crop physiology students. A quantitative approach helped to teach the students the principles of how crops grow and develop. The students were shown how the interaction of many parameters caused results that were different than they would expect from a single, fairly straightforward relationship between variables. In natural systems, the quantification of complex interactions, which include the effect of a large number of parameters, is possible only with computer models. The method included a lecture portion where crop physiological principles were discussed with the aid of a flow diagram of the crop unit of SWB. The demonstration of the J. Nat. 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8 SWB model was easy to set up, and required only a computer and data projector. The homework exercise was developed around topics familiar to the students. The SWB model provides the chance to create practical homework assignments on many different topics. The method allows the lecturer to cover more material in a given amount of time compared with traditional teaching methods, and is also beneficial for model developers to test the user-friendliness of their models. The classroom demonstration of the SWB model was well received by the students. The results of the homework were encouraging for both the instructors and students. In particular, the principles of mechanistic crop growth and development modeling appeared to be well understood. More attention should be given to explaining the process of leaf senescence, the effect of water stress on leaf aging, and the implications on the partitioning of energy at the surface. APPENDIX List of Symbols DM Dry matter production (kg m -2 ) DWR Dry matter/water ratio (Pa) E Evaporation (mm) E c Radiation use efficiency (kg MJ -1 ) EMDD Emergence day degrees (d C) ETo Penman-Monteith grass reference evapotranspiration (mm) FI evap Fractional interception of radiation by photosynthetically active and senesced leaves FI transp Fractional interception of radiation by photosynthetically active leaves f l Leaf partitioning factor FLDD Day degrees at end of vegetative growth (d C) f r Fraction of dry matter partitioned to roots GDD Growing day degrees (d C) Hc Crop height (m) Hc max Maximum crop height (m) HDM Harvestable dry matter (kg m -2 ) K Canopy solar radiation extinction coefficient LAI Leaf area index LAIage i Age of leaf area index generated on day i (d C) LDM Leaf dry matter (kg m -2 ) MTDD Maturity day degrees (d C) PART Leaf-stem partitioning parameter (m 2 kg -1 ) PET Potential evapotranspiration (mm) RD Root depth (m) RDM Root dry matter (kg m -2 ) RGR Root growth rate (m 2 kg -1/2 ) rpf Reproductive partitioning fraction R s Solar radiation for a particular day (MJ m -2 ) SDM Stem dry matter (kg m -2 ) SI Stress index SLA Specific leaf area (m 2 kg -1 ) SWB Soil Water Balance model T Actual transpiration (mm) T avg Daily average air temperature ( C) T b Base temperature ( C) T cutoff Cutoff temperature ( C) TDM Aboveground dry matter (kg m -2 ) T f Temperature factor for light-limited crop growth T lo Temperature for optimum light-limited crop growth ( C) TransDD Day degrees of transition period from vegetative to reproductive growth (d C) Transl VPD wsf ylai Factor determining translocation of dry matter from stem to grain Vapor pressure deficit (Pa) Water stress factor Leaf area index of senesced leaves REFERENCES Allen, R.G., L.S. Pereira, D. Raes, and M. Smith Crop evapotranspiration. Guidelines for computing crop water requirements. Irrigation and Drainage Pap. 56. FAO, Rome, Italy. Annandale, J.G., N. Benadé, N.Z. Jovanovic, J.M. Steyn, and N. Du Sautoy Facilitating irrigation scheduling by means of the Soil Water Balance model. Water Research Commission Rep. K5/753/1/99. Water Research Commission, Pretoria, South Africa. Baker, J.M Use and abuse of crop simulation models. Agron. J. 88:689. Boote, K.J., J.W. Jones, and N.B. Pickering Potential uses and limitations of crop models. Agron. J. 88: Campbell, G.S., and R. Diaz Simplified soil water balance models to predict crop transpiration. p In F.R. Bidinger and C. Johansen (ed.) Drought research priorities for the dryland tropics. ICRISAT, India. Campbell, G.S., and J.M. Norman An introduction to environmental biophysics. 2nd ed. Springer, New York. Campbell, G.S., and C.O. Stockle Prediction and simulation of water use in agricultural systems. p In D.R. 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Conf. on Evapotranspiration and Irrigation Scheduling, San Antonio, TX. 3 6 Nov ASAE, St. Joseph, MI. Tanner, C.B., and T.R. Sinclair Efficient water use in crop production: research or re-search? In H.M. Taylor et al. (ed.) Limitations to efficient water use in crop production. ASA, CSSA, and SSSA, Madison, WI. Torbert, H.A., M.G. Huck, and R.G. Hoeft Simulation of soil plant nitrogen interactions for educational purposes. J. Nat. Resour. Life Sci. Educ. 23: Willmott, C.J Some comments on the evaluation of model performance. Bull. Am. Meterol. Soc. 63: J. Nat. Resour. Life Sci. Educ., Vol. 29, 2000