AOBPreview originally published online on January 3, 2006
Annals of Botany 2006 97(3):377-388; doi:10.1093/aob/mcj048
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ALAMEDA, a Structural–Functional Model for Faba Bean Crops: Morphological Parameterization and Verification
Departamento de Producción Vegetal: Fitotecnia, Escuela Técnica Superior de Ingenieros Agrónomos, Universidad Politécnica de Madrid, Ciudad Universitaria, 28040 Madrid, Spain
* For correspondence. E-mail ines.minguez{at}upm.es
Received: 2 August 2005 Returned for revision: 29 September 2005 Accepted: 21 November 2005 Published electronically: 3 January 2006
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
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• Background Plant structural (i.e. architectural) models explicitly describe plant morphology by providing detailed descriptions of the display of leaf and stem surfaces within heterogeneous canopies and thus provide the opportunity for modelling the functioning of plant organs in their microenvironments. The outcome is a class of structural–functional crop models that combines advantages of current structural and process approaches to crop modelling. ALAMEDA is such a model.
• Methods The formalism of Lindenmayer systems (L-systems) was chosen for the development of a structural model of the faba bean canopy, providing both numerical and dynamic graphical outputs. It was parameterized according to the results obtained through detailed morphological and phenological descriptions that capture the detailed geometry and topology of the crop. The analysis distinguishes between relationships of general application for all sowing dates and stem ranks and others valid only for all stems of a single crop cycle.
• Results and Conclusions The results reveal that in faba bean, structural parameterization valid for the entire plant may be drawn from a single stem. ALAMEDA was formed by linking the structural model to the growth model ‘Simulation d'Allongement des Feuilles’ (SAF) with the ability to simulate approx. 3500 crop organs and components of a group of nine plants. Model performance was verified for organ length, plant height and leaf area. The L-system formalism was able to capture the complex architecture of canopy leaf area of this indeterminate crop and, with the growth relationships, generate a 3D dynamic crop simulation. Future development and improvement of the model are discussed.
Key words: Faba bean, architecture, allometric relationships, structural-functional modelling, virtual plants, parameterization, L-systems, growth model
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
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Most process crop models consider the canopy as a homogenous medium within which plant organs and leaf area are not spatially described. Functional–structural plant models (FSPMs) allow the inclusion of geometrical and topological specifications and organ inter-relationships that is not possible in most crop models. Together with equations that drive physiological process, these models can represent the growth processes of the individual plant organs in response to their microenvironments. This ability is especially important in early stages of crop cycles when canopies are most heterogeneous. Short time steps, less than the one day characteristic of current crop process models, can then be implemented for all processes (e.g. Connor and Fereres, 1999
). These models can also simulate plant competition and be used to perform virtual experiments (Prusinkiewicz, 2004), including at the organ level (Birch et al., 2003), to improve understanding of interactions between various growth processes (Godin and Sinoquet, 2005).
Several FSPMs have been reported, the AMAP family of models (Atelier de Modélisation de l'Architecture des Plantes) and the Lindenmayer systems (L-systems) being the main approaches. The AMAP family (Reffye et al., 1997) was developed to study tree architecture and includes water transport and assimilate partitioning. The L-systems were introduced by Lindenmayer in 1968 as a mathematical theory of multicellular development based on generative grammars (Lindenmayer, 1968). They have been applied to plant modelling (Prusinkiewicz and Lindenmayer, 1990; Prusinkiewicz, 1994), and an L-system describing the basic structure of a faba bean plant was described in Díaz-Ambrona et al. (1998). Examples of FSPM L-system-based models are those of Fraxinus pennsylvanica (Prusinkiewicz et al., 1993), ADEL-maize (Fournier and Andrieu, 1998), cotton (Hanan and Hearn, 2003), soybean (Pachepsky et al., 2004) and rice (Watanabe et al., 2005). These models were developed with various procedures to simulate growth, including empirically fitted functions and interception and use efficiency of radiation.
The construction of a functional–structural crop model requires: (a) an accurate description of geometrical and topological architecture of the plant and canopy at a scale appropriate to the objectives of the model; and (b) functions describing growth processes of individual organs.
The first requirement is a morphological study, because the main morphological knowledge of crop species is ‘still framed within the monographic studies from the first half of the 20th century’ (Moulia et al., 1999). For faba bean, such modelling requires descriptions that do not exist in the literature, in terms of number and final organ size, and time and rate of organ appearance. Information on time course evolution of organs is especially scarce. Some descriptions of faba bean morphology can be found in Millet (1970), Hebblethwaite (1983), Bond et al. (1985) and Duc (1997), but, to our knowledge, there is no coherent data set describing a faba bean canopy in detail.
Some of the process functions to meet the second requirement can be found in functional simulation models of the faba bean crop that do not simulate crop structure (Stützel, 1995a, b; Manschadi et al., 1998a, b; Boote et al., 2002; Turpin et al., 2003; and the simpler model by Dennett and Ishag, 1998; Ishag and Dennett, 1998).
Also available is the SAF (Simulation d'Allongement des Feuilles) model (Durand et al., 1999). The leaf model of Festuca arundinacea developed by Fournier (2001) links the growth model SAF to an L-system, although this is done at the tiller level only, without the simulation of a whole plant or canopy. The SAF model is based on sigmoid equations used for modelling aspects of the growth of kiwi fruit (Gandar et al., 1996) and it has been adapted recently to simulate elongation and phyllochron (PH) of grass leaf in Fournier et al. (2005). Renton et al. (2005) present a similar approach linking a canonical model to an L-system.
The aim of this work is to develop and verify a functional–structural model of the faba bean crop, using the L-system approach. A morphological description of the faba bean provided the equations and parameterizations needed for an L-system linked to the SAF model to simulate the growth of internodes, petioles and leaflets of faba bean. It contributes to a need, identified by Birch et al. (2003), for the development of models of the growth of vegetative organs that have generic application across species and can be aggregated to form plant and crop structural models.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
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Morphological study
Faba bean (Vicia faba L.) were sown in autumn (November 1999, referred to as A99, and October 2000, A00) and winter (March 2000, W00, and February 2001, W01) at the Escuela Técnica Superior de Ingenieros Agrónomos, of the Universidad Politécnica of Madrid, Spain (altitude: 595 m, 40°26'36'' N). The crops were harvested in June/July. Seed of the cultivar Alameda were sown at commercial crop densities at 0·14–0·15 m intervals in rows 0·2 m apart in one 6 x 8 m plot on each sowing date, on an Entisol of the great group Arent. Cultivar Alameda has an indeterminate growth habit and has been used as a reference for this crop in Central and Southern Spain. The plots were irrigated and a thermal cover was used in November 1999 to ensure the emergence under the low temperatures at that time. Phenological events (emergence, organ appearance, flowering, maturity and senescence), air temperature and photosynthetically active radiation (PAR) at 2 m height were recorded for every sowing date. Thermal units (TUs) and photothermal units (PTUs) were calculated above a base of 0 °C as done previously by Stützel (1995a) and Boote et al. (2002)
. PAR interception, relative to incident, was measured on several occasions during each field experiment at successive 10 cm vertical positions within the canopy, with three replications per measurement taken with a 0·8 m long ceptometer (Sunfleck Ceptometer, Decagon, Cambridge, UK), over an area of 0·3 m2 that included 10 plants (Ridao et al., 1996).
In the 1999–2000 cycle, 19 plants from each plot were digitalized during the crop cycle to generate their 3D structures (Hanan and Room, 1997). The 3D cartesian coordinates and Euler orientation angles of nodes (one point), petioles, leaflets, flowers, pods (two points per structure) and stem cross-section (four points) were measured every 2–4 weeks, depending on the crop growth rate, registering up to 1000 points per plant and measurement. Digitizing was performed with the POL95 software (Adam, 1998) and the digitizer Fastrak 3SPACE model (by Polhemus Incorporated, Colchester, VT, USA). The phyllotactic (
) and pitch angles (
), defined in Fig. 1 were used to describe leaf positions.
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Destructive measurements in all plots were made every 2 weeks to determine the number and position of organs, internode length and leaf area. The number of senescent leaves was recorded in A99. Internode lengths were measured separately for every sowing date and for primary, secondary and tertiary stems. For all sowing dates, in both growing seasons, leaf area was measured in 0·10 m deep canopy layers above a soil area of 0·3 m2. Leaf area was also measured per leaf on all stems (average of six plants) with a ‘Leaf Area Meter' Delta T MK2 model (by Delta T Devices, Cambridge, UK). Leaf area index (LAI) was determined with a ‘Plant Canopy Analyser’ model LAI-2000 (by LI-COR Incorporation, Lincoln, NE, USA; LI-COR, 1991
) previously calibrated for faba bean.
Data processing and analysis
Floradig software (Hanan and Room, 2000) was used for calculating the lengths of the digitized objects. The number and lengths of vegetative organs were studied as functions of both thermal time and node rank. Linear and non-linear regressions were used to determine the parameters of functions describing relationships between the number of organs and thermal time and of alometries between plant components. Microsoft Office Excel 2003 (Copyright 1985–2003 Microsoft Corporation) was used for characterization of the regressions by the R2 coefficient and the root mean s.d. Means and s.d. of maximum length of flowers and pods, and leaf angles, were also calculated. Total leaf length (TLL) was defined as the sum of petiole (P) and leaflet lengths (Fig. 1) looking for a measure that considered all leaf parts, independently of their position.
Modelling
In an L-systems-based model, the canopy is constructed by aggregation of individual plants, themselves built of connected modules (Table 1, Production 0). The ‘Production rules’ are statements that define the dynamics of each module and the addition of new modules at each time step. 3D simulations of canopy structure can be generated. The time step is chosen according to the requirement of the functions that define the physiological processes. In this way, the dynamics of each plant organ are simulated, providing, for example, a spatially localized estimation of leaf area. Variation within and between plants can then be simulated.
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ALAMEDA operates at the leaflet scale to explore the improvement that L-systems can provide to crop modelling by including a detailed description of leaf surfaces. Therefore, buds, internodes, the parts of the petiole limited by the attachment of each pair of leaflets, leaflets, inflorescences, pedicels, flowers and pods, canopy and plant were defined as modules. In the context of L-systems, variables are various time meters, temperature, organ number, dimension, etc., and can be used as ‘parameters’ ascribed to each plant module, depending on their specific function in each statement of the L-system. Parameters make possible the exchange of information across the L-system, and between canopy and environment, and determine the exact position and function of every plant component or module, at every moment. ALAMEDA contains a parametric, environment-sensitive, L-system, in which certain parameters define age and environment, as chronological (t, dt), thermal and photothermal time, while others define the rank of each organ and its phenological stage, shape, size evolution and position angles. The model is also stochastic because variation among and within plants is considered through probabilities and random variables. This applies for stem number per plant, leaf angles, number of flowers and pods per node, and, progression of number of leaflets per leaf.
The ALAMEDA code can be found in Ruiz-Ramos (2003) and is also available from the authors (Ruiz-Ramos and Mínguez, 2004). Table 1 shows a simplified version of certain code statements. The programming structure includes a main L-system to simulate the stems, and two sub-L-systems to simulate leaves and reproductive organs separately.
Growth and development
A plant can have up to four stems in the model, generated by the apical and lateral buds (Table 1, Production 1). An apical bud accumulates TU until the next PH has been fulfilled (Table 1, Production 2); from this moment, the bud will produce a new internode, a leaf and an alternate bud (Table 1, Production 3). One sub-L-system drives leaf development and growth, and each leaf will be composed of a petiole and the leaflets (Table 1, Productions 3 and 6). Leaves bear a different number of leaflets depending on their position on the stem and the existence of flowers. Another sub-L-system generates flowers and pods once a reproductive module reaches flowering or pod setting time, in PTU, respectively. Flower and pod size is described by a sigmoid curve function of TU after senescence, fitted empirically up to their maximum size. The buds of secondary stems follow the same dynamics as the main stem after applying a delay between stems, expressed in TU.
In ALAMEDA, the growth of vegetative organs is simulated by linking the L-system to the empirical growth model SAF (Durand et al., 1999). Briefly, SAF consists of differential equations describing the elongation rate (Gandar et al., 1996) of each organ partitioned into three one-dimensional zones, i.e. division (DZ), elongation-only (EOZ) and mature (M). In each internode, petiole or leaflet, the change in length of EOZ is the balance between the inflow from DZ, the expansion of tissues within EOZ itself and an outflow to M, and, at each step, effluxes are proportional to the length of DZ and EOZ, respectively. These dynamics produce a sigmoid pattern of growth. The relative elongation rates and the steady rate of ageing of the system are linear functions of temperature. Each organ is given an initial length related to the size of the corresponding apical meristem (Lyndon, 1998), resulting in a rising initial organ length from plant base to the top. SAF is very sensitive to these initial lengths, and final organ length depends strongly on them.
Within ALAMEDA, a sub-routine called SOLVFABA (equation SOLVer for FABA bean) links the growth model SAF to the L-system (Table 1, Productions 4, 7, 8 and 9). It solves the differential equations of SAF for every vegetative organ until their senescence and updates the values of the DZ, EOZ and M compartments, as well as the total length of the organ at every time step. SOLVFABA is organized in three parts: Solvfabastem.h and Solvfabaleaf.h, which apply the Euler method to calculate the SAF equations for internodes and leaf components respectively, and SolvfabaTLL.h that calculates leaflet and petiole dimensions, and leaf area from TLL (Fig. 1). The expression of the evolution of morphological dimensions as a function of TU provides the link to the temperature-based model SAF.
Simulation
The graphical interpretation is updated at each time step by the ‘Homomosfism’ production rules, drawing each internode longer as compartment sizes increase (Table 1, Production 5). For model construction, L-system programming, iteration and visualization were performed with the L-Studio (Prusinkiewicz, 2000) and Cpfg (Mech, 1998) software. Model programming combines Cpfg standard syntax and part of C language.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
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Morphological relationships and model parameterization
The parameterization for growth and phenology is summarized in Table 2 and for canopy composition in Table 3. The number of data sets from which each input was calculated is also specified.
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Phenology
Timing of phenological events was introduced as constants, as follows. Secondary stems appeared with a successive delay that was expressed as multiples of the PH (‘DEL2’, ‘DEL3’ and ‘DEL4’ in Table 2); flowering was simulated according to PTU, so for synchrony, other phenological events were expressed both in TU and PTU as well as progress rates of reproductive organ setting (Table 2).
Stems and internodes
Stem number per plant was introduced in the model according to the mean probability distribution of the four data sets from field experiments (Table 3). The internode length distribution followed the same pattern for all stems of the same sowing date, but the location of the longest (maxima) internodes within the stem varied among sowing dates, as can be observed in Fig. 2. This plot compares stems 1–3 of the A99 to A00 sowing, illustrating the crop plasticity. Although this version of ALAMEDA simulates internode growth without specifying the growing season, final plant height was within the reported values of 0·5–2 m (Bond et al., 1985) and plants displayed the general morphology described by Millet (1970).
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Number of leaves and leaflets
Alternate leaves, one leaf per node, appeared from the third node of the main stem, and from the first node of the other stems. Figure 3 shows the progression of leaf production and maximum leaf number on the main stem as a function of thermal time, as derived from all the field experiments. A simple linear regression (Fig. 3) was chosen because its slope provides an estimate of leaf appearance rate, and its inverse, the PH, expressed in TUs. This method provided PHs of 57·8, 59·2 and 66·7 °Cd for stems 1, 2 and 3, respectively. These values were obtained for stages of development until approx. 900 °Cd, i.e. commencement of pod growth, when the PH was constant, (Fig. 3, equation b). The average PH used as an input in the model is shown in Table 2. Final number of leaves were expressed as random variables depending on the stem rank (‘LEAVES’ and LEAVESf' in Table 3). The number of senescent leaves could also be explained by linear regression with thermal time, and its corresponding rate was obtained; this rate was used as an input to the model (‘PS’ in Table 2).
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The number of leaflets per leaf increased from two up to six, occasionally to eight, from the bottom to the top of the plant, as also reported by Duc (1997)
, with intermediate regions with a constant number of leaflets. Even numbers of leaflets were more frequent, but odd numbers were not uncommon. Secondary stems for all sowing dates presented the same distribution of leaflet number as the main stems from leaf 2 upwards, with an offset of one leaf (Table 3, input ‘Chi – 1’). The distribution of the number of leaflets was expressed as a rounded-up average of the main stem values together with the s.d., to include variation among plants, using the mean data from stems 1–3 from A99 and W00 sowing dates (‘Chi’ variables in Table 3). The progression of the number of leaflets could also have been represented by a logarithmic function of the leaf number [y = 2·7Ln(x) – 1·9; R2 = 0·9; root mean square deviation (RMSD) = 0·42] but it was not chosen for this version of the model.
Leaf relationships
All vegetative organs followed parallel patterns, as shown in Fig. 4 for the distributions of TTL, petiole length (P) and length of the first part of the petiole (A), and those shown in Fig. 2 for internodes of A99. They revealed the same region of maxima within the stem, pointed out by the vertical lines in Fig. 4. P comprised 30 % of TLL for basal leaves, increasing up to 40 % for upper leaves in all stems as is shown in Fig. 5 (regression a). In the model, the variables corresponding to the ‘Petiole length’ section in Table 4 were calculated with this allometric relationship and the equation of ‘COEF’ (R2 = 0·91; RMSD = 5·7). This equation expresses the variation of the ratio A/P with leaf rank and is applied to A for leaves with more than two leaflets. Leaflet lengths (‘fols’ and ‘fol’) were derived from TLL (Table 4).
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Leaf area on stems of the same sowing date showed parallel curves of similar shape. Comparing curves from different sowing dates, maximum leaf area (between 62 and 78 cm2) corresponded to the region of leaves 14–19 of the main stem regardless of sowing date. In the case of secondary stems, this maximum was displaced 1–3 leaves downwards. These maxima correspond to the mid- and upper canopy layers where light interception is maximum (results not shown). This stability in the location of maximum leaf area was not observed for internode or leaf lengths. A basic allometric relationship was found between leaf area and TTL for every stem for both sowing dates and was introduced in the model (‘LA’ equations in Table 4). In Fig. 5, two of the equations tested, linear (b) and logarithmic (c), are presented for the main stem. The logarithmic equation provided a better fit, but values of leaf area were negative for small values of TLL so the linear equation (b) was chosen as the input for the main stem (‘LA’ for ‘Stem 1’ in Table 4). This effect was weaker for secondary stems, and the logarithmic fit was chosen. R2 coefficients were equal to 0·84 and 0·7, and RMSD equal to 4·1 and 6·6, respectively, for stems 2 and 3 (‘LA’ for ‘Stem2’ and ‘Stem 3’ in Table 4).
Leaf angles
The phyllotactic and pitch angles describe leaf position in the canopy. No significant differences were found for phyllotactic angles (test F at 95% confidence level) for the various stems of the same sowing date, while an offset of 30° was found between stems of autumn (A99) and winter (W00) sowings (Table 3). From these angles, three and four dominant phyllotactic orientations arose, for autumn and winter sowings, respectively, but with a large s.d. The pitch angle of leaves changed as a function of leaf number along the stem, but there were no significant differences (test F at 95% confidence level) between stems or sowing dates. A linear regression (R2 = 0·77; RMSD = 4·8) corresponding to the W00 sowing from the main stem was the input for the model (equation for ‘pitch’ in Table 3).
Reproductive development
Reproductive structures presented the following topology and geometry: the first flower occurred 1–2 nodes beyond the region where the number of leaflets per leaf increased and this offset was introduced depending on the stem rank (‘four’ and ‘four1’ variables in Table 3). These values, and those referred to the first pod, were greater for the main stems, decreasing with stem rank, and varying within 1–3 nodes. Each inflorescence produced 3–5 flowers and from two to five pods, in agreement with Duke (1981). Reproductive organ length was always decreasing with stem, with a mean decrease of 10% for flowers and between 8 and 25 % for pods, measurements including pedicels. Mean pod lengths were smaller than those reported by Duke (1981) (pods ranging from 8 to 20 cm long) and showed a greater variability than flower size. The model inputs related to first leaf-bearing reproductive structures; their number per node and length are shown in Tables 2 and 3. Variations among stems and s.d. data provide a realistic description of the canopy.
SAF parameterization
Parameterization of the evolution of organ length and leaf area was made with data only from the A99 data set. Maximum dimensions of vegetative organs (maximum internode length and maximum TTL) were not an input for the model. According to the above results, the parameterization considered that all stems of a plant behaved similarly; therefore, all internodes and all leaves were simulated with one set of SAF parameters determined empirically (shown in Table 5). This comprised one sub-set for internodes and another sub-set for leaves, consisting of coefficients to compute the linear variation with temperature of the steady rate of ageing of the system and of the relative elongation rates. Table 4 shows other equations used to solve SAF within the L-system: the functions to calculate the initial length of every organ (‘LINI’ for internodes and ‘LINIf’ for leaves) that were fitted empirically, and equations to update the values of internode and leaf length at every time step (‘L’ and ‘TLL’, respectively).
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Simulation
3D-simulation and numerical outputs
3D dynamic simulations are shown in Fig. 6A in vertical projection with plants sown at field densities (rows 0·2 m apart and 0·14–0·15 m within the row). Visualization is easier for plants at greater spacing (0·4 m x 0·2 m) (Fig. 6B and C). Flowering, at 1130 °Cd, and delay between stems are shown in Fig. 6B, while pod setting and progress of senescence from the base of the plant are shown in Fig. 6C (1350 °Cd). The simulation of one stem with the internodes at different elongation stages is presented in Fig. 6D. The numerical output of the model can be customized so that chosen variables and parameters for each plant component can be exported to text files at the frequency required.
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Verification
Verification of the model was made for plant height, organ length, leaf area and LAI, and was carried out with the three data sets not used in the parameterization of these variables. 3D simulation provided a visual verification, since perception by the human eye of the virtual canopy is itself a test of radiation interception and reflection (Adam et al., 2004
). The simulated crop reached the observed final plant height for both sowing dates (Fig. 7), although the model underestimated the progress of plant height, with internodes elongating slowly during the middle of the crop cycle.
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Figure 8A shows the verification of the maximum area of leaves for the main stem of the W01 sowing date. Leaf area from various stems is presented in Fig. 7 together with total leaf area of three plants from the different sowing dates. The evolution of leaf area is presented for the main stem in Fig. 8B. For the W01 sowing, leaf area was underestimated during the middle part of the cycle. This sowing presented the smallest PH (45 °Cd compared with the average 58 °Cd) so that the model, parameterized with a mean PH, generated leaf area more slowly than was observed in this case.
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LAI evolution was simulated taking into account the stem number distribution in each plot. The results are shown in Fig. 8C together with measured data of A00 and W01. In A00, simulated data matched measured data considering their s.d.; however, in W01, the sensitivity to PH could also be observed as mentioned above, producing an underestimation of LAI in the central part of the cycle. Results for the simulation of radiation interception were obtained for the first two versions of ALAMEDA (Ruiz-Ramos et al., 2000
, 2001). Computation was made externally to the model, revealing no significant differences (at the 95% confidence level) between measured and simulated values of incident PAR on layers within the canopy. A linear regression between measured (x) and simulated values (y) was also obtained (y = 0·83x – 67·9; R2 = 0·98; RMSD = 57·3), indicating that underprediction occurs at low observed values, the model is responsive (slope = 0·83) and the scatter is low (high R2). |
DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
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This work has focused on obtaining simulations of faba bean canopies at field densities using field experiments specifically designed for the purpose. The digitizing technique was able to capture, using up to 1000 points per plant, the detailed geometry and topology of the faba bean canopy, and describe its development in time. Where possible, the newly collected information was compared with previous reports, revealing general agreement with them.
The morphological study stressed the high plasticity of faba bean plants, which makes it difficult to obtain universal relationships on organ number and size, for all stems and ranks. However, allometric functions relating leaf area to leaf length, the number of leaflets per leaf, the PH, the leaf pitch angle, and the location of maximum organ length and leaf area per stem were applicable to all sowing dates and stem rank. Another group of relationships was valid for all stems of individual crop cycles. Information on morphological parameters such as length, leaf area and angles, valid for the entire plant, may be drawn from the main or a single secondary stem.
All descriptive equations obtained were useful for a coherent encoding of the plant structure. Further testing may be needed, since to formulate the structural part of the model, every plant organ had to be initialized and described specifically from the beginning. Further study of other plant parts may reveal useful relationships and allometries, and is needed for generalization of some of the relationships found. Watanabe et al. (2005) have used an allometry to calculate rice leaf blade width, and other leaf allometries have been found across various species of dicotyledons, including legumes such as soybean (Pachepsky et al., 2004), although it has not been reported for faba bean. Burton (2004) has suggested that the ubiquity of this form of relationship implies that the underlying processes of leaf growth are quantitatively similar across species.
Results including variation among stems and organs provide a description suitable for ‘realistic representation’ of the crop by an L-system-based model (Prusinkiewicz, 1994), that simulates canopy by aggregation of non-identical plants. However, the problem of a more mechanistic modelling of the number of organs in indeterminate crops is complex and still remains. We found that the number of stems in faba bean ranged from two to seven in a density trial (results not shown) and varied across sowing dates even at the same density, so further, more detailed, work is needed. Process crop models of legumes include parameters that define the maximum/minimum number and size of organs depending on the cultivar. Also, the number of reproductive nodes in peas has been studied by Dumoulin et al. (1994) and modelled by Roche et al. (1998). In contrast, in ALAMEDA, the maximum size of vegetative organs is not an input, but the result of the application of a growth model. Considering the results overall, improvements can be made in modelling leaf and stem number matching the indeterminate growth habit of the cultivar Alameda. This will require more accurate estimation of the PH and organ growth duration. Future versions of the model could include the specification of sowing season and genotype to accommodate this plasticity (Pachepsky et al., 2004). The number of stems per plant could then be calculated based on the corresponding stem distribution, or as a function of genotype descriptors (Stützel, 1995a). This could also allow application and testing of other results valid for different sowings, e.g. internode length distrubution, and phyllotaxis effect on radiation interception by the canopy.
ALAMEDA localizes the leaf area spatially, vertically through internode length simulation and horizontally through phyllotaxis simulation, but application of a radiation interception model to these outputs should clarify the improvement that this kind of model may offer to crop modelling (e.g. Chelle and Andrieu, 1998). Daily mean temperature was the only environmental variable included in the present work since it was the only variable considered in the growth model that was incorporated. As implementation of shorter time steps is now possible, higher temporal resolution of environmental variables could be considered in the near future. Also, a vertical profile of temperature, or temperature at the organ level, would permit a more accurate computation of the thermal time experienced by each organ (Birch et al., 2003). Further linkages with other process models, such as radiation interception, water balance and photosynthesis, will require the introduction of other variables. ALAMEDA has been built to allow for these linkages and accept inputs within the canopy either through a vertical profile or at each organ.
Certain physiological processes of existing functional models of faba bean can be included in sub-routines invoked by the L-system. An example of this methodology has been shown here by linking the L-system to an SAF model of application across various species, using parameters as information messengers between the models. Process models use variables representative of the mean state of a crop or a process at every time step (e.g. biomass in g m–2), while the L-system can deal with values for each plant organ. For instance, leaf area could be derived from the specific leaf area varying within the plant and during the crop cycle (Stützel, 1995b). However, the inclusion of several processes in a structural model still remains a challenge (Prusinkiewicz, 2004).
3D dynamic simulations have proven a useful tool in parameterization, providing visual feedback in detecting modelling gaps or mistakes, and have practical applications (Hanan and Hearn, 2003). In this work, parameterization of the most descriptive part of the model, the representation of reproductive organs, relied heavily on visual simulations. Phyllotaxis (approx. 120°), number of stems per plant, soil coverage and other canopy characteristics can then be analysed further by processing of 3D images as in VEGESTAR (Sinoquet et al., 1998; Adam et al., 2004).
Summarizing, five classes of plant components and organs (internodes, petioles, leaflets, flowers and pods) have been simulated, three of them functionally. For a simulation of nine plants, the model has calculated the lengths of approx. 550 internodes, 950 petioles and 2000 leaflets, i.e. approx. 3500 organs, illustrating the powerful and integrative nature of the L-system approach. The amount of data that the model can provide even in this early stage, and considering its future development, shows the need for designing specific software and statistical tools to deal with the numeric output (Godin and Sinoquet, 2005).
Finally, L-system formalism can codify the architecture of complex and indeterminate plants in ways that reflect both generality and variation of plants in a canopy built by aggregation of a limited number of non-identical plants. This work has shown the utility of an L-system, functional–structural model of faba bean, ALAMEDA, and has discussed ways to incorporate further process models. The L-system provides the basic conceptual and programme structures within which functional relationships can be connected, playing a role comparable with that which physical plant structure provides for physiological processes.
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We thank Dr B. Andrieu and Dr J.L. Durand from INRA (France) for their very useful comments. This work was financed by the CICYT through project number PB97-0569 of the Spanish Ministerio de Educación y Cultura. M.R.-R. had a grant from the Ministerio de Ciencia y Tecnología.
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LITERATURE CITED |
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