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Plant and Soil 190: 235–246, 1997. c 1997 Kluwer Academic Publishers. Printed in the Netherlands.
Modelling and simulation of the architecture and development of the oil-palm (Elaeis guineensis Jacq.) root system II. Estimation of root parameters using the RACINES postprocessor
Christophe Jourdan1 and Herv´e Rey2 1
D´epartement des Cultures P´erennes and 2 Unit´e de Mod´elisation des Plantes, Centre de Coop´eration Internationale en Recherche Agronomique pour le D´eveloppement (CIRAD), B.P. 5035, F-34032 Montpellier Cedex 1, France. 2 Corresponding author Received 26 June 1996. Accepted in revised form 27 February 1997
Key words: 3-D numerical models, Elaeis guineensis, RACINES postprocessor, root architecture, root development, simulation, stochastic models, voxellization
Abstract A stochastic model of oil-palm (Elaeis guineensis Jacq.) root system architecture and development has been developed. This model enabled us to create 3-D numerical models of complete root systems by simulation. The application of a postprocessor software, called , to these 3-D numerical models, provided an estimation of some parameters of plant root systems. The objective of this paper is to present oil-palm root characteristics as possible outputs of the application of this RACINES software. The outputs described in this article cover (i) spatial distribution of roots under plantation conditions, (ii) the estimation and distribution of total root biomass, per root type or per soil horizon and (iii) the location and quantification of absorbent surfaces. The computing techniques used were based on voxellization of space and creation of 3-D virtual sceneries exactly reproducing observed planting designs. By comparing the results of observations and simulations for spatial distribution (by trench wall density maps) and root biomasses (by real and virtual sampling) we were able to carry out additional numerical validations of the model. Introduction The advent of increasingly powerful computers has enabled the generation of increasingly complex tree structures: fractals, combinatorial trees, etc. (Fran¸con, 1991; Fran¸con and Lienhardt, 1994; Prusinkiewicz et al., 1988, Prusinkiewicz and Lindenmayer, 1990). New developments based on L-systems are now treating plants with more elaborate growth rules (Kurth, 1994; Shibusawa, 1994). AMAP (Plant architecture modelling workshop) models attempt to integrate qualitative (Hall´e and Oldeman, 1970; Hall´e et al., 1978) and quantitative (Reffye, 1979) botanical knowledge recently acquired in the field of plant architecture. A complete processing chain has thus been designed
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(Jourdan and Rey, 1997a), comprising morphological and architectural observation of the plant, acquisition of standing plant measurements, calculation of meristem functioning and growth simulation (Reffye et al., 1993). All these various stages have their own rules and all of them are indispensable for satisfactory and realistic simulation of a plant’s architecture. The software packages derived from this multidisciplinary approach can construct 3-D numerical models of plants and stands, which can serve as a basis for various applications (Dauzat, 1994; Reffye et al., 1995). Different mathematical models of root system growth and architecture have recently been developed with a view to modelling and simulating behaviour observed in situ, partially or totally representing the architecture and distribution of the root system in the soil, or even examining root system functioning. These
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236 models can be applied to various fields, including (i) studying the effects of environmental factors (Jones et al, 1991; Tardieu and Pellerin, 1991), particularly growth in a compacted soil (Grant, 1993a,b) or spatial distribution of the root system and root disease control (Pag`es et al., 1995), (ii) the aspects of root architecture on both the efficiency and potential of soil exploitation (Berntson, 1994; Fitter et al., 1991), and lastly (iii) combining growth models and water uptake; (Bland and Jones, 1992; Clausnitzer and Hopmans, 1994; Protopapas and Bras, 1987) or mineral uptake (Kelly et al., 1992) models. With oil palm, its large-scale cultivation raises various questions regarding, for example, root biomass quantification, or the precise location of different root types, especially their absorbent zones. The overall model of oil-palm root system architecture previously described (Jourdan and Rey, 1997a) was used to produce 3-D numerical models by simulation. The objective of this paper is to present oil-palm root characteristics in order to try to answer the previous queries. These characteristics are obtained as possible outputs of the application of a postprocessor software (called ) which is able to extract data from the 3-D numerical models.
Methods Once created, the 3-D numerical models described (Jourdan and Rey, 1997a), are processed in various ways to provide oil-palm root characteristics. The RACINES software was developed for this purpose.
Figure 1. Voxel space and partitioned oil-palm root system.
partitioned and represented by a set of voxels. A voxel is then said to be occupied when it is crossed by one or more root axes. The size of the elementary voxel is determined by the user (Figure 2) and may contain all or part of the root system. Using the partitioned virtual scenery, the RACINES software can supply a certain amount of data concerning either voxel characteristics (size, exact location accessible by its absolute coordinates in space) or the characteristics of what it contains (particularly the root type occupying the voxel, the root length, diameter, etc.). Spatial distribution
Voxel space technique Using an appropriate visualization software, a virtual scenery containing one or more root systems was created from one or more 3-D numerical models generated by AMAPsim. The scenery was then partitioned using the voxel space technique. This procedure was originally developed to cope with interactions between the above-ground parts of trees and their environment (shading, interference between branches, between crowns, obstacles) during their growth (Blaise, 1991; Green, 1989). According to Green (1989), a voxel space is a region of 3-D space partitioned into identical cubes called or . This technique is applicable to root systems (Figure 1). The root axes contained in the voxel space are thus
Root density maps were created from field observations on several trench walls and for palms of various ages. This was done using a 1 m1 m grid divided into 10 cm10 cm elementary squares, installed vertically at different distances (0.2 m, 0.5 m, 1 m, 1.5 m and 2 m) from palms aged 3, 10 and 20 years. Using the visualization software, a virtual grid, identical to the one used in the plantation, was created, along with virtual sceneries reproducing the study designs in the field. It was thus possible to put 3-D numerical models of the root systems inside these virtual sceneries. All these data were used as inputs in the RACINES postprocessor software. The outputs obtained consisted of simulated root density maps. Model validation was then performed by comparison of the simulated and observed maps.
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Figure 2. Root system and different sizes of the voxel used. The voxel may contain the entire root system (A) or just part of it limited by a given soil horizon (B) or by vertical core sampling (C).
However, this validation was not obtained by individually comparing the elementary squares of the real maps and the simulated maps, since the spatial distribution of roots observed in the field was highly variable. As our stochastic model was able to restore this variability, we were led to work on a population of observed and simulated maps, comparing mean root densities. The latter were calculated by 10-cm horizons for all the grids located at a given distance from a palm. Through joint development of several root systems, virtual scenery simulation bringing into play the true planting design also makes it possible to check the conformity of soil occupation, in the topsoil or subsoil, with observations in the field. Although root competition is not modelled, the model results enabled to determine the palm age from which competition for space among root systems of neighbouring trees may occur. Estimation of root biomasses We created an annex data file in which we indicated for each root type (Jourdan, 1995a) the observed root dry matter per length unit (). By way of the partitioned virtual scenery and the associated linear root-biomass data file, the RACINES software was then used to estimate total root biomass or biomass per root type for the complete simulated root system(s), but also per soil horizon or vertical core sample, depending on the partition carried out. In addition, a considerable amount of root biomass data is available, mainly from Dutch auger samples (Ouvrier, 1995; Rey, 1988; Ruer, 1968). A possible validation of the model was thus based on a comparison of simulated and observed root biomasses. We were able to do this by producing core samples
or virtual trench wall profiles in a simulated plantation. The biomass values per root type and per horizon extracted from RACINES were then compared to values observed in the field. Estimation of lengths and absorbent surfaces A study was also made of radicle absorbent zones by video-densitometry of dye indicator (Jaillard et al., 1996; Jourdan, 1995a). It involved detection of oilpalm root proton-excretion zones using image analysis software. Characterization of the absorbent zones was carried out indirectly by putting forward the hypothesis that the proton excretion zone of roots is an effective marker of their mineral nutrient uptake zone (Clarkson, 1991). According to the results obtained concerning the radicle and various morpho-anatomical studies (Jourdan, 1995a), we have assumed that the length and the localisation of the absorbent zones of the other oil-palm root types were proportional to their diameter. On the 3-D numerical model, each root was identified in space and its diameter was known. It was then possible to calculate the lengths and absorbent surfaces by simulation, irrespective of palm age. The RACINES software provided a detailed result for each root type. For example, it was possible to establish the relationship between the absorbent surfaces of tertiary and quaternary roots, characterized as “absorbent roots” by Purvis (1956) and those of primary and secondary roots, characterized as , and thus to estimate the percentage of the absorbent part of the root system or of a soil horizon for a plantation of a given age.
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Figure 3. Simulation of a 7-year-old plantation seen from above (A) and from the side (B) revealing competition for space in the upper horizons. The palms are planted in 9 m equilateral triangles.
Results Spatial distribution Simulations displaying a virtual oil-palm plantation planted in a conventional design where the palms are set out in 9 m equilateral triangles (i.e. a density of 143 palms per hectare) can be used to characterize spatial occupation of the topsoil and subsoil (Figures 3a and 3b) by root systems. They also reveal the age at which the roots of neighbouring palms start competing with each other. Under the conditions found in the Cˆote d’Ivoire, and with the planting material observed, root competition for space occurred in the topsoil as early
as 5 years of age (Jourdan and Rey, 1997b) between horizontal primary roots (RI H). For an 11-year-old plantation, competition also spread downwards via the secondary roots (Figure 4) in the direction of downward vertical growth (RII VD). With a virtual grid, we were able to estimate the number and position in space of each root type present, irrespective of plantation age. The simulation result could be displayed in the form of a root density map (Figure 5). The model was validated using a histogram representation, giving the observed and simulated mean root-density values, per 10 cm soil horizon. Figure 6 shows the results of mean root density comparisons for the primary roots (RI) of 5 oil palms
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Figure 4. Simulation and display of a virtual scenery containing two 11-year-old palms, planted 9 m apart, seen from the side. The root systems shown comprise primary and secondary roots only. Note the zones of contact between the crowns of the above-ground part and between the root systems near the soil surface.
Figure 5. Root density map obtained by simulation 1 m from the stem of a 10-year-old palm. Each elementary square represents 10 cm 10 cm of soil.
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Figure 6. Mean primary root density profiles obtained on simulated trench walls (blank bars) and observed trench walls (shaded bars). The results are given per 10-cm thick horizon and for the entire width of the grid (= 1 m) at different distances from the stipe, 20 cm (C, F), 50 cm (A),1 m (D, G), 1.5 m (B) and 2 m (E, H) for oil palms aged 3 years (A, B), 10 years (C, D, E) and 20 years (F, G, H).
240
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241
n-1, in order to assess the root biomass produced in a
Figure 7. Variations in the simulated total root biomass of 11 individuals drawn at random from the distribution (black dots) and the simulated mean biomass (continuous line) over time. The observed mean root biomass ( ) of more than 10 individuals is given for information.
#
aged 3, 10 and 20 years and for various distances from their stipes. These results enabled validation of the RI rooting profile in the field. At less than 50 cm from the palms, the RIs were primarily distributed with a high density between 20 and 60 cm in depth. More than one metre from the palms, the RIs mostly occupied the horizons between 20 and 40 cm. Estimation of root biomasses The linear root biomass was determined for all root types, so as to estimate the biomass of the simulated 3-D numerical models. It was thus possible to estimate the total root biomass of juvenile palms. We simulated 11 different individuals drawn from the distribution and calculated their total root biomass approximately every 30 days (Figure 7). The simulated total root biomass increased exponentially at the end of the juvenile phase. According to our simulations, the RIs were responsible for this rapid increase from the 8th month of cultivation. At the end of the juvenile phase, the biomass of the RIs was much greater than that of the other roots. However, 2 years after germination, RI biomass seemed to stabilize, whereas RII biomass increased sharply to reach 4 kg/tree at 3 years, whereas the biomass of the other roots was around 1.5 kg (Jourdan, 1995a). For example, a simulated estimation of root biomass in palms aged 4 and 16 years gave 21 and 385 kg per tree respectively. For a hectare of plantation at a density of 143 palms ha 1 , total root biomasses thus become around 3 and 55 tonnes ha 1 respectively. We calculated the total root biomass of an oil palm, eliminating all self pruning between year n and year
year (Table 1). We also sought to estimate self pruning by comparing the root biomasses produced (total or annual) with and without self-pruning (Table 1). The percentage increase in annual root biomass produced between 2 successive years was very high in the first 4 years of the oil palms’ life, but its intensity decreased as time went by. At the same time, the share of root biomass pruned each year, estimated by the model, increased to reach 80% at 4 years. Lastly, the percentage increase in the root biomass produced with self pruning was also very high in the first 3 years, with a tendency to decrease sharply thereafter, reaching 46% at 4 years. Location and estimation of absorbent surfaces The experiment involving video-densitometry of dye indicator enabled us to define the position and size of the absorbent zones for each root type. The zone is located in the immediate vicinity of the apex (e.g. 0.5 to 1 cm behind the root cap for the radicle), i.e. in the zone with limited cell differentiation (cell elongation zone) (Jourdan, 1995a). Once the absorbent zones of the different roots had been located, it was possible to display them on the 3-D numerical models (Figure 8) and to accurately reconstitute an oil palm plantation, taking into account the planting design. It was possible to estimate the oil palm age at which the roots of neighbouring oil palms start competing for nutrients. For palms planted at the usual density, simulations revealed that some roots (RI H) of a 4-year-old palm began to compete in the topsoil with the same type of roots from neighbouring palms. At 5 years, space colonization was almost total to a depth of 1 m. At 7 years, the surface horizons were totally colonized and competition between secondary roots was established deeper down. By simulation, it was possible to calculate the absorbent surface of a complete oil-palm plantation. For a hectare of 5-year-old oil-palm plantation, the absorbent surface of the tertiary plus quaternary roots, qualified as absorbent roots, was 5 times (Table 2) that of the primary plus secondary roots, qualified as exploratory roots. These results also showed us that out of the total root surface, 23% is absorbent, most of which (83.7%) is ensured by tertiary roots (28.9%) plus quaternary roots (54.8%). As regards the total surface of each root type, it was also possible to see (i) the very small but not insignificant share (less than 10%) of absorbent zones on primary and secondary roots,
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242 Table 1. Estimation of the root biomass produced with and without self-pruning for one oil palm during its first 4 years. (a): rate of increase between year n and year n-1. (b): percentage of self-pruned annual root biomass compared to the annual root biomass produced
Years since germination
Annual root biomass (kg) Without Self-pruned self pruning
Total root biomass (kg) Without Self-pruned self-pruning
1 2 3 4
0.17(-)a 3.65(+2047%) 18.42(+405%) 54.74(+197%)
0.17 3.82 22.24 76.98
0.05(29.4%)b 0.88(24.1%) 11.14(60.5%) 44.08(80.5%)
0.05 0.93 12.07 56.15
Figure 8. Display of absorbent zones (white) on a 3-D numerical model of a partial simulated root system aged 90 days.
(ii) the larger share of these zones on RIIIs (26%) and (iii) the major share on RIVs (55%). It was therefore reasonable to assume that nutrient uptake in the oil palm is primarily via the quaternary roots and more moderately via the tertiary roots, but the role played in this function by the primaries and secondaries should not be overlooked.
Discussion and conclusion Spatial occupation Spatial distribution is portrayed by simulations and displays (Figures 3 and 4) using 3-D numerical mod-
els placed in virtual scenery that reproduce real growing conditions. An initial quantitative validation of the model was based on comparing the root density maps shown in Figure 6. For 3-year-old palms (Figures 6a and 6b), the model slightly overestimated the number of RIs in the horizons deeper than 40 cm. This was particularly the case with profiles located 50 cm from the palms (Figure 6a), where 96% of the RIs primarily occupied the upper horizons. These roots corresponded to RI Hs emitted by the palm as soon as it was planted in the field. For these palms, the total number of roots in the mean simulated and observed profiles were nonetheless very similar since a slight underestimation of the model was seen for the surface horizons. For 10-year-old palms, and for the different
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Figure 9. Simulated 8-year-old oil-palm plantation (planting density: 143 palms ha from below.
1)
with root systems comprising primary roots only. Seen
Table 2. Estimation of the total and absorbent root surfaces of a 1-ha 5-year-old oil-palm plantation
Branching order
Surface (m2 ha 1 ) Total of the root Of the absorbent system zones
Absorbent surfaces (%) % % by branching on total order
RI RII RIII RIV Total
1887 1445 1594 1461 6388
1.86 1.91 6.69 12.71 23.17
119 122 427 812 1481
positions tested (Figure 6c, d and e), the model generally underestimated the number of RIs, irrespective of the depth. However, the general trend of the simulated and observed root profiles at 0.2 and 1 m from the palms was identical. In fact, the number of RIs in the horizons between 0 and 40 cm in depth and situated 0.2 m from the palms (Figure 6c) amounted to 59% for simulated roots and 60% for observed roots. At 1 m from the palms, and still in the 0-40 cm horizons (Figure 6d), 65% RIs were found by simulation, whereas 61% actually existed. However, at 2 m from the palms (Figure 6e), the model did not satisfactorily reproduce RI distribution. At that distance, the horizontal primary roots mostly (64%) occupied the 10-40 cm horizons, which was poorly portrayed by the model (46%). For
6.28 8.44 26.81 55.58 23.17
20-year-old palms (Figures 6f, g and h), the model slightly overestimated the number of RIs; especially in the horizons less than 40 cm down. For the horizons more than 40 cm down, the correspondence between the mean simulated and observed values was relatively good, irrespective of the profile position with respect to the oil palms. All in all, these comparisons revealed that the model portrayed the vertical and horizontal distribution of primary roots relatively well. However, the 3-D numerical model display (Figure 3b) showed a slightly underestimated root density in the upper horizons, and two hypotheses could explain this phenomenon. Either the number of roots produced by the model was too small, or the geometry, particularly the directions of
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244 root growth, was poorly portrayed. The first hypothesis was discarded as the numbers of simulated and observed emitted roots tallied well (unpublished data). Greater detail, essentially in the geometric parameters, will therefore be incorporated in future in order to “fine-tune” the model. Root biomasses The second quantitative validation approach was based on a comparison of root biomasses simulated by the model and calculated by RACINES postprocessor, with those observed in the field by different methods, by different authors and under different cultivation conditions. In both cases, the biomass values took self pruning into account. The total biomass simulation results for juvenile oil palms are compared to observed root biomasses in Figure 7. In the first year of cultivation, simulated total root biomass was perfectly comparable to observed total root biomass and the difference between the two curves was not significant. Other simulations at different palm ages were made in accordance with root biomass data available in the literature. Root biomass measurements were made by Ouvrier (1995), in the Cˆote d’Ivoire (at Dabou) under similar soil conditions to those found at La M´e (Jourdan and Rey, 1997b), taking samples with a Dutch auger. For 4-year-old palms, the simulated biomasses (3 t ha 1 ) were higher than the observed biomasses (1.4 t ha 1 ). However, simulation took into account the total root system, which was not the case with auger samples, which were limited in depth to 1 m and in distance to few metres from the palms. In particular, the biomass of the RI VDs and RII VDs was not taken into account in the field observations. Root biomass observations have been carried out on palms aged 16 years from germination at Dabou (Braconnier and Caliman, 1989) and at La M´e (Rey, 1988) by taking samples with metal drawers down to a depth of 6 m. Observed total root biomasses were 58.8 t ha 1 at Dabou and 31.4 t ha 1 at La M´e and by simulation, the calculated total root biomass was 55.0 t ha 1 . In order to exactly reproduce the root biomass sampling conditions under which Rey worked (1988) at La M´e, we used our visualization software to produce a virtual scenery representing an exact replica of the design used for the observations. In this scenery, we used the voxel space technique to create a virtual trench identical to the one dug by Rey, in which biomass calculations were made. The simulated root
biomass obtained under these conditions and scaled down to 1 ha of plantation was 25.4 t ha 1 (as opposed to 31.4 t ha 1 observed). As far as self pruning is concerned, Table I shows that the percentage increase in the annual root biomass produced with self-pruning decreases sharply, reaching 46% at 4 years. This result will need to be validated since Dufrˆene (1989) estimated from root biomass calculations by Ruer (1968) and Rey (1988) that the share of renewed roots amounted to 37% for 13-yearold palms. As we have already put forward strong hypotheses on the self pruning process (Jourdan and Rey, 1997a), we decided to make a sensitivity analysis to test this process. We varied the time lapse between root death and self pruning by more or less 50% for every root type. Simulations revealed a fairly weak response of total root biomass to the “time to self-pruning” parameter (Jourdan, 1995a). In fact, a 50% increase or decrease in the self pruning time lapse corresponded to a 10% increase or 20% decrease in total root biomass respectively. Model sensitivity to the “self pruning” parameter is therefore low within the range of values that we fixed, which supports the hypotheses put forward. Absorbent surfaces One of the major assets of 3-D numerical models lies in the physiological and functional applications they offer. The video-densitometry of dye indicator technique enabled us to locate the excretion zones very accurately (to within a few mm or a few cm depending on root type), in the immediate proximity of the root tip. These measurements were validated by a detailed anatomical analysis (Jourdan, 1995a), which revealed how root suberization takes place in line with the distance from the apex, though further, more precise physiological measurements (micro-electrodes) are needed to confirm the results obtained. According to these informations, root system simulation provided us with results on the distribution of absorbent zones in space and on the quantification of their surface. To sum up, competition in an oil palm plantation between the roots of neighbouring palms occurs at 5 years and the total absorbent surface is 1,481 m2 , representing 23% of total root surface (Table 2). According to various authors (Lambourne, 1935; Purvis, 1956; Ruer, 1968; Wright, 1951), a distinction can be made between exploratory roots and absorbent roots, though exploratory roots also play a role in absorption by their apical tip. Our results confirmed
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245 this idea, since primary and secondary oil-palm roots, considered as exploratory roots, possess 1/5 of the total absorbent surface of the root system (Table 2). Prospects for model application Looking ahead from the applications available with the current version of the model and of the RACINES postprocessor, many future applications can be envisaged, leaving the field wide open for various data processing and agronomic prospects. Among the data processing prospects, the mathematical model of root system architecture could be combined with the functional model of oil palm production developed in the Cˆote d’Ivoire by Dufrˆene (1989). In the latter model, the roots were characterized by total biomass only. Our results provide new elements in terms of biomass distribution, growth dynamics, spatial colonization and exploitation. They therefore offer the possibility of (i) linking root functioning to that of above-ground parts (also modelled and simulated by the same model, see Figure 4), (ii) studying growth correlations within the whole plant and (iii) linking the localization of the absorbent zones with a mineral uptake model. In many mineral uptake models, only a few morphological and geometric characters are mentioned: mean lengths and/or mean root density, mean diameter, mean “half-distance” between roots (Baldwin et al., 1973; Barber and Silberbush, 1984; Claassen et al., 1986; Kelly et al., 1992; Smethurst and Comerford, 1993), but no overall root architecture is indicated and taken into account in the calculations. Mathematical modelling of root system architecture, as described here for oil palm, but also like that developed for maize (Pag`es et al., 1989), peach (Pag`es et al., 1992) or Hevea (Le Roux, 1994), may provide greater accuracy for distribution, and especially for quantification, of mineral uptake. It will be necessary to interface this architecture model with mineral absorption models in future, for accurate quantification of water and mineral nutrition in this type of large-scale perennial tree crop. Environmental effects on oil-palm root system architecture and growth were not examined in this work. New data processing tools currently being developed will soon enable us to cover interactions between roots and the environment. Indeed, this new generation of software packages combines voxel space techniques and simulation of the growth of all the root meristems at the same time (parallelism technique according to Blaise (1991)). It will also enable the use of rules for
decisions taken throughout the growth process, thus making it possible to account for these interactions. However, soil/root or root/root interactions are sometimes very difficult to observe and quantify in the field. With the combined use of parallelism technique and decision-making rules reflecting hypotheses about these interactions, it will be possible to conduct virtual trials via simulation. Indeed, the 3-D numerical models produced could be compared to field observations, which will validate or refute the hypotheses put forward. Through prior simulations of virtual sceneries that comply with the actual planting design, and take the entire palm into account (Figure 9), virtual trials will also enable, among other things, (i) optimization of field trials (simplification of observation protocols, problem issues more precisely defined, time-saving, etc.), (ii) estimation of optimum planting densities (in monocultures and/or with intercrops, of the same age or not) taking into account the respective development of root systems, (iii) optimization of fertilizer applications (where?, when?, how much?, etc.), (iv) the effect of various agricultural techniques (subsoiling, pruning, crop management sequences, etc.). Apart from the various agricultural applications they will generate, mathematical modelling, along with plant architecture and functioning simulations, will remain powerful synthesis tools that will help increase our knowledge, through which future research is bound to progress.
Acknowledgements We thank Kouam´e Brou and the Institut des Forˆets - D´epartement des Plantes Ol´eagineuses (IDEFORDPO) for their hospitality in Cˆote d’Ivoire, Philippe de Reffye, Jean Fran¸cois Barczi, Fr´ed´eric Blaise and Yann Gu´edon for designing and developing the modelling and simulation software in France. The authors also wish to thank P Biggins for the English translation of this manuscript.
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