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Plant and Soil 190: 217–233, 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 I. The model

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: Elaeis guineensis, root architecture, root development, stochastic models, simulation, 3-D numerical models Abstract The objective of this work was to model the architecture and growth dynamics of the oil-palm root system. The morphological and functional unit of the root system, called “root architectural unit” and its development sequence enabled us to establish the basis of a mathematical formalization of the root system architecture. The topology of the branched structures and the processes of growth, branching and mortality were described and modelled by stochastic processes (graph model, automata, laws of probability). The models obtained were then combined with geometrical parameters in an overall mathematical model: the reference axis. Simulation of this model provided 3-D numerical models. Validations of the overall model based on comparing the 3-D numerical models with observed root systems, appeared satisfactory. Introduction Early studies of root systems were purely descriptive and generally involved anatomical and morphological descriptions (Kubikova, 1967). The analysis of how the different botanical entities described were formed and how they functioned was then looked at in more detail. With the observation and analysis techniques developed in recent years (Fitter, 1991; Pag`es and Aries, 1988), it is now possible to study complex root systems and understand how they function. In order for them to be easily accessible to agronomists, the results obtained have to be synthetized, explained and quantified. Mathematical modelling is the most appropriate tool for carrying out these different operations. Several types of model have been developed over the last two decades, involving root system growth and architecture as much as its functioning. A dis-



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tinction can be made here between structural static models (Gerwitz and Page, 1974; Henderson et al., 1983), and dynamic models of root system growth and development (Brouwer and de Wit, 1969; Chen and Lieth, 1993; Jones et al., 1991; Lungley, 1973; Porter et al., 1986; Rose, 1983), models of water uptake (Herkelrath et al., 1977; Lafolie et al., 1991; Rowse et al., 1978; Taylor and Klepper, 1975) and mineral nutrient uptake (Baldwin, 1975; Baldwin et al., 1973; Barber and Silberbush, 1984; Claassen et al., 1986; Habib et al., 1989; Passioura, 1963), combined growth and uptake models (Barnes et al., 1976; Bland and Jones, 1992; Huck and Hillel, 1983; Protopapas and Bras, 1987), and lastly 3-D root-architecture models (Berntson, 1994a; Diggle, 1988; Fitter et al., 1991; Le Roux, 1994; Nielsen et al., 1994; Pag`es and Ari`es, 1988; Pag`es et al., 1989). Root architecture mathematical models have mostly been developed very recently and enable quite realistic three-dimensional simulations.

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218 The mathematical modelling used in our work is a synthetic tool capable of accounting for the different stages in the life of a plant. This tool is based on stochastic processes (automata, laws of probability for example) which were initiated after work by de Reffye (1979) on the modelling of coffee tree above-ground architecture, then further developed on other plants (Costes et al., 1992; Jaeger and Reffye, 1992; Reffye et al., 1988, 1991). These models are based on an analysis of meristem functioning and are capable, at any given moment, of accounting for the plant structure described. We propose to apply these techniques to the oilpalm root system. The purpose of this article is therefore to present a mathematical model that accurately represents the architecture and development of this root system. The model comprises several stochastic models, which describe the various processes involved in root system edification. By simulation, it is possible to create 3-D numerical models that can be displayed on a computer screen. These 3-D numerical models enable subsequent numerical processing that lead on to agronomic applications (Jourdan and Rey, 1997a).

Material and methods Material and study site Our studies were conducted at the La M´e research station in south-eastern Ivory Coast. The variety of oil palm chosen was a reproduction hybrid (L2T  D115D) from the C1001F family commonly used in commercial plantations. Several categories of oil palm age, from germination to 20 years, were observed in order to define the root architectural unit of the oil palm (Jourdan and Rey, 1997b). The results obtained showed that following a transitional juvenile phase, eight different morphological types of roots were distinguished according to their development pattern and state of differentiation: primary vertical roots (RI VD) and primary horizontal roots (RI H), secondary horizontal roots (RII H), upward growing secondary vertical roots (RII VU) and downward growing secondary vertical roots (RII VD), superficial (sRIII) and deep (dRIII) tertiary roots and quaternary roots (RIV). This root polymorphism enabled us to define a morphogenetic gradient, on which a modelling process was based.

The developer’s approach The general problem was to formally identify and represent the different elements observable in the root system that would enable modelling, then simulation. Observations and architectural analysis in situ were the first (Figure 1–1) and second (Figure 1–2) stages of the process (Jourdan and Rey, 1997b). They provided a collection of data that needed structuring. The third stage was therefore to create a computer data file or code file (Figure 1–3) that formally represented the branched structure described (Rey et al., 1997). The fourth stage was the actual mathematical modelling (Figure 1–4) of root system architecture using AMAPmod software (Reffye et al., 1995). It was during this stage that the laws accounting for growth, mortality and branching processes in the different types of roots, along with their variability, were established. The fifth stage (Figure 1–5) then consisted in constructing a specific reference axis (Reffye et al., 1991) using all the knowledge acquired in the preceding phases: root architectural unit, values of the mathematical law parameters and of geometrical parameters, variations in these parameters over time. It served for the simulation software, also called >, The sixth stage was simulation (Figure 1–6) of the root system architecture by the AMAPsim growth generator (Reffye et al., 1995). The result of simulations consisted of 3-D numerical models that were computer representations (data files in matrix form (Jaeger, 1987)) of the three-dimensional structures of root systems. These 3-D numerical models were displayed using specialized tools involving computer images. The simulations also enabled the generation of code files in a format permitting remodelling then validation. The seventh stage was therefore validation, which is essential since it confirms model credibility. Several validations are possible (Figure 1-7): Visual > based on a comparison between simulations and field observations (Figure 1–7a), quantitative > by comparison between the parameters of the different laws arising either out of modelling, or remodelling to test the pertinence of the mathematical laws used (Figure 1–7b), and two quantitative validations: (i) comparison of root impact charts both simulated and observed on trench walls and (ii) comparison of simulated and observed root biomasses using the RACINES postprocessor software (Jourdan and Rey, 1997a). These two validations lead on to agronomic applications (Figure 1–8) which are the eigth and final stage of the process.

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219

Figure 1. Experimental procedure involved in creating 3-D numerical models.

The models used Root topology Formal representation of oil-palm root system topology was carried out using a model derived from the graph theory (Gondran and Minoux, 1990). A graph model can be defined by a set of entities, called >, by a set of relations, called > which exist between the nodes and by the properties of these edges. This model served to define an appropriate language (or code) dedicated to plant description (code file). It was designed by Blaise (1991), developed by Rey et al. (1997) on the above-ground parts of trees, and applied to describing the oil-palm root system (Jourdan, 1995). For this root system, we had to define the nodes, the edges and their properties. Within the root system, the absence of morphological markers similar to those usually seen in stem parts (node, internode, etc.) led us to seek a concept that would enable its formalization, which is based on the definition of a structural entity that enables integral coding of the topology and geometry of the system,

Figure 2. Diagram of a branched system discretized by the elementary length unit.

and its evolution with time. In this case, the entity consisted of the minimum distance observed between two

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220 successive branches for each root type, as the “fundamental distance” separating two branches in tomato roots (Newson et al., 1993). The root axis could then be split into discrete elementary length units (Figure 2). Likewise, the inter-branch unit, representing the portion of a root axis between any two successive lateral roots, could be broken down into one or more elementary length units. In our work, elementary length units of an axis are formalized by nodes of the graph model. Two relations exist between these elementary length units: (i) to characterize the growth process of an axis, we defined > edges between two successive elementary length units and (ii) to characterize the branching process, we defined edges between the elementary length units of a main axis and its laterals. Each elementary length unit was then coded and attributes, enabling characterization of the root type, physiological condition, length, diameter, angle of insertion and direction of growth, were added. This set of values was then used to explore the structure and extract data samples. The data were finally used to elaborate and assess models for processes involved in the formation and development of the oil palm root architecture. Growth process Oil-palm root growth has been described by dynamic monitoring in minirhizotron (Le Roux and Pag`es, 1994; Riedacker, 1974) in the case of young oil palms and in field rhizotrons (Jourdan and Rey, 1997b) for oil palms in plantations. Modelling of root axis growth was based on the theory developed by de Reffye (1979) on the above-ground parts of Coffea robusta. A population of root axes of the same type was observed on many trees (Jourdan and Rey, 1997b). The additional growth of each root axis (number of elementary length units) was measured. Time was then discretized (one day time interval) and, for each root axis observed, the number of elementary length units formed was allocated to each time interval. Thus, on each observation date, the total length of the root axes was known. The mean and variance of distribution of the number of elementary length units were calculated and the meanvariance relation established. It was then possible to precisely describe the growth by a stochastic process (Reffye et al., 1988). In fact, if the relation was linear, and considering the case of a sample with limited dispersal (unimodal distribution) and with variance smaller than the mean, the distribution could be adjust-

ed by a binomial law. Meristem functioning could then be assimilated to a succession of independent random draws, each afforded the same probability of producing a new elementary length unit. From then on, probability calculations could be used to model this functioning. At any moment the distribution of the number of observed elementary length units on these axes was a binomial law of parameters N and p. N is the number and p the apparition probability of new elementary length units. The mean m of this law is:

m = Np and the variance  2 :

 2 = Np (1

p ) = m (1 p )

Once m and  2 were known, it was possible to calculate the N and p parameters of the binomial laws modelling the growth process of each root type. Mortality process Root axis growth is not continuous. There are temporary or definitive halts in the growth process. The root mortality process leads to the disappearance of roots following an ageing phase, or senescence, of varying lengths. Root mortality can be characterized by studying the length and the number of surviving roots (Hendrick and Pregitzer, 1993). In the absence of early and convincing morphological markers of oil-palm root mortality, we employed this easy to use approach. An axis was considered to be dead when none of the partial reiterations continued their growth. Mortality could therefore be characterized by the final length of the axis, and it was this criterion that we formalized. Based on field observations, and for each root type, we defined length categories in which roots had varying degrees of probability (Pm ) of dying. Each category took into account the background of the process, i.e. the number of dead roots in the previous length categories. Root mortality was modelled through their vitality. To do this, a survival probability (Ps = 1 – Pm ) was allotted to each elementary length unit and the probability varied depending on the length category, the root type and the age of the palm considered. This formalization could be written in a recurrent form: Ps (i) =

1

!N

i

ri

1

1 Ni

1

Pi

1 j 1 rj

=

where Ps (i) = the survival probability of an elementary length unit in length category i,

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221 tem, we modelled the branching process of the other root types throughout its life span using a homogeneous two-state first-order Markov chain only, which graph is shown in Figure 3. Matrices of initial (II) and transition (P) probabilities and the deduced branching probability (Q) are given below: Initial probability matrix: 

II = Figure 3. Graph of the transition probabilities for a two-state Markov chain formalizing the branching process.

Self-pruning process After death, an axis may persist for varying lengths of time in the branched structure before self pruning. This important criterion was very difficult to estimate in the field due, once again, to a lack of significant markers of root age. We therefore attempted to formalize this process by putting forward the hypothesis whereby each dead axis could not self prune until it only bore dead lateral formations. According to this hypothesis, the self-pruning process would occur after a process of mortality by exhaustion, which would first affect the peripheral roots (RIV), then the intermediate roots (RIII then RII), ending with the primary roots (RI). Branching process Many authors have tried to understand, explain and model the root branching process. Unlike phyllotaxy in the above-ground parts, no rhizotaxy has been clearly detected in most plants, and particularly palms. The root system branching process has therefore often been considered as a random process. Based both on this hypothesis and the root axis discretization into elementary length units, we attempted to model the process using a stochastic model, Markov chains. In a first stage, the branching process of the radicle was modelled by a homogeneous two-state first-order Markov chain (Jourdan et al., 1995a). In a second stage, a homogeneous four-state first-order Markov chain was used to represent the succession of different root types borne by the radicle (Jourdan et al., 1995b). In a third stage, given the architecture of the oil-palm root sys-



Transition probability matrix: 

P= ri = the proportion of dead roots in length category i, in relation to the initial population, Ni = rank of the last elementary length unit in length category i, where No = 1 (Ni – Ni 1 represented the size of the current length category).

0 1

P00 P01 P10 P11



Branching probability: Q=

P11 ) P00 ) + (1 P11 ) (1

(1

Branching time lapse The branching time lapse is the time taken by a mother root to initiate a daughter root up to the appearance of the latter on its surface. This time lapse can be expressed in terms of length through the rate of growth of the mother root, hence as the number of elementary length units. The branching time lapse was therefore represented as the distance between the last daughter root to emerge and the apex of the mother root, which constituted the length of the unbranched apical zone (LUAZ). We therefore formalized this LUAZ by a law of the number of elementary length units making it up. Geometrical parameters The geometrical parameters of a root system can be characterized by the angles of emission of the different roots in space, along with changes of direction (reorientation). The angles of root emission are of two types: radial angle and angle of insertion (Figure 4). The first is considered in the radial plane of the bearing structure, i.e. the horizontal plane if the bearing axis is vertical. The literature shows that branching usually occurs opposite a xylem pole (Abad´ıa-Fenoll et al., 1982; Bell and McCully, 1970; Mallory et al., 1970). The radial angle therefore depends on the number of xylem generators or poles. In oil palm, as in all Monocotyledons (Esau, 1965; Fahn, 1990; Riopel, 1969), the roots are characterized by a multitude of xylem poles and several woody poles may be connected to the lateral (Jourdan, 1995), as has been shown in Monstera (Guttenberg,

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222

Figure 4. Geometrical parameters: characterization of the insertion and radial angles for the oil-palm root system.

1940). The value of the radial angle is therefore not very predictable, so we chose to determine it by a random draw between 0 and 2 . The second type of angle, the angle of insertion, is considered in a longitudinal plane, i.e. in a vertical plane if the bearing axis is vertical. It was determined by the mean (and variance) values obtained from observations. In order to account for gradual changes in direction, we used “Young’s pseudo-module” to give a positive (or negative) geotropic curve to the emitted roots (Blaise, 1991). Sharp changes in direction, due to all kinds of obstacles (pebbles, soil surface, etc.), along with root interactions with each other or with the soil (moisture, compaction) have not been taken into account in the current version of the model. Overall model: reference axis The overall mathematical model of oil-palm root system architecture combines the set of models described above, which characterize the processes of growth, mortality and branching of root axes, along with their geometrical parameters. A final process, involving root axis differentiation, also has to be taken into account if root morphogenesis is to be satisfactorily translated. Indeed, for a detailed study of plant architecture, e.g. for accurate simulation, the notion of branching order is sometimes found to be insufficient. In this case, it is best to consider typology, or the state of axis differentiation, rather than just their topology. The structure of an axis is in fact the result of meristem functioning and expresses the state of differentiation, or even the physiological state of the meristem at the moment of differentiation. The set of

values for the biological characteristics of an elementary entity (elementary length unit, inter-branch unit, axis) at the time of its formation can be used to define a posteriori, the physiological age of the meristem that gave rise to it (Barth´el´emy et al., 1997). The parameters that describe the different processes are also directly dependent upon another variable, time. Mathematical modelling consequently has to integrate the time factor. Thus, plant growth amounts to getting two evolutive phenomena to cohabit: time and physiological age. Simulation of growth involves simulation of meristem functioning. It is therefore based on the topological arrangement of the entities formed per unit of time, on their differentiation state and on their spatial distribution. These notions have been integrated in the overall model, so as to reproduce the complete ontogenesis of the root system described in both space and time. The overall mathematical model chosen to formalize the notion of physiological state of the root meristems was a graph model. So, the different root types with their topological, geometrical and morphogenetic characteristics were formalized by the nodes of the overall mathematical model. The relation between each root type describes the possibility for one root type to bear another root type with an associated transition probability. The edges were said to be oriented, since switching from one root type to another was in one direction only, in increasing order of differentiation state. This type of oriented graph model combined with transition laws defined a finite “left-right” automaton in mathematical terms, called the reference axis (Reffye et al., 1991).

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223



Table 1. Growth rate (mean standard deviation) of eash root type on 10-yearold oil palms Root type RI H RI VD RII H RII VU RII VD sRIII dRIII RIV

Growth rate (cm d 0.29 0.29 0.20 0.15 0.20 0.07 0.10 0.10

1)

 0.13  0.13  0.04  0.10  0.04  0.04  0.05  0.02

Figure 5. Relation between the mean (m) and the variance ( 2 ) of radicle length samples during the growing period: 0–90 days.

Results Modelling We applied this theory to the oil-palm root architecture. Each node of the reference axis of the oil-palm root system is represented by a state of differentiation, i.e. a root type, characterized by a set of growth, branching and mortality laws. The edges and the associated probability are represented (i) by the values of the growth model for the > relation and (ii) by the values of the probabilities of branching, given by the Markov chain models, for the relation. Model validation Of the four types of validation carried out, only two are developed in this paper, the two others were covered in Jourdan and Rey (1997a). The first is a qualitative > and the second is a quantitative > which test the pertinence of the mathematical laws used. It was based on “remodelling” the code files resulting from the simulations. The method consisted first in carrying out a large number of simulations. For each simulation, a code file corresponding to the 3-D numerical model obtained was generated in the standard coding format (Figure 1-7b), the same as the format used to code field observations. All the code files were assembled in a single file to obtain a data base that statistically reproduced the variability of the previously modelled population. It thus proved possible to remodel the simulated population. The parameters of the mathematical laws were estimated for the simulated population. Validation then consisted in comparing the parameters of the discrete laws obtained before and after simulation.

Growth process Oil palm roots are characterized by continuous, steady growth with no apparent rhythm. The fastest mean growth rates are observed in young palms (between germination and 1 year) for RI (1 cm day 1 ), the slowest are seen in adult palms (around 10 years) for RIII and RIV roots (0.1 cm day 1 ). Under conditions in the Ivory Coast, RI roots in adults have an annual mean growth rate of around 0.3 cm day 1 , corresponding to an increase of around 1 metre per year (Table 1). The mean-variance relation was established for the length samples over time. For the radicle (Figure 5), the mean m) was linked to variance ( 2) by a significant linear regression (R2 = 0.86), which enabled us to use a binomial law B(N, p) to model this process. This relation was of the type:

 2 = 0:21m now, for a binomial law

 2 = (1

p )m

hence 0:21m = (1

p )m

and p=1

0:21 = 0:79

For the other root types, growth observations in field rhizotrons over a period of at least 6 weeks, showed that the distribution of observed lengths could

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224 Table 2. Survival probability (Ps ) for each elementary length unit of 3-year-old RI VDs. Pm : mortality probability of the successive categories; Pcm : cumulative mortality probability; % Length: percentage of the maximum length of the root type; Nb E: corresponding number of elementary length units from the base of the root Pm 0 Pm 0 % Length 0 Nb E 0 ps

0,1 0,1 36 396 1

1

0;1 1 0




0,1 0,2 44 484 1 396 0

=0.999941

1

1

0;1 0 ;1




0,2 0,4 56 616 1 484

=0.999706

396

1

1

0;2 0;2




0,2 0,6 64 704 1 616

484

= 0.999521

Figure 6. Changes in the cumulative mortality probability (Pcm ) depending on the maximum root length percentage. The mortality curve arises from an adjustment by a third-degree polynomial function (R2 = 0.92).

also be simulated by a binomial law with constant probability for all the observation periods, with a mean value p = 0. 77 ( 0.05). Given the equivalence between the values obtained for the radicle and the roots of adult palms, but also given the uniformity of growth for all the roots, we attributed to each elementary length unit a constant apparition probability of p = 0.8. It was thus possible to model apical meristem functioning by a regular succession of clock signals, each assigned a constant probability of 0.8 of giving a new elementary length unit. Mortality process The mortality curves for each root type were obtained from an adjustment by a third degree polynomial function (R2 = 0.92). The trend of the mortality curves was identical for all root types. It was possible to identify three phases (Figure 6): (i) an initial phase, accounting for 40% of the final length during which the probability of root mortality (Pm ) was low, (ii) a second phase during which Pm increased sharply (from 40 to 80% of

1

1

0;2 0 :4




0,2 0,8 72 792 1 704

=0.998987

616

1

1

0 ;2 0 ;6




0,1 0,9 80 880 1 792

=0.998269

704

1

1

0 ;1 0 ;8

0,1 1,0 100 1100




1 880

=0.998269

792

1

1

0;1 0;9




1 1100

=0

final length), and (iii) a final phase where Pm stabilizes on the remaining 20% of the final length. As for the growth process, the mortality process was managed step by step, taking the elementary length unit as the discretization step. The aim was therefore to calculate the probability of survival for each elementary length unit according to its rank on the root. Cumulated mortality probability (Pm ) was divided up arbitrarily to define root length segments in which the number of elementary length units was known (% of final length) and, for each of them, the probability of mortality (or survival) remained constant. For example, the maximum length of the RI VD at 3 years was estimated at 5.5 m (for a mean length of around 3 m), i.e. 1,100 elementary length units of 0.5 cm. Division into mortality categories (Pm ) for this root type (Figure 6) then made it possible to calculate the probability of survival (Ps ) for each elementary length unit (Table 2). These calculations were carried out for all the other types of roots. The results for 10-year-old palms are given in Table 3. Self-pruning process Under minirhizotron growing conditions, RIV longevity proved to be 3 to 4 weeks. During growth monitoring, some dead RIV self-pruned around two months after their emission date. These roots would therefore seem to self prune a month after their death. For the other roots, in the absence of significant criteria under both rhizotron and field conditions, the selfpruning of dead axes was formalized by putting forward strong hypotheses. We explain below the hypothesis of a mortality process by exhaustion. This explanation concerned the lateral formations of the radicle, but was also applicable to the other root types. On an RIII, we saw that an RIV could persist for two months at the most before self pruning. For RIII, we fixed the length of the period between death and self-pruning arbitrarily at 60 days, so that they self-pruned when they were no longer bearing any living RIV. By re-editing the

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880

225 Table 3. Survival probability (Ps ) for each elementary length unit of each root type on 10-year-old oil palms. Pm : mortality probability of the successive categories; Pcm : cumulative mortality probability; % Length: percentage of the maximum length of the root type Pm Pcm % Length

0 0 0

0,1 0,1 36

0,1 0,2 44

0,2 0,4 56

0,2 0,6 64

0,2 0,8 72

0,1 0,9 80

0,1 1,0 100

Ps Ps Rs Ps Ps Ps Ps Ps

1 1 1 1 1 1 1 1

0.999775 0.999941 0.999415 0.999902 0.999958 0.999268 1 1

0.998868 0.999706 0.997060 0.999509 0.999790 0.996326 0.992163 1

0.998158 0.999521 0.995217 0.999201 0.999658 0.9945025 0.932390 0.898174

0.996109 0.998987 0.989915 0.998312 0.999272 0.987409 0.975677 0.898174

0.993357 0.998269 0.982821 0.997116 0.998763 0.978572 0.961344 0.898174

0.993357 0.998269 0.982821 0.997116 0.008763 0.978572 0.961344 0.898174

0 0 0 0 0 0 0 0

RI VD RI H RII H RII VU RII VD sRIII dRIII RIV

same reasoning for the RIII borne by RII, we fixed the length of the period before RII self pruning at 90 days. The estimated time values, expressed as a number of days, taken between root death and self-pruning for all root types in the adult phase, are 200, 140, 110 and 30 days for RI, RII, RIII and RIV respectively. Branching process In the branching process, daughter roots are initiated in sequence and continually near the apex of the mother root, then emerge at varying distances from this apex. The mother roots are therefore characterized by a zone bearing daughter roots and an unbranched apical zone. Branching is said to be acropetal: the youngest roots are found furthest from the base of the root. Daughter root distribution on the mother axis is more or less regular, no change in branching density along the axes has been observed; oil-palm root branching is diffuse. The oil-palm root branching process was formalized by a stochastic model of the non-oriented automaton type represented by Markov chains. First, for the radicle (Jourdan et al., 1995a) and the other root types at the adult stage., we used an automaton with 2 states (unbranched and branched) that was homogeneous (process independent of the rank of the states on the bearing axis) and of order 1 (the state of rank n only depended on the state of rank n-1). Then we used Markov chains with 4 states in order to account for distribution of the lateral roots produced (Jourdan et al., 1995b). The transition probabilities from one state to the other characterized the succession of daughter roots on the mother root, taking into account the immediate surroundings of these roots; they were therefore

conditional probabilities. The transition probabilities therefore took into account not only the state of the preceding elementary length unit, but also the type of borne axis. The automaton was therefore capable of accounting for the succession of different types of lateral formations borne by the axis that initiated them. The radicle branched with a probability Q = 0.33 which remained constant throughout its life span. If the elementary length unit is 0.5 mm, the probability of finding 2 successive RII on the radicle was low (p11 = 0.22), and it was not rare to see unbranched zones of more than 2 mm long. The RI branching probability in the juvenile phase decreased with time (0.11 < Q < 0.36), the RII borne being spaced further and further apart. The transition and branching probabilities for RI H and RI VD emitted by palms aged 3 and 20 years old remained constant. For example, the different parameters of the Markov chain are given in Table 4, for all root types and for 10-year-old palms. The very low values for the probability of finding 2 successive RII (0.02 < P11 < 0.1) characterized the diffuse and not very dense branching of the RI in plantations. A close relation was established between the branching parameters and the diameter of the RII in the adult phase. For example, RII H with a diameter of less than 2 mm are more branched (Q = 0.39) than RII VD with a diameter of 2 mm or more (Q = 0.22). The sRIII were more branched than the dRIII. The sRIII of palms aged 10 and 20 years were the most branched in the root system.

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226 Table 4. Branching probability, initial and transition probabilities of each root type on 10-year-old palms Initial probabilities



Root type

Branching probability

o 1



Transition probabilities   Poo P01 P10 P11

(Q)

 RI H

0.17

RI VD

0.20

RII H

0.39

RII VU

0.48

RII VD

0.22









 sRIII

0.46

 dRIIII

0.31

0:85 0:15 0:92 0:08 0:85 0:15 0:71 0:29 0:94 0:06 0:87 0:13 0:95 0:05





























  RIV

0

Branching time lapse The histogram showing the lengths of the unbranched apical zone (LUAZ) for the set of radicles as a whole is unimodal (Figure 7). This sample was adjusted by a shifted binomial law B(d, N, p). The parameters of this law were the minimum observed LUAZ (d = 1 cm), the maximum observed LUAZ (N = 6 cm) and the probability (p = 0.42). The mean length of the unbranched apical zone of the radicles was 3.5 cm  1.5 which represents, with an average growth rate of 0.44 cm day 1 a mean branching time lapse of 8 days. In other words, radicle segments bearing the first RII are 8 days old. The LUAZ of the other root types was stable over a long period for a given palm age. The LUAZ distributions were also adjusted by a shifted binomial law B(d, N, p) for which the parameters and sample characteristics (mean, standard deviation) are given in Table 5.

0 0

0:80 0:20 0:97 0:03 0:77 0:23 0:95 0:05 0:60 0:40 0:61 0:39 0:45 0:55 0:60 0:40 0:75 0:25 0:88 0:12 0:51 0:49 0:58 0:42 0:65 0:35 0:78 0:22



0 0 0 0

















Figure 7. Histogram of unbranched apical lengths (LUAZ) of radicles and adjustment by a shifted binomial law B(d, N, p) (d: minimum observed LUAZ, N: maximum observed LUAZ, p: probability).

Geometrical parameters The geometrical parameters were not subjected to in-depth modelling; their various mean values were

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227 Table 5. Mean length (m) of the unbranched apical zone (LUAZ) and parameters of the shifted-binomial laws B(d, N, p) adjusted to the distributions for the root type on 10-year-old oil palms Root type Minimum (d) RI H RI VD RII H RII VU RII VD sRIII dRIII RIV

4.0 3.0 1.0 1.0 1.0 0.5 0.5



LUAZ (cm) Mean (m) standard deviation ()

 2.3  2.5  1.8  1.5  1.9  0.6  0.5

12.0 10.0 3.0 3.5 3.1 1.6 1.5

directly introduced into the model, associated with probabilities of variation around these values. In the model, primary roots were drawn at random for insertion in the root-soil plate, at angles of between 15 and 90  from horizontal, as seen in the field. The insertion angles of the different lateral roots of all types, other than RI, were at 90  , with a probability p = 0.05 of varying by 9  around this value. As regards the radial angle, the primary roots were uniformly distributed around the palms. As no rhizotaxy could be detected for all the root types, a random draw between 0 and 2 was adopted to account for it. The main directions of growth of each root type were previously described in the architectural analysis (Jourdan and Rey, 1997b). Among the geometrical parameters common to all the roots, two other phenomena were taken into account in the model: (i) as root growth was not strictly rectilinear, we attributed to each root a probability of slight random deviations and (ii) as each oil palm root retains a constant diameter throughout its life span, the module for axis thickening over time was not used. Synthesis in the reference axis The reference axis, an overall model combining the previous models, allowed simulations that were scheduled from 1 day to a maximum of 25-year-old palms with 1-day calculation intervals. The reference axis was constructed by arranging the different root types in order on the theoretical axis with integration of the root-soil plate (Figure 8a). The ordering arrangement was based on an increasing degree of differentiation in parallel with plant ontogenesis, from “exploration roots” (RI) to “exploitation roots” (RIV).

Probability Maximum (n)

(p)

18.0 16.0 6.0 7.0 7.0 4.0 4.0

0.57 0.54 0.46 0.47 0.37 0.35 0.30

The rate of primary root emission was governed by the number of roots emitted per day with a given probability. The number was calculated so as to reproduce the total number of roots seen to be present on the root-soil plate throughout the lifetime of the palm: approximately 300 roots at 4 years, 3,000 at 10 years and 7,000 at 20 years. Each root type kept the same degree of differentiation, or in other words the same “physiological age” irrespective of the age of the oil palm (Jourdan, 1995). This particularity of oil palm roots made it possible to develop an oriented automaton that took into account the structure of the architectural unit already discovered (Jourdan and Rey, 1997b). As a result, each state of the automaton was represented by a single root type. The transition probabilities from one state to another formalize the branching process, i.e. the probability for a given root type of bearing one or more others. Thus, with this automaton, it was now possible to reconstitute the architecture of the oil-palm root system and its evolution over time (Figure 8b). The initial conditions are entered using the editor of AMAPsim software. For each state of the automaton, the values of the different estimated parameters obtained by modelling the various processes are given. In order to describe overall development of the oilpalm root system using just a few observed oil-palm ages, from 0 to 1 year, 3, 10 and 20 years in our case, we carried out linear interpolations between the values taken by the parameters of the processes for each root type at these specific ages. Therefore the characteristics of the parameters for the intermediate ages were automatically defined.

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228

Figure 8. Reference axis modelling the oil-palm root system (A). Diagrammatic representation of root system architecture based on the reference axis (B).

Simulation and display By simulation, the AMAPsim growth generator softarare was used to generate 3-D numerical models that characterized root system architecture. The 3D numerical models were displayed with appropriate software, LANDMAKER (Reffye et al., 1995). The results of simulations are given for palms aged 1 month (Figures 9a, b), 1 year (Figure 9c) and 5 years (Figure 9d). The variability in the observed root populations was restored by using a “seed"” i.e. a random number that served to initialize simulation. Each 3-D numerical model then resulted from a comparison between this random number and the probability values of the previously entered laws. A set of simulations could thus restore the natural variability of the initial population. Validation Qualitative > of the models was based on a visual comparison between the simulations and drawings or photos of the field situation. This approach, which was done exclusively to the appreciation of the observer, offered the advantage of being rapid and easy to implement, but it was subjective. The various simulations described in this article gave a fairly accurate picture of reality. In particular, the 3-D numerical models satisfactorily reflected the hierarchization of the root system into vertical and hori-

zontal exploration roots, and exploitation roots. The ontogenesis sequence of the different root types over time were also well respected. The quantitative > by remodelling described here was intended to test the ability of the simulation software to transcribe the introduced parameter values, without modifying them, and the coherence of root system architecture restitution by the reference axis. For the branching process, we compared the Markov chain parameters, for different root types and for palms of different ages, obtained before and after simulation. The comparison of the Markov parameters (initial II and transition P probabilities) for observed (obs) and simulated (sim) 1-month-old radicles can be presented as: 

IIsim

=

IIobs

=



0:45 0:55 0:33 0:67





and Psim

=

and Pobs

=





0:51 0:49 0:81 0:19 0:53 0:47 0:80 0:20





The correspondence between the values of the branching parameters obtained before and after simulation was highly satisfactory, for both the radicle and for the other root types, irrespective of palm age. Consequently, simulation restored the previously characterized branching process well.

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229

Figure 9. Display of 3-D numerical models of an oil-palm root system at 1 month (whole: A, close-up: B), 1 year (whole: C) and 5 years (where only the RI and RII are shown: D).

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230 Discussion and conclusion

Mortality process

In a stochastic model, the biological hypotheses are not expressed explicitly, unlike in so called deterministic models. Such a model is therefore not explanatory, it is simply descriptive, though this does not prevent interpretation of the biological phenomena it describes, or justification of the modelling hypotheses put forward. Hypotheses concerning the growth, mortality and self pruning processes were taken into account when developing the mathematical model.

Given the lack of significant senescence markers in oil palm, we characterized the root mortality process by the “final” length of the axes. This parameter was satisfactorily observed for short roots, generally due to their short life span (radicle, RIII and RIV). On the other hand, for roots running into metres, we determined it by taking into account their maximum observed or estimated length. In fact, for RI VD and RII VD, our observations in trenches, one of which went down to a depth of 4 m, failed to reach the maximum rooting depth of 19- year-old palms, but earlier observations at La M´e had revealed the presence of RII VD at a depth of almost 6 m in a trench 4 m away from the palms (Rey, 1988). These values were confirmed at Dabou (plantation 100 km from La M´e) by Braconnier and Caliman (1989). It should be remembered that the soil at the study site has no apparent discontinuity and is considered to be generally homogeneous down to a depth of at least 7 m. We therefore “limited” rooting to a depth of 7 m. For the RI H, and also because of a lack of observations on oil palms at the end of their exploitation, we arbitrarily fixed a “limit of lateral extension” that we estimated at 25 m. These values need to be confirmed with a view to simulating oil palms aged 10 years and over under the same soil and climatic conditions as those found at La M´e.

Growth process For a given palm age and each root type, the growth rates incorporated into the model corresponded to the mean value of growth rates observed at different times of the year. The means integrated instantaneous growth rates that fluctuated depending on the season. Indeed, during the dry season (especially in January and February), we observed a smaller number of roots in the field rhizotrons and their growth rate was slow, whereas in the rainy seasons (May–June and October–November), there were many roots with high instantaneous growth rates. The model, which took into account mean steady growth over the year did not reflect these seasonal fluctuations. Seasonal root growth variations were also observed by Ruer (1968) and Dufrˆene (1992). Under the conditions at La M´e, Dufrˆene (1992) observed higher RIII and RIV growth rate during the months of June–July and November-December, which coincided with the end of the rainy seasons, whereas Ruer (1968) revealed the existence of a “growth peak” in December–January, which he correlated with the mean effective sunshine in the previous two months. On the other hand, under the more humid conditions found in Indonesia, root growth is far more spread out over the year (Dufrˆene, 1992). These two authors quantified growth according to root biomass results. It needs to be checked whether this seasonal biomass increase was linked to an increase in growth rate, or to greater root emission and branching, or to a combination of these two parameters. Root production, which seems to be greater at the end of the rainy seasons, must be linked to the physiological condition of the entire palm which, at these periods of the year, is at the end of the maximum bunch yield phase.

Self pruning process The self pruning process can be characterized by the time taken for dead roots to become separated from the structure that bore them. We chose to parameterize it by putting forward the following hypothesis: an axis only self prunes once all its lateral formations have selfpruned, a little along the lines of a process of mortality by exhaustion of the structure. The short peripheral axes die and self-prune first, then the bearing axes, etc. According to this principle, borne roots have a shorter life than the axis that bears them, which is also usually larger in diameter. This fits in with Marshall and Waring’s model (1985) according to which roots have a predetermined life span, have a finite quantity of available carbon, in starch form, and only die once the stock has been exhausted. Small diameter roots, which have a smaller stock than large diameter roots, die first. However, Vogt and Bloomfield (1991) pointed out that an opposite model exists, according to which roots have an indeterminate life span and only die when environmental conditions become unfavourable.

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231 Although many biotic factors (hormones, carbon balance between sources and sinks, etc.) and abiotic factors (soil temperature and moisture content, mineral content, etc.) involved in the mortality process have been identified, the events responsible for root senescence are still not clearly understood (Kramer and Kozlowski, 1979). Quaternary roots observed in a minirhizotron had a life span of around 3 to 4 weeks and self pruned a month after they died. Based on these data and the hypothesis put forward above, we estimated the time taken for self pruning of the other root types. The estimated values, particularly those for RI in the field (200 days) ought certainly to be confirmed by additional observations, since there is currently no precise information in the literature to validate these values. Consequently the sensitivity of the overall model to the values of parameters of this self-pruning process has been tested in Jourdan and Rey (1997a). The reference axis By using the reference axis as the overall mathematical model, we were able to numerically characterize the architecture of the oil-palm root system, and its development over time. This type of stochastic model falls into the category of 3-D root-architecture mathematical models that have been developed in the last ten years (Berntson, 1994a; Diggle, 1988; Fitter et al., 1991; Le Roux, 1994; Pag`es et al., 1989). These models take into account the geometrical parameters of each root, unlike the root growth models developed a decade earlier, such as so-called empirical models (Gerwitz and Page, 1974), numerical models (Lungley, 1973) or algebraic models (Hackett and Rose, 1972; Porter et al., 1986; Rose, 1983). Other growth models based on application of fractal geometry to architectural analysis of root systems have been developed recently (Fitter and Stickland, 1992; Nielsen et al., 1994; Shibusawa, 1994; Tatsumi et al., 1989; Van Noordwijk et al., 1994). Although some of these models enable the definition of “branching rules”, the carbon allocation scheme, total root length, or their distribution according to diameter, they do not take into account, more especially, a case where several distinct root types are borne on the same axis (case of the oil palm radicle, juvenile RI and RI H). Moreover, they are often limited to a two-dimensional description of the branched systems observed (Berntson, 1994b). One of the major limitations of the present model is the very large size of the computer files for 3D numerical models and the considerable calculating

capacity required (for graphic stations only). A 3-D numerical model of the complete root system of a 1year-old oil palm (40 RI) takes up 40 Megabytes, at 4 years (300RI) it takes up 1 Gigabyte. Several simplification procedures have been undertaken, mostly involving geometrical parameters, but any simplification entails a loss of information: the compromise to be found therefore depends on the application sought. For example, simplification may consist in only keeping the most representative parts of the system and then seeking one or two factors of proportionality. However, the excessive size of the 3-D numerical models is mainly due to the considerable size of the oil-palm root system (several billion roots of all types at 10 years for example!) The different 3-D root-architecture models found in the literature usually involve annual plants with a relatively small root system, such as Zea mays (Pag`es et al., 1989), wheat (Diggle, 1988), Senecio vulgaris (Berntson, 1994a) or Phaseolus vulgaris (Nielsen et al., 1994). Models exist for perennial plants, but simulations have very often been limited to seedling root systems, as is the case with Hevea brasiliensis (Le Roux, 1994; Pag`es et al., 1995), Prunu spersica (Pag`es et al., 1992) or Pinus radiata (Brown and Kulasiri, 1994). As with most of the root system architecture models published to date, our simulations, which were generated by the AMAPsim software, do not restitute environmental effects on the root system, but interactions between the soil system and the root system are many and very often induce behavioural changes in roots (Fitter, 1991). It is difficult to take these interactions into account, because they are complex and little is known about them, and because the simulation tools used are not appropriate. Using the voxel space, which has already been used for many forestry applications (Blaise and Reffye, 1994; Reffye et al., 1995) and even for root systems (Brown and Kulasiri, 1994), seems to be a very promising technique for taking plant-environment interactions into account. The overall mathematical model used in this work combines several stochastic models and is a probabilistic model. Although it restitutes root architecture more or less realistically, it is nevertheless incapable of accounting for growth correlations (carbon flow) that exist within the whole plant, but also within the root system (Brouwer, 1981). More generally, whole plant functioning is not covered in this type of model, which remains primarily descriptive. Be that as it may, the simulation results obtained through 3-D numerical models open up the way for numerous practical

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232 and agronomic applications, such as an estimation and distribution of total root biomass, location and quantification of the absorbent surfaces and an estimation of the useful soil volume for nutrient uptake (Jourdan and Rey, 1997a).

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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