AOBPreview originally published online on November 23, 2007
Annals of Botany 2008 101(8):1207-1219; doi:10.1093/aob/mcm272
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Computing Competition for Light in the GREENLAB Model of Plant Growth: A Contribution to the Study of the Effects of Density on Resource Acquisition and Architectural Development
1 Ecole Centrale Paris, Laboratoire MAS, Châtenay Malabry, F-92295, France
2 INRIA-Futurs, DigiPlante, Châtenay Malabry, F-92295 France
3 INRA, UMR AMAP, Montpellier, F-34000 France
4 CIRAD, UMR AMAP, Montpellier, F-34000 France
* For correspondence. E-mail paul-henry.cournede{at}ecp.fr
Received: 24 February 2007 Returned for revision: 24 May 2007 Accepted: 22 August 2007 Published electronically: 23 November 2007
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX 1: PARTITIONING THE...APPENDIX 2: COMPUTING THE...ACKNOWLEDGEMENTSLITERATURE CITED |
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Background and Aims: The dynamical system of plant growth GREENLAB was originally developed for individual plants, without explicitly taking into account interplant competition for light. Inspired by the competition models developed in the context of forest science for mono-specific stands, we propose to adapt the method of crown projection onto the x–y plane to GREENLAB, in order to study the effects of density on resource acquisition and on architectural development.
Methods: The empirical production equation of GREENLAB is extrapolated to stands by computing the exposed photosynthetic foliage area of each plant. The computation is based on the combination of Poisson models of leaf distribution for all the neighbouring plants whose crown projection surfaces overlap. To study the effects of density on architectural development, we link the proposed competition model to the model of interaction between functional growth and structural development introduced by Mathieu (2006, PhD Thesis, Ecole Centrale de Paris, France).
Key Results and Conclusions: The model is applied to mono-specific field crops and forest stands. For high-density crops at full cover, the model is shown to be equivalent to the classical equation of field crop production ( Howell and Musick, 1985, in Les besoins en eau des cultures; Paris: INRA Editions). However, our method is more accurate at the early stages of growth (before cover) or in the case of intermediate densities. It may potentially account for local effects, such as uneven spacing, variation in the time of plant emergence or variation in seed biomass. The application of the model to trees illustrates the expression of plant plasticity in response to competition for light. Density strongly impacts on tree architectural development through interactions with the source–sink balances during growth. The effects of density on tree height and radial growth that are commonly observed in real stands appear as emerging properties of the model.
Key words: Functional–structural plant models, GREENLAB, competition for light, Beer–Lambert Law, plant plasticity, dynamical system
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX 1: PARTITIONING THE...APPENDIX 2: COMPUTING THE...ACKNOWLEDGEMENTSLITERATURE CITED |
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Functional–structural plant growth models combine the description of the structural dynamics of plants and of the ecophysiological processes governing resource acquisition and repartition. Readers are referred to Sievänen et al. (2000) or Prusinkiewicz (2004) for the presentation of general concepts and reviews, and to Vos et al. (2007) for the presentation of recent progress. Models of this type usually propose a very detailed representation of the individual plant architecture (at least the above-ground part), giving the geometric description of all organs (internodes, leaves, fruits, flowers, etc.); for example see M
ch and Prusinkiewicz (1996) or de Reffye et al. (1997). Such an accurate representation allows the study of ecophysiological processes at the level of organs. Good examples are given by Fournier and Andrieu (1998) on maize, Gautier et al. (2000) on white clover or, more recently, Wernecke et al. (2007) on barley. The detailed geometry is interesting as it allows determination of the microclimate sensed by individual organs (see Chelle, 2005). For the light local environment, light models estimate the radiative fluxes received by each organ by modelling the radiative exchanges in the whole structure (see Soler et al., 2003 for an example of the radiosity method, or Chelle and Andrieu, 2007, for a review). Leaf photosynthesis (e.g. Allen et al., 2005), leaf transpiration (e.g. Dauzat et al., 2001) or other phenomena are computed accordingly.
However, the exact geometric representation of real canopies remains attainable only through the very laborious work of plant digitalization (e.g. Sinoquet et al., 1998). Moreover, digitalization only allows represention of the particular plants studied, and most results are difficult to generalize. Thus, light models are usually used on virtual canopies generated by 3-D geometric simulations of crops or stands (Chelle and Andrieu, 1998; Soler et al., 2003). These virtual canopies may quite satisfactorily mimic some global geometric properties of real canopies, such as the gap fraction (see Fournier and Andrieu, 1998), but their detailed geometries necessarily differ, which may entail a significant bias in the computation of radiative fluxes. Whilst the use of light models to compute accurately the radiative exchanges in canopies is of obvious interest for the understanding of plant biological processes, their application to model the growth of real plant populations remains to be achieved.
Quite paradoxically, there is a real need to study the extension of individual-based functional–structural models to plant populations, especially in terms of competition for light. The first attempts were mostly geometrical. Spatial interactions of branches belonging to different trees are used to control their development. The original idea dates back to Mitchell (1975), and later inspired Mch and Prusinkiewicz (1996) and Blaise et al. (1998). Kurth and Sloboda (1999) used neighbouring plant geometries to estimate the shade at each shoot and infer local development rules. In Evers et al. (2006) and Wernecke et al. (2007), the effect of plant density is modelled only for tillering. To our knowledge, only LIGNUM (Perttunen et al., 1996) uses an explicit model of competition for light to compute resource acquisition. An empirical function gives the shading effect. However, this function is empirical and specific (given for a particular Scots pine stand in specific environmental conditions). Wider generalization seems difficult.
In contrast, models of competition for light have been broadly studied in individual-based, process-based or empirical models used in forest science. The literature on the topic has abundant since Bella (1971) and Mitchell (1975). For empirical models, competition indices are usually defined to characterize the growth condition of each tree. Readers are referred, for example, to Corral Rivas et al. (2005), who present the most usual indices and compare them, and to Pretzsch et al. (2002) for the definition of a 3-dimensional index based on tree crown models. Sorrensen-Cothern et al. (1993) present a process-based model integrating competition for light and plasticity in crown development. Even though the architecture is not described in detail, the model is in essence very similar to what we believe should be done at the level of functional–structural models, since it is based on the Beer–Lambert law for resource acquisition and on the relative growth demand of tree sectors for biomass allocation. This model has inspired the definition of our method to compute competition for light in the GREENLAB model.
This paper begins with the presentation of the main functional concepts used in the GREENLAB model, its resource allocation equation for the individual plant, and how to extrapolate this equation to an explicit formulation of interplant competition for light according to density, both for the context of homogeneous crops or tree stands and for spatially heterogeneous plant populations. Only the case of intraspecific competition is considered in this paper, since we do not study interactions between different species; this field of research would imply more conceptual ecological studies. We refer the reader to de Wit (1960) for pioneering works on this subject or to Damgaard (2004) for more recent approaches.
Our model of competition is applied to determine the production of field crops according to density, and its validity is discussed by comparing it to the classical equation of field crop production (Howell and Musick, 1985). The advantages of our approach to account for local effects such as uneven spacing or variability in plant productions resulting from unsynchronized emergence or seed variability are highlighted.
The application of the model to trees illustrates the expression of plant plasticity in response to competition for light. Density strongly impacts on tree architectural development through interactions with the source–sink balances during growth.
For the sake of clarity, mathematical derivations are described in separate appendices at the end of this paper.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX 1: PARTITIONING THE...APPENDIX 2: COMPUTING THE...ACKNOWLEDGEMENTSLITERATURE CITED |
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In this section, we present the model of resource acquisition used in the functional–structural model of plant growth GREENLAB (de Reffye and Hu, 2003; Yan et al., 2004) and detail how we extrapolate this model to stands. We also recall some basic concepts about plant plasticity that expresses themselves in a competitive context, particularly for trees through the interactions between functional growth and architectural development.
A source–sink model with a common pool of reserves
The main hypothesis used to compute functional growth is that the biomass produced by each leaf is stored in a common pool of reserves and redistributed among all organs according to their sink strengths. The initial seed and the leaves are sources. Leaves, internodes, fruits, roots and rings (resulting from plant secondary growth) are sinks. Secondary growth is the process controlling the increase of branch diameters and is known to be a very important phenomenon required to properly describe density effects in tree stands (e.g. Deleuze and Houllier, 1995; Deleuze, 1996). The Pressler Law specifies that ‘the increment of diameter in a particular point on the stem is proportional to quantity of foliage above the point’ (Mitchell, 1975). In GREENLAB, it is assumed to be true for every branch and the secondary growth is computed in a very efficient way thanks to an algorithm based on substructure instantiations (Kang et al., 2002). It is possible to construct all the stacks of rings not only for the trunk, but also for every branch or twig (see Fig. 1). This method to determine secondary growth is completely compatible with the common pool of reserves. The quantity of foliage above each internode in the tree architecture is computed thanks to the structural factorization, and it is used to weight the internode sink for cambial growth. In this way, it is not necessary to handle an exhaustive exploration of the plant topological structure, in contrast to what is done for most simulation models (e.g. Perttunen et al., 1996; de Reffye et al., 1997).
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In the latest version of the model, the balances between sources and sinks represented by the ratio of available biomass to the organs' demand are used to control the interactions between functional growth and architectural development. This allows the dynamical represention of the plastic response of the plant to environmental constraints (cf. Mathieu, 2006 and Mathieu et al., 2006). In the context of competition for light in tree stands, this interaction plays a key role, which will be detailed later.
Model of resource acquisition for the individual plant
Barthélémy and Caraglio (2007) see the development of a plant as the expression of a precise and ordered sequence of morphogenetic events, namely the construction of new botanical entities. In GREENLAB, the time unit to compute the ecophysiological functioning (resource acquisition and allocation) is chosen to coincide with the time unit of this morphogenetic sequence and is called the growth cycle. The new botanical entity set in place at each growth cycle is called the growth unit (or shoot). Therefore, the individual plant is described as a discrete dynamical system. At growth cycle n, the empirical equation of its biomass production Qn is given by
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n is a known function of the environmental conditions at cycle n, Sn is the leaf surface area of the plant at cycle n and β, 
are so-called hidden model parameters, estimated from experimental data by model inversion, as detailed by Guo et al. (2006).
An analogy can be drawn between this empirical equation and the classical relationship linking production of field crops with transpiration. Howell and Musick (1985) showed that biomass production is proportional to crop transpiration, T. It is driven by the potential evapotranspiration, the soil water content and the exposed green leaf surface area, denoted by Sl,n. If En denotes at cycle n the product of the potential evapotranspiration modulated by a function of the soil water content, Tn is estimated as:
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The exposed green leaf surface area (per unit surface area) can be seen as the chance of hitting a leaf when casting a vertical light ray through the vegetation canopy. Note that it is equal to 1 minus the so-called azimuthal gap fraction. It is evaluated using the Beer–Lambert extinction law (Nilson, 1971):
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Finally, the biomass production per unit surface area Qc,n is:
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To compare the production equation of the individual plant (eqn 1) and that of the field crop, we use a very common concept developed in forestry models (see for example Bella, 1971, or Bonan, 1988). The two-dimensional projection of space potentially occupied by the plant onto the x–y plane defines a characteristic surface denoted by Sp. Equation (1) may thus be written:
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Despite this analogy, it is important to note that in our model the parameters of the individual-based production equation are retrieved by inverse methods, (see Guo et al., 2006; Ma et al., 2007a). Parametric estimation from experimental data from maize crops seems to show that taking Sp as a constant parameter during the whole of plant growth is good enough to calibrate biomass production for small mono-stem plants.
Extrapolating the resource acquisition model to stands
The ‘local LAI’ analogy can help us to derive a model of competition for light in stands. As done by Bella (1971), Gates et al. (1979), Bonan (1988) in the context of dendrometric forestry models, or more recently by Bauer et al. (2002) in ecological modelling and Alsweis and Deussen (2005) for computer graphics, we assign each individual in the stand a disk-shaped zone, Sp, representing its assumed crown projection (or zone of influence). Competitive neighbours are defined as two plants having overlapping disks. However, unlike these authors, we do not compute competition indices but instead estimate the exposed green leaf area, Sl, for each plant using the Beer–Lambert extinction law, so as to evaluate its individual biomass production.
The computation of the exposed photosynthetic foliage area in eqn (5) is based on the assumption of vertical irradiation, which is justified by the time unit considered to compute photosynthesis (at least several days), and uses the so-called Poisson model: Nilson (1971) demonstrated, citing both theoretical and empirical evidence, that the gap fraction can generally be expressed as a decreasing exponential function of the LAI, even when the random turbid medium assumptions generally associated with the Poisson model are not satisfied.
Therefore, in order to compute the exposed photosynthetic foliage area of every plant, we consider the Poisson model for each one individually, using Sp. We assume a stochastic spatial distribution of the foliage area, such that the projection of this distribution onto Sp is uniform. If we consider dS to be an infinitesimal surface element of Sp, the stochastic distribution of the number of foliage layers covering it can be approximated by a Poisson model of parameter k(S/Sp) (by approximation of the binomial model). It is straightforward to get the gap fraction from this model, but not to compute the exposed foliage area of each plant, as we need to consider more closely what happens in the zones where the surfaces Sp overlap.
For this purpose, we partition the crown projection surface of each plant according to the competing neighbours, as shown in Fig. 2 (which represents a simple example of partition) and in Appendix 1 (which gives the mathematical details). Then, on each surface element of the partition, we compute the expectation of the exposed leaf surface area according to the foliage area distribution of all the competitors involved for this surface element. The formula is quite straightforward for spatially homogeneous stands and is presented in Appendix 2. In the case of heterogeneous stands, it is also necessary to consider the vertical distribution of the foliage area in the crown to study the chance of domination of each plant competing for light, and to combine the Poisson models. Assuming that the vertical distribution is uniform, we can derive the exact formula. It only depends on the parameters of the Poisson models, kj(Sj/Sp,j), of all the competitors, their total heights and the heights of their crown bases. As the exact formula involves infinite series, we propose in Appendix 2 an approximated equation. The comprehensive mathematical computation can be found in Cournède and de Reffye (2007).
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S2 It is important to note that this method is very general and would apply when computing resource acquisition in any individual-based plant growth model, as long as the model does not consider the local environmental conditions of every individual leaf in order to determine photosynthesis.
From this competition model, we deduce the biomass production of every plant at each growth cycle, resulting in an incremental reserve pool available to all growing organs, namely sinks (see Yan et al., 2004, or Guo et al., 2006 for details). Mathieu et al. (2006) showed that the ratio of available photosynthates to the sum of all competing sinks can be chosen as a key variable to control the architectural development of trees. In a competitive context, resource allocation (and thus the above ratio) may be strongly affected. As a consequence, the plasticity of plants expresses itself as they adapt their architectures in response to environmental constraints.
Modelling plant plasticity in a competitive context
Density has a strong impact on both plant development and plant growth. In this paper, we restrict ourselves to competition for light and we do not take into account competition for water or nutrients. The quantity of light received by plant foliage and its quality are both affected by density. Mutual shading always induces a decrease in biomass production, but plants usually adapt themselves to environmental constraints in order to compensate functionally for the inevitable reductions in total plant growth and biomass that occur under resource limitation (e.g. Sultan, 2000). The response varies with the type of plant. Functional traits (sink and source relationships) or architecture can be modified. Such flexibility allows the plant to optimize its growth strategy.
Physiological plasticity
We do not detail in this paper the plasticity of functional traits and we refer the reader to Sultan (2000) for a general presentation of the phenomena. However, we will briefly note some experimental observations in order to illustrate the ability of a plant to adapt its strategy, and note how GREENLAB model parameters may be affected. It is important to note that these physiological adaptations have crucial consequences on biomass production and repartition, and thus need to be taken into account along with the computation of light interception (described in the previous section) when modelling growth in a competitive context.
For example, density experiments made on maize (Ma et al., 2007b) have revealed a stability of stem development (but disappearance of tillers), a diminution of the specific leaf weight and a good stability of source and sink balances. For sunflower (Rey, 2003) or tomato (Dong et al., 2007) stem development is also stable, leaf thickness and secondary growth are shown to decrease, while source and sink balances are strongly modified: internodes and petioles increase their relative sink strengths and the fruit demand diminishes. If the environmental conditions are not too extreme, plants tend to stabilize their heights and their leaf surface areas.
Tree architectural plasticity
We first note some basic botanical concepts underlying the model of architectural development in GREENLAB. We refer readers to Barthélémy and Caraglio (2007) for a comprehensive review.
1. Tree structure organization
As explained by Barthélémy et al. (1997), the architecture of a plant can be seen as a hierarchical branching system in which the axes can be grouped into categories characterized by a particular combination of morphological parameters. Thus, the concept of physiological age (PA) was introduced to represent the different types of axes. Usually, we need less than five physiological ages to describe the axis typology in a given tree. Axis vigour decreases when physiological age increases. The main trunk is of physiological age 1, main branches are of physiological age 2, and the largest physiological age corresponds to the ultimate state of differentiation of an axis: it is usually short, without branches.
In trees, a high variability between shoot structures can be observed, especially regarding the number of phytomers and the branching patterns. A growth unit is composed of phytomers of different kinds. Each one generally bears an axillary bud of a specific physiological age that will give birth to an axis of the corresponding type. This organization leads to substructure instances that repeat a lot of times in the tree structure (Fig. 3). Reiteration occurs when the physiological age of an axillary bud is the same as that of its bearing axis.
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Phytomers are arranged according to botanical rules, such as acrotony, which defines the prevalent development of lateral axes in the distal part of a parent shoot. For example, in a growth unit belonging to the trunk (PA = 1), the lowest phytomers usually have no branch. They are followed by a sequence of phytomers bearing branches: first short ones (PA = 4), then more-and-more vigorous ones (PA = 3, PA = 2). The uppermost phytomer may eventually bear a reiteration (PA = 1; see Fig. 4).
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2. Plasticity of growth units
The variations in the morphology of growth units according to environmental conditions or tree age have been described for beech trees (Nicolini, 2000), spruce trees (Deleuze, 1996) and oak trees (Heuret et al., 2000). Growth units of beech trees may vary from two phytomers to 15 phytomers, depending on plant age and light conditions. In stressed conditions, only the unbranched basal area develops. Progressively, sequences of phytomers with axillary buds of bigger physiological ages will appear. At the same time, the lengths of the corresponding sequences increase (Nicolini, 2000).
In GREENLAB, the number of phytomers of each kind and the apparition of branches are controlled by the ratio of available biomass to the sum of all competing sinks (Mathieu, 2006). The dynamic evolution of this ratio according to ontogeny and/or environmental conditions allows for reproduction of the evolution of growth unit sizes and branching patterns that are commonly observed on trees. Simple illustrations of the plasticity of growth units and of acrotony are given in Fig. 4.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX 1: PARTITIONING THE...APPENDIX 2: COMPUTING THE...ACKNOWLEDGEMENTSLITERATURE CITED |
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The method for computing interplant competition for light in the GREENLAB model was implemented in DigiPlant software developed at the Ecole Centrale Paris, (see Cournède et al., 2006). Several tests were performed for its validation, from homogeneous crops to heterogeneous tree stands.
Biomass production of field crops
We show that for high-density crops, our model is strictly equivalent to the classical formula of field crop production given by Howell and Musick (1985). From eqn (4), giving the biomass production per unit surface area Qc,n, we deduce the average production per plant, Qn. Let d denote the crop density (i.e. the number of plants per unit surface area), and define Sd = 1/d. Sd can be considered as the average area potentially available to each plant. Thus,
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Comparing with the individual-based resource acquisition equation of the GREENLAB model (eqn 5), we see that both equations are strictly equivalent if the parameter Sp is set to Sd. However, analysis of experiments presented by Ma et al. (2007b) showed that this does not work properly for low densities or at the early stages of growth. Typically, we can consider that when Sd is substantially bigger than the crown projection surface area, then the plant can be considered as isolated and production should no longer vary with Sd. This condition is not satisfied by eqn (9), as this implies a very slow stabilization of production when Sd increases. In Fig. 5, the biomass production according to Sd/Sp computed with the competition model presented in this paper (from eqn B1, Appendix 2) is compared with the Howell and Musick production function (from eqn 9); this is done for two different leaf surface areas. It can clearly be seen that our model coincides with the classical crop production equation for high densities and stabilizes more quickly when Sd increases. As we consider disk-shaped crown projection surfaces (Sp), the plant can be considered as isolated as soon as the distance to its closest neighbours is greater than twice the radius of Sp, which corresponds to Sd/Sp = 4/
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A drawback of our formulation is its (at least apparent) complexity in the intermediate range of densities, between the high densities where the Howell and Musick equation holds and the case of free growth where the same type of equation applies, except that Sp replaces Sd.
In the case of homogeneous stands, we would prefer to have a consistent formulation of the form:
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Sd when Sd is small (high densities) and f(Sd) = Sp for Sd/Sp 
4/ (open-grown plants). Preliminary tests made on maize (Ma et al., 2007b) and tomato (Dong et al., 2007) in different density conditions seem to prove that for a given Sd (i.e. for a given density) there is a Sp,exp (empirical parameter obtained by model inversion) such that the production equation is well-approximated for all growth cycles by:
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It is straightforward to apply the method of computation to spatially heterogeneous crops. It may potentially account for uneven spacing caused by poor germination and for competition between plants of different sizes resulting from late emergence or differences in seed biomass. The importance of these effects on production variability has been pointed out by Pommel and Bonhomme (1998). A specific experiment is currently being carried out on sugar beet in partnership with the French institute for research on sugar beets (ITB, Paris) in order to quantify emergence variability and to test the competition model in this heterogeneous case.
In Figs 6 and 7, the ability of the model to cope with the effects of poor germination on production is illustrated by simulation of the growth of three sugar beet plots. Plot A is regular with no missing plants, Plot B has two missing plants and Plot C has four missing. The GREENLAB parameters used for the simulations were obtained by parametric estimation from preliminary experiments carried out by the ITB. Figure 7 shows the evolution of the ratio of biomass production in Plot B and Plot C with respect to biomass production in Plot A. At the early growth cycles, the ratios of biomass production equal the ratios of plants in the plots (Plot B to Plot A, 23/25 = 0·92; and Plot C to Plot A, 21/25 = 0·84). When the effects of competition increase, these ratios increase, which means that the production deficit due to the missing plants is partly compensated by the growth of others.
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Evolution of Sp
Whilst it seems reasonable to consider a constant Sp for crops, this hypothesis should not remain valid a priori in the case of trees, considering their architectural transformations and the evolution of their crown sizes. We thus propose to link Sp to the foliage area using an allometric relationship of the form:
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( 
[0;1]) are system parameters that are variety dependent. For the isolated plant, we thus have:
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When = 0, we get back to the usual GREENLAB equation (cf. eqn 5). The case = 1 corresponds to geometrical growth and potentially leads to an infinite spatial occupation. When 0 
< 1, we can prove that either Qn tends to zero, or that, after some time, the plant reaches a stable production regime and Qn remains within given bounds, before the plant's end-of-life decay. Consequently, Sn also remains within bounds and a maximal crown size is reached, which defines a maximal projection surface area Sp,max that can be computed theoretically. Several numerical tests tend to show that, under specific conditions, Qn can be approximated by:
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Plastic response of trees to density
The effects of density on tree growth and tree development are different from that of a simple reduction of incident light intensity (which would imply decreasing En in the GREENLAB production equation). Light intensity decreases from top to bottom inside the canopy. It triggers a diminution in biomass acquisition for the lower branches for a constant total demand of organs. The ratio of biomass production to organs' demand (Q/D) decreases. Consequently, the life span of the new branches is smaller, and some axillary buds abort. It allows stabilizing of carbon allocation to the remaining organs. Among others, Lanner (1985) observed that from the viewpoint of source–sink physiology ‘the amount of elongation occurring in a stem unit is relatively insensitive to less than drastic environmental variation’. Trees tend to maintain the length of their growth units. As a consequence, it is broadly observed that, within a reasonable range, stand density has very little effect on growth in height (Smith, 1962; Baskerville, 1965; Sjolte-Jorgensen, 1967). The clear responsiveness of radial growth to spacing is also very well known. As tree crowns get smaller with increasing density, diameters of tree boles decrease. Lanner (1985) suggested that source–sink relations in trees may help explain why radial growth is strongly influenced by spacing while height growth is not.
As a step of the validation process of our approach, we aimed at reproducing these results as emerging properties of the combination of the competition model and of the model of interactions between functional growth and architectural development
We present some simulation results obtained with a virtual tree resulting from previous studies (Mathieu, 2006). The physiological parameters are set approximately, since the parametric estimation of the GREENLAB model for big trees is still under investigation. However, the different botanical phases of the tree architectural development are faithful to the classical concepts presented by Barthélémy and Caraglio (2007).
We first present the results of simulations of homogeneous stands composed of this model tree, for a wide range of densities. In Fig. 8, the patterns of tree height and basal diameter are consistent with the expected results: tree diameter is very sensitive to density, in contrast to tree height. In Fig. 9, some of the simulated trees are represented to show the impact of density on architectural development and to illustrate plant plasticity. It is interesting to note that all the trees are generated with a completely deterministic system using the same endogenous parameters and the same climatic conditions; only the competition conditions differ. Interactions between functional growth and architectural development will induce important variations in growth unit sizes, branching thresholds and branch life spans.
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As the competition model is very general, it can be used in a large variety of cases, such as spatially heterogeneous stands or dynamically evolving stands, for example to test thinning strategies. In Fig. 10, a schematical presentation is made of the simulation of a simple heterogeneous clump of trees.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX 1: PARTITIONING THE...APPENDIX 2: COMPUTING THE...ACKNOWLEDGEMENTSLITERATURE CITED |
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The proposed model of competition appears like a generalization of the resource acquisition equation of the GREENLAB model, and is shown to be consistent with the classical field crop production of Howell and Musick (1985). It is interesting to note that the same arguments would still hold for the light-use-efficiency approach developed by Monteith (1977) to estimate carbon uptake by vegetation. The model is very general and can be applied to both even-aged and uneven-aged plant populations, in both spatially homogeneous and heterogeneous conditions.
Our approach is in keeping with the GREENLAB strategy to analyse plant growth in interaction with the environment at an integrative level. The objective is to estimate from experimental data and by inverse methods the endogenous (hidden) model parameters driving the source–sink relationships among all plant organs, but at the scale of the whole plant. This approach has proved to be very efficient and very stable for a large variety of cultivated plants. Ma et al. (2007a) present the results of this analysis for maize over several experimental seasons, with very stable parameters.
In this research context, the first objective of the derived model of competition for light is to allow the analysis of the competition conditions through which the plant has grown. Likewise, it should allow us to isolate the mechanistic effects of competition for light (typically the decrease of resource acquisition due to mutual shading) from the physiological or morphological effects of plant adaptation to their constrained environments.
Whilst the model properties seem promising, the validation process on real plants is still in its early stages, and several problems still remain to be solved. First, we need to compare the results of the model calibration on experimental data with previous calibrations carried out using the production equation (eqn 1), which do not integrate competition. If we use this empirical equation in competitive contexts, we can deduce by model inversion an empirical Sp,exp and, at least for several tests made on homogeneous crops (maize, Ma et al., 2007b, and tomato, Dong et al., 2007), the calibration results are satisfactory. This leads us to study the link between this Sp,exp and the crop density, and also with the more general competition model. The comparison between the simple model and its generalized form will help us determine clearly their respective domains of validity.
Besides its usefulness in terms of plant growth analysis, the competition model, when it is coupled with the model of interaction between functional growth and architectural development introduced by Mathieu et al. (2006), can serve as a tool for theoretical botany in order to study the expression of plant plasticity and the emergence of specific botanical phenomena, such as reiterations or sympodial growth. A particular effect of the interaction between growth and development is that trees adapt their sizes automatically according to their neighbours' for functional reasons. It is not necessary to consider geometrical interactions as in previous studies based on spatial competition (e.g. Mch and Prusinkiewicz, 1996, or Blaise et al., 1998) in which tree crowns stop their development because of mutual contact. This method presents two drawbacks: first, the computing cost of spatial detection, and second, the fact that the system behaviour is forced. With our approach, the density effect can be estimated quickly thanks to the system dynamical equations, which implicitly contain the competition effect for each plant.
However, some underlying hypotheses of the proposed model may be too restrictive for some applications. First, because vertical irradiance is assumed, the model will not be able to account for some aspects of stand structure, such as a possible growth advantage for trees on the southern-most edge of a clump over those on the opposite edge. Taking into account angular irradiance would be possible by adjusting the Beer–Lambert equation. Second, we do not consider anisotropic development of the crown, in order to avoid heavy geometric computation. This simplification is clearly wrong locally, as shown by the modular theory of growth (cf. Franco, 1986). However, as the functioning is defined at the whole-plant level, the modelled tree will behave globally like the particular anisotropic one. For specific applications, modular development could be implemented using models of spatial arrangement (e.g. Sumida et al., 2002). Finally, the computation of the exposed photosynthetic foliage area assumes a stochastic spatial distribution of the foliage area, such that the projection of this distribution onto Sp is uniform. This hypothesis of uniform distribution may be too simple when faced with real plants. However, in GREENLAB, Sp is estimated from experimental data. Basically, it corresponds to using a theoretical distribution of leaf area whose vertical projection onto Sp is uniform and which is equivalent in terms of global light interception to the real distribution of leaf area in the plant crown. Likewise, to compute the probabilities of dominance between neighbours, we also give ourselves a vertical stochastic distribution of foliage and the theoretical calculation is only done in the uniform case (Appendix 2). Sinoquet et al. (2005) showed that deviation from randomness is mainly due to spatial variations in leaf area density within the canopy volume. It may thus prove useful to study a more general case of stochastic distribution, as this may be strongly related to crown geometry, and perhaps to link these stochastic distributions to existing crown models (e.g. Pretzsch et al., 2002).
An interesting extension of this work would be its application to mixed crops and multi-species plant communities. Even if the general framework used to compute the exposed leaf area would still hold, this would not be sufficient since different plant species have different strategies to obtain their necessary share of the resources in order to grow and reproduce (Damgaard, 2004). For example, trees of different species will produce their new leafy shoots at different periods in spring, which may have some important effects on light interception. Interactions between plant species have been investigated in ecology, often by performing two-species competition experiments (e.g. de Wit, 1960). The empirical concept of interspecific competitive forces between different plant species has been introduced; however, the way to make the link with our method to compute competition in the context of functional–structural growth model still remains to be studied.
In 1975, Mitchell had already foreseen the importance of simulation techniques to study the evolution of forest systems and their ‘response to initial spacing, thinning, genetic selection, fertilization, damages...’. For this purpose, individual-based models were presented as the most promising means of study, as they were more detailed and faithful to reality than whole-stand models, despite their computational load. Today, owing not only to the progress of computers but also to the mathematical formalisms that have been introduced (e.g. the structural factorization of plants in Cournède et al. 2006), it is possible to model and simulate the growth of plant populations at the level of individual leaves and internodes without any computational restriction. Moreover, new mathematical formalsims open up opportunities for new applications, as they make it possible to estimate model parameters from experimental data by model inversion, and optimization and control techniques can be used to improve cultivation methods.
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APPENDIX 1: PARTITIONING THE CROWN PROJECTION SURFACES |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX 1: PARTITIONING THE...APPENDIX 2: COMPUTING THE...ACKNOWLEDGEMENTSLITERATURE CITED |
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Let (Pi)i
[1,N] denote the set of plants in the stand and 
(i) denote the neighbourhood of the plant Pi, defined by: |
| (A1) |
Let 
[(i)] be the set of all subsets of (i) and let 
[(i)]. We define S(i,) by:
|
| (A2) |
is the complementary part of Sp,j.
All the Sp,j being disk-shaped surfaces, the exact formula giving S(i,) can be computed theoretically. However, the formula becomes quite laborious in the case of heterogeneous stands when Card() 3.
Hence:
|
| (A3) |
)} for 
((i)) is a partition of Sp,i. |
APPENDIX 2: COMPUTING THE EXPOSED FOLIAGE AREA OF THE PLANT Pi |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX 1: PARTITIONING THE...APPENDIX 2: COMPUTING THE...ACKNOWLEDGEMENTSLITERATURE CITED |
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From Appendix 1, we deduce that the exposed foliage area Sl,i of the plant Pi is given by:
|
| (B1) |
) is the probability that dS, an infinitesimal surface element of Sp,i, is covered by the foliage of the plant Pi, and that moreover Pi dominates its neighbours in over dS; that is to say that the upper foliage element covering dS belongs to Pi.
We also denote Pc,i the probability that dS, an infinitesimal surface element of Sp,i, is covered by the foliage of the plant Pi. With the Poisson model, we know that
|
| (B2) |
and we prove that |
| (B3) |
In the case of spatially heterogeneous stands, we need to assume a vertical stochastic distribution of the foliage area in the plant crowns in order to determine Pd(i,). For uniform vertical distributions, we can combine the Poisson models of the competitors to derive the exact formula (after laborious calculations). It involves infinite series and only depends on the Pc,j of all the competitors, their total heights, Hj, and the heights of their crown bases, hj (see Cournède and de Reffye, 2007 for more details). The result is not presented here as we prefer using the following approximation:
|
| (B4) |
|
| (B5) |
= min[Hi, max(hi, hj)] and β = max[hi, min(Hi, Hj)]. Even though it may appear rather complex, this equation usually simplifies.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX 1: PARTITIONING THE...APPENDIX 2: COMPUTING THE...ACKNOWLEDGEMENTSLITERATURE CITED |
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We thank the anonymous reviewers and the handling editor Thierry Fourcaud for their helpful comments on this manuscript.
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LITERATURE CITED |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX 1: PARTITIONING THE...APPENDIX 2: COMPUTING THE...ACKNOWLEDGEMENTSLITERATURE CITED |
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