Performance of trees in forest canopies: explorations with a bottom-up functional–structural plant growth model


  • F. J. Sterck,

    Corresponding author
    1. Utrecht University, Plant Sciences, PO Box 80084, 3508 TB Utrecht, the Netherlands;
    2. Wageningen University, Forest Ecology and Forest Management Group, PO Box 342, 6700 AH Wageningen, the Netherlands
      Author for correspondence: Frank Sterck Tel: +31 317 478046 Fax: +31 317 478078 Email:
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  • F. Schieving,

    1. Utrecht University, Plant Sciences, PO Box 80084, 3508 TB Utrecht, the Netherlands;
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  • A. Lemmens,

    1. Wageningen University, Forest Ecology and Forest Management Group, PO Box 342, 6700 AH Wageningen, the Netherlands
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  • T. L. Pons

    1. Utrecht University, Plant Sciences, PO Box 80084, 3508 TB Utrecht, the Netherlands;
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Author for correspondence: Frank Sterck Tel: +31 317 478046 Fax: +31 317 478078 Email:


  • • Here we present a functional–structural plant model that integrates the growth of metamers into a growing, three-dimensional tree structure, and study the effects of different constraints and strategies on tree performance in different canopies.
  • • The tree is a three-dimensional system of connected metamers, and growth is defined by the flush probability of metamers. Tree growth was simulated for different canopy light environments.
  • • The result suggest that: the constraints result in an exponential, logistic and decay phase; a mono-layered-leaf crown results from self-shading in a closed canopy; a strong apical control results in slender trees like tall stature species; the interaction between weak apical control and light response results in a crown architecture and performance known from short stature species in closed forest; correlated leaf traits explain interspecific differences in growth, survival and adult stature.
  • • The model successfully unravels the interaction effects of different constraints and strategies on tree growth in different canopy light environments.


Trees grow to larger sizes and produce more complex crowns than other plants, but use the same ‘building blocks’ as other plants. These building blocks are the metamers, each of them consisting of an internode (including shells of xylem, cambium, and phloem), a node, one or more leaves, one apical meristem (apex) and an axillary meristem in the axil of each leaf (White, 1979; Bell, 1991; Room et al., 1994). Metamers represent the local units from where a tree grows: carbohydrates (further referred to as ‘carbon’) are acquired by the leaves, which can be used for the maintenance of living tissues, invested in radial internode growth, and in the production of new metamers. The question arises how metamers respond (grow, rest, die) to tree status (internal factors) and environment (external factors) and, in turn, how they jointly produce a developing, integrated, complex, three-dimensional (3-D) structure such as a tree. There is no easy answer to this question because metamer responses involve different phenomena (e.g. branching, apical growth, radial growth, death, dormancy) and magnitudes (e.g. branch rates, growth rates, Sterck & Bongers, 2001; Sterck et al., 2003), and metamer responses result from interactions among different factors, for example carbon availability (King, 1991a; Perttunen et al., 1996), phyto-hormones (Kramer & Kozlowski, 1979; Wilson, 2000), ontogeny (Halléet al., 1978; Sterck & Bongers, 2001), position (Halléet al., 1978; Sterck, 1999), genotype (Halléet al., 1978; Coen, 2004), acclimation potential (Stoll & Schmid, 1998; Sprugel, 2002), and light conditions (King, 1991a; Sorrensen-Cothern et al., 1993; Sterck et al., 2003). Functional–structural models are needed to integrate the effects of these different metamer responses into growing, 3-D tree structures.

A number of functional–structural plant models have been developed over the past decades (Room et al., 1994; Mäkelä, 2003; Mäkeläet al., 2004). Static tree models with a strong physiological – morphological basis describe the 3-D structure and light environment of plants and calculate the carbon economy for tree saplings (Warren-Wilson, 1981; Pearcy & Yang, 1996; Sinoquet et al., 2004) and simple taller trees (Chazdon, 1986). These models evaluate how morphological and physiological traits contribute to the crown architecture and carbon economy of trees (Valladares et al., 2000, 2002), but not how this affects long-term tree growth and development. Tree growth models relate whole-tree carbon economy to three–dimensional growth (Sorrensen et al., 1993; Paccala et al., 1996; Perttunen et al., 1996; Bartelink, 1997; Mäkeläet al., 2004). In these models, the ‘mechanisms’ (physiological, developmental, morphological properties) were defined at the lower organizational levels (e.g. leaves, metamers, shoots, or branches), and whole plant parameters were added to produce growing trees of realistic size and shape (Mäkeläet al., 2004). Virtual plant models apply formal growth or branch rules, for example l-systems and fractal geometry (Aono & Kunii, 1984; Prusinkiewicz & Lindenmayer, 1990; Mech & Prusinkiewicz, 1996; West et al., 1999; Enquist, 2002), to describe three dimensional structures, but often lack a strong physiological, morphological or developmental basis. In this paper, a new functional–structural tree model is presented that analyses the responses to light for, for example, tropical rain forest conditions, and responses to other environmental factors and hormonal processes are excluded at this stage. The model calculates the carbon acquisition and carbon expenses at the metamer level, integrates these processes to the whole plant carbon economy and, ultimately, simulates growing trees as specified by growth and death rules at the metamer level, within the limits of the available carbon reserves. By contrast to earlier tree-models, the whole tree carbon economy and three-dimensional growth are fully emergent properties of the joint action of the metamers (there are no whole plant parameters set a priori). The model thus provides a theoretical tool to integrate metamer responses into growing 3-D tree structures and, most importantly, to separate the effects of various constraints and plant traits on tree growth in a light limited environment, such as a tropical rain forest.

For tropical rain forests, broad patterns emerge on how species differ in crown shape, leaf distribution, branch patterns, growth and survival in relation to the adult stature and shade-tolerance of the species. The processes that underlie these broad interspecific patterns still belong to ‘the secrets’ of the invisible, long, histories of forest-trees. Light is expected to play an important role as it is a most variable, limiting, resource in these forests, and many rain-forest trees indeed die, or only just survive, under the given light-driven carbon limits (Clark & Clark, 1992). The canopy is thus the most obvious constraint on a tree's carbon economy. Canopies vary from a closed forest to forests with gaps of different sizes. The associated light levels range by two orders of magnitude (Chazdon & Fetcher, 1984), and strongly affect plant architecture, physiology, growth and survival (e.g. Veneklaas & Poorter, 1998). In addition, a closed canopy exhibits a vertical gradient with light levels ranging by the same order of magnitude. A second major constraint is the shading of lower leaves by higher leaves in the same tree. This so-called ‘self-shading’ may affect growth and plant architecture (Horn, 1971; Sterck et al., 2001; Poorter et al., 2003), but, except for saplings (Valladares et al., 2000, 2002), there is little proof of the role of self-shading in the broader, interspecific, performance-patterns.

Growth strategies reflect different ‘evolutionary – ecological choices’ as exhibited by different genotypes, species or functional groups. We compare model-trees that represent genotypes, species or functional groups differing in development or leaf traits. The development traits reflect the physiological, morphological, and developmental processes that affect the flush probability of metamers: Apical dominance enables an apex to control the activity of lower lateral meristems (e.g. Brown et al., 1967; Cline, 1997), apical control reduces the growth of lateral shoots relative to the leader shoot (Wilson, 2000), and light-foraging facilitates the growth of shoots in more favorable light conditions (Stoll & Schmid, 1998; Sprugel, 2002). These processes contribute to the structural development of any tree and, probably, to the differential growth-patterns among functional groups, species, genotypes, and con-specifics (Wilson, 2000; Kramer & Borkowski, 2004; Sterck, 2004).

The leaf traits in the present study reflect the strong correlations among different leaf traits in tropical forest tree communities, as well as in other plant communities (Reich et al., 1992; Wright et al., 2004): maximum assimilation and specific leaf area decrease with increasing leaf life span, and thus counteract the contribution of a longer leaf life to the carbon economy of a tree. Short leaf life traits seem most advantageous in productive habitats (Chabot & Hicks, 1982; Williams et al., 1989). Here, leaves ‘pay back’ their costs rapidly and negative shading-effects are avoided by producing new leaves higher, in more favorable light conditions. Long leaf life traits seem advantageous in less productive habitats. Here new leaves face low, but less variable, light levels and can only pay back their costs by extending their life span. In such conditions, mechanical and physiological protection decrease the maximum assimilation and specific leaf area, but contribute to a longer, functional, leaf life. It can be hypothesized that trees with a short leaf life maintain a positive carbon economy and achieve competitive superiority in open conditions, where high assimilation and low carbon expenses per leaf area maximize their growth rate. In shade conditions, the long leaf life trees are the only ones that pay back their leaves in time, and therefore persist better in darker conditions (Kitajima, 1994; Kobe et al., 1995; Veneklaas & Poorter, 1998). The present paper explores the effects of the constraints (light environment, carbon economy), the development traits (apical control, light response) and the correlated leaf traits on tree growth using a new functional–structural plant model. The results will be discussed in the context of broad interspecific patterns among rainforest tree species.

Methodology: The Model

Plant structure

In the model, a tree is a three-dimensional structure of connected metamers. Each metamer consists of a segment (node plus internode), a petiole, a leaf-blade, an apical meristem, and an axillary meristem (Fig. 1). Segments and petioles are defined as cylinders, leaf-blades as ellipses, and the two meristems as points at the top of a segment. The three-dimensional tree structure is defined by the coordinates of each cylinder and ellipse of the tree in the positively oriented orthonormal basis (e1, e2, e3) (north, west, zenith), the ‘world-coordinate system’.

Figure 1.

A schematic design of the three-dimensional tree structure. (a) One metamer: the vertical line represents the woody pipe in the segment (node plus internode); the dots on top of the segment represent the apical and axillary meristem; the horizontal line below the axillary meristem is the woody pipe in the petiole; the petiole is connected to an ellipsoid leaf-blade; the dotted cylinder at the stem basis represents the segment that consists of one living pipe in this one-metamer system (not drawn to scale). (b) A simple tree consisting of three metamers. Note the increase in the segment radius due to an increasing number of woody pipes. (c) The same tree, but leaf 1 was dropped and the appending pipe died (dotted line), but remains part of the first segment. (d) The same tree, but the metamer of leaf 3 was dropped. Note that there are now two dead pipes running through the lowest segment.

For each cylinder, we define a (local) positively oriented orthonormal basis (c1, c2, c3) (Fig. 2). Here c1 is oriented in the direction of the central axis of the cylinder; c2 of a segment points to the attachment of the petiole; c2 of a petiole lies ‘left’ of c1 in the (e1, e2)-plane; c3 = c1 × c2. The (c1, c2, c3) basis is defined by the coordinates of its origin in the world-system, and the azimuth and elevation of c1. Every cylinder has its own (c1, c2, c3)-specifications, length and radius, and is thus coordinated in the world-system (see Pearcy & Yang, 1996, for similar calculations for YPLANT).

Figure 2.

The orientation of the local-bases of metamer-components. The origin of the local bases are coordinated in the world-system (e1, e2, e3), see as an example, the vector from e to B1. B1 is the basis of the cylinder on top of it, representing a segment. B2 is the basis of a segment on top of the first segment, that is in apical direction. Bases B3 and B4 are located at the same coordinates as B2, but were drawn separately to keep the orientation of different bases visible. They represent a lateral segment and a petiole, respectively. Note that the lowest segment, the petiole and the leaf-blade (all dashed lines) belong to the same metamer.

For each leaf-blade, we define a (local) positively oriented orthonormal basis (l1, l2, l3) (Fig. 2). The (l1, l2, l3) origin corresponds with the point where the petiole is attached. Here l3 is defined as the leaf normal for the upper side of the leaf, l1 is placed along the length axis (midrib, from leaf basis to tip), and l2 is placed along the width axis of the leaf. The (l1, l2, l3) basis is defined by the coordinates of its origin in the world-system, the azimuth and elevation of l3, and the azimuth of l1. Every leaf-blade has its own (l1, l2, l3)-specifications, and with the length and width (see Pearcy & Yang, 1996, for similar calculations for YPLANT) it is coordinated in the world-system.

The whole metamer is thus coordinated by first the local bases of segment, petiole, and leaf-blade, second the coordinates of these bases in the world-system, and third the dimensions of the segment, petiole, and leaf-blade. The three local bases of each metamer are calculated from the local segment-basis, that is of the connected, proximal, metamer. Since for every leaf, petiole and segment we know the explicit orientation with respect to the world-system, phyllotaxis and elevations can be easily implemented.

Finally, every leaf-blade is connected by a woody pipe to the tree basis (Fig. 1). Such a pipe has a constant cross-section and runs from the leaf-blade basis, through the petiole, through all the connected segments, down to the tree basis. The petiole radius equals the pipe radius. The cross-sectional area of a stem segment equals the sum of all cross-sectional areas of the pipes running through it (Watson & Casper, 1984; Franco, 1985).

In a tree, segment, petiole, and leaf-blade are grouped in one object, that is the metamer. In turn, the metamers of one tree are grouped in a nested-list. At the highest level, the list encompasses the whole tree and starts at branch order 0 (the main stem); at the next level, the lists represent the different branches that start at branch order 1. In theory, this pattern may be repeated over and over. The three-dimensional structure of a tree is thus defined by a (nested-) list and the coordinate-specifications of each metamer-component (segment, petiole, leaf-blade) in that list.

The production of metamers

A model-tree develops when it produces new metamers, or when it drops metamers or leaves. At each time step, such development depends on a number of specifications at the metamer level.

The model quantifies the flush probability of each metamer (i.e. the probability that a metamer produces one or more new metamers, plus the attached pipes) at every time-step (Figs 3, 4). The flush probability depends on probabilities that relate to the metamer position, the axillary inhibiting factor, the local light level and, indirectly, the carbon status of the tree. First metamer position: the probability PP is set to 0 for metamers that have no apex, and set to 1 for metamers with a free apex (Fig. 3a). Second the axillary inhibiting factor: PN of a metamer is reduced with a certain fraction from one branch-order to the next (Fig. 3b). Third local light response ability: PL scales linearly with light levels between a minimum and maximum daily light level (Fig. 3c). Fourth the carbon status: the carbon effect PC is implemented by drawing at random metamers until the carbon pool, gained over the previous time step, is exhausted (see section ‘carbon economy’). We can now summarize the effects of these factors on the flush probability PF by the product of probabilities (for all codes in equations, Appendix 1, available online as supplementary material):

Figure 3.

The flush probability is the product of three probabilities. (a) The metamer position. Metamers with an apex have full flush potential, metamers without have no flush potential. (b) The branch order N. Flush probability decreases with a constant fraction per branch order. Examples: fraction = 0.75 (open dots), and 0.25 (black dots). (c) The daily photosynthetic light intensity (mol m−2 d−1). Flush probability increases linearly with increasing light between a minimum and maximum daily light level.

Figure 4.

The flush options of the presented simulations are represented by arrows. PF, flush probability; PA, probability of producing metamer at apical position. PAL, probability of producing one metamer at apical position, and one at lateral position.

image(Eqn 1 )

When a metamer is selected for flushing, it produces one or more new metamers according to a selected production rule or l-system (Aono & Kunii, 1984; Prusinkiewicz & Lindenmayer, 1990). In presented simulations, a flushing metamer has a probability PA to produce a new metamer at the apex (the branch extends), and PAL to produce a new metamer at the apex and another new metamer at the axillary metamer (the branch extends and ramifies, see Fig. 4). Note that with every new metamer a woody pipe is produced, running from leaf blade basis to tree-basis. This means that the segments between a new metamer and the tree-basis increase their cross-section area with the cross-section area of the new pipe, and that they adjust their cylinder radius accordingly. This so-called ‘pipe-model assumption’ ignores radial growth responses to local mechanical stresses (Morgan & Cannell, 1987; Mattheck, 1991), but is considered a valid simple model to link extension with radial branch growth in growing trees (Shinozaki et al., 1964; Mäkelä, 1986, 2003; Sterck, 2004). We chose this production rule, and not a more complex one (e.g. in Halléet al., 1978; Prusinkiewicz & Remphrey, 2000), since it is simple and still creates a wide variety of growing, 3-D trees.

The loss of metamers and leaves

Loss rules are also evaluated per metamer at every time step. A leaf and appending pipe die when the leaf has passed its maximum age, or when the net photosynthesis is negative (leaf respiration exceeds gross photosynthesis, e.g. Givnish, 1988, see section ‘carbon economy’). The segment and attached meristems survive as long as there are living pipes in it, that is as long as the segment or more distal segment supports one or more leaves. A tree thus drops metamers and whole branches and, ultimately, dies when it has no leaves anymore (Fig. 1; King, 1994).

The carbon economy of a tree

The production of each new metamer requires carbon; the tree pays the costs of the new leaf-blade and appending pipe. These costs are calculated as follows:

image(Eqn 2 )

Here, for a leaf and connected pipe i, Ci is the carbon-cost (kg), F is the C-mass to biomass ratio, ALi is the leaf area, SLAi is the specific leaf area, ρPi is the pipe density, APi is the pipe cross-sectional area, and LPi is the pipe length. The constant 1.45 reflects that 45% of the C invested in the new leaf or pipe is lost as ‘growth-respiration’ (Poorter & Villar, 1997).

To determine the carbon income of the tree we reason as follows. Let R describe the total maintenance costs per day (leaves and living pipes). Let Σ Pi describe the gross photosynthesis rate per day, summing photosynthesis for the individual leaves in the tree. Given time step Δt (in days), the carbon status (or pool) at time t is given by:

image(Eqn 3 )

Here inline image is the remaining carbon in the pool at time t−1. A tree thus does not store carbon. The respiration of pipes (R) was calculated on a mass-based respiration rates, whereas the respiration of leaf was scaled with the maximum photosynthetic capacity. This approach ignores the effects of temperature (Ryan & Waring, 1992) and other factors (Amthor, 2000; Gifford, 2003), but was considered a simple, appropriate, tool to compare model-trees.

The gross photosynthesis per day Σ Pi,t−1 sums the photosynthesis of individual leaves (more precisely, the leaf-blades). For an individual leaf i, let Pi describe the gross photosynthesis per day, Ai the area, Pi(x) the gross photosynthesis rate per area per day at the leaf center x, and Pi(x, t) the instantaneous gross photosynthesis rate per area at that leaf center x at time t. For units, see Table 1 and Appendix 1. We can now write:

Table 1.  Fixed parameters settings in the presented simulations
  • a

    leaf length and leaf width;

  • b

    pipe length is defined by the length between leaf blade basis and root (Shinozaki et al., 1964), and segment length (length of the internode plus node) is fixed at 0.2 m;

  • c

    c pipe cross-sectional radius is calculated from a fixed leaf area to pipe cross-sectional area ratio (set to 20000);

  • d

    approximately the average leaf density value for multiple species, Mexican forest trees (Bongers & Popma, 1990), and the average wood density value for various multiple-species data sets (Reyes et al., 1992; Niklas, 1994);

  • e

    ‘golden’ phyllotaxis for spiral organization (Bell, 1991);

  • f

    variable, see Table 2.

  • g

    ∼ mean maintenance rate for xylem (Penning de Vries, 1975; Veneklaas & Poorter, 1998); production of new tissue costs ∼145% of the carbon investment in structural biomass, i.e. 45% of the carbon is lost by respiration (Poorter & Villar, 1997).

Length0.20 & 0.10a0.02bm
Cross-section radius 0.00050.0005cm
Density300d600d600dkg m−3 biomass
Maintenance costsf60.0g60.0gnmol C mol−1 C s−1
Construction costs1.451.451.45mol C mol−1 C
image(Eqn 4 )
image(Eqn 5 )

For leaf i, the gross photosynthesis rate at leaf center x, Pi(x) is the integral of the instantaneous photosynthetic rate PAi(x, t) over the time interval between sunrise (sr) and sunset (ss). PAi(x, t) is a function light intensity I(x, t) and is given by the nonrectangular hyperbola:

image(Eqn 6 )

(Johnson & Thornley, 1984), where Φ is the quantum yield (the initial slope of the light response curve, mol CO2 mol−1 photon), θ is the curvature factor (0 ≤ θ ≤ 1), PCAi is the asymptotic light-saturated gross photosynthetic rate (mol m−2 s−1), and I(x, t) is the light intensity (mol m−2 s−1) at the leaf center. The parameters Φ, θ, and PCAi are set as constants for each model-tree (Appendix 1), but PCAi was varied among trees (see the section ‘Parameter settings’ below). Photosynthetic parameters were constant over the whole leaf life span, as suggested by earlier studies on rain forest trees (Raaijmakers, 1994; Rijkers, 2000, but see other patterns for other plants, Chabot & Hicks, 1982; Kikuzawa, 1991). The light intensity I(x, t) at the center of the individual leaf depends on the solar track, the light extinction by a surrounding canopy, and the light extinction by the tree-crown (self-shading).

Light intensity

The total light intensity at the center of the leaf at time t is given by:

image(Eqn 7 )

Here Is(x, t) is the direct light and Id(x, t) the diffuse light at leaf center x at time t. For the direct and diffuse light we can write:

image(Eqn 8 )
image(Eqn 9 )

where τcs, φs) and τts, φs) are the transmission coefficients of the canopy and tree-crown for the direct light coming from the solar arc, <ess, φs), l3> corrects for the angle by which the leaf surface with normal l3 is hit by the light coming from e(θs, φs), and fss, φs) is the direct light coming from the solar arc. In the diffuse light equation, τcd, φd) and τtd, φd) are the transmission coefficients of canopy and tree-crown for diffuse light coming from the sector with azimuth φ and elevation θ, <edd, φd), l3> corrects for the angle by which the leaf surface with normal l3 is hit by the light coming from (θd, φd), and Idfd, φd) is the diffuse light intensity coming from the solid arc cos θ dθd dφd. We further evaluate these equations in three steps: the direct light intensity without shading effects, the diffuse light intensity without shading effects and, ultimately, the shading effects by canopy and tree crown on the direct and diffuse light intensities.

The light intensity coming from the solar arc fss, φs) can be written as:

image(Eqn 10 )

where fsc is the solar constant (1360 Wm−2), τa is the transmission coefficient for the atmosphere, and θs is the solar elevation angle (Campbell, 1977). The constant 0.45 refers to the 45% of the solar constant that is received as direct photosynthetic active radiation.

The total diffuse light intensity in (e1, e2)-plane (earth surface) coming from the hemisphere approximates 10% of the direct light intensity from the solar arc, under clear sky conditions (Pearcy & Yang, 1996):

image(Eqn 11 )

On a normalized hemisphere (radius set to 1), the light intensity is considered constant and is given by:

image(Eqn 12 )

This constant light intensity is needed in order to consider the light extinction from different directions. When we divide the hemisphere in sectors, we can write:

image(Eqn 13 )

Here I(θ, φ) = Ih is the spherical light intensity (mol photons m−2 s−1) that comes from the centroid at azimuth angle φ and elevation angle θ. The sector surface is calculated (dθ dφ cos θ), correcting for the change in surface area with increasing elevation (cos θ). The dot-product <e(θ, φ), l3> corrects for the angle by which a leaf surface is hit by the light coming from e(θ, φ). Ultimately, we may integrate the light intensities of the different arcs over the hemisphere. Such an integration is useful when we consider the light extinction in the different hemispherical sectors.

The amount of light that is ultimately received by the leaf depends on the shading effects of the surrounding canopy, and the tree-crown (self-shading). The canopy is characterized by a height and a spherically homogenous leaf distribution, with leaves being infinitely small. In addition, there may be a gap with a certain radius (Fig. 5). The tree structure is simplified for the calculation of the light environment. The bounding box of the tree-crown is divided in 10 × 10 × 10 = 1000 blocs of equal size, and the leaf area in each box is calculated (Fig. 5). For a leaf, the light transmission that comes from an arc, that is from one of the sectors or for the sun, is determined by the different path lengths with constant leaf densities in the direction of the center of the arc. From some given leaf, let LAI’i describe the projected leaf area in the direction of the arc with azimuth φ and elevation θ, for a block lying on the path from that leaf in the direction of (θ, φ). Then:

Figure 5.

Light interception illustrated for one leaf of a tree in a canopy with a gap. The canopy is characterized by a height, homogeneous leaf distribution, and a LAI. The tree-crown is surrounded by a bounding box, with 10 × 10 × 10 boxes of equal size (for clarity, fewer blocks are shown). Diffuse light is calculated for different sectors of the hemisphere, direct light is calculated for the line pointing towards the sun (solid line). Note that lines may intercept the canopy and/or tree-crown, in case of which transmission of light is reduced (see Methodology section).

image(Eqn 14 )

with λi the projected leaf area density in that block, and Li the path length through that block (Fig. 5). The transmission coefficient τ(θ, φ) can be written as:

image(Eqn 15 )

Where K is the absorption coefficient, set to 0.86 (Appendix 1, Valladares et al., 2002). Furthermore, for paths outside the bounding box of the tree crown and only in the canopy, the transmission coefficients can be calculated in the same way:

image(Eqn 16 )

with λc the projected leaf area density in the canopy into the direction (θ, φ), and Lc the path length through the canopy for that direction (Fig. 5). Thus, the total transmission coefficient τ(θ, φ) is just:

τ(θ, φ)  = τt(θ, φ)τc(θ, φ)

And in a similar way the transmission coefficient in the direction of the solar beam is calculated. We used this light extinction procedure, rather than the more detailed interception models (see, e.g. YPLANT, Pearcy & Yang, 1996), to avoid excessive calculation times in trees that rapidly produced thousands of leaves.

We now have all the elements to solve Eqns 8 and 9, and can thus calculate the total light intensity at the leaf center x at time t. From this we can solve the instantaneous photosynthetic rates at the leaf center in response to light (Eqn 6) for the hours between sunrise and sunset, and thus calculate the leaf gross photosynthesis per day (Eqn 4). By following the same procedure for all the leaves in a tree, the gross photosynthesis rate per day was calculated for the whole tree, and the net acquired carbon per time step. This is crucial, since the acquired carbon per time step is the major ‘engine’ of tree growth in the model.

Parameter settings

Model-trees thus grew according to growth and death specifications at the metamer level, within the limits of available carbon. A few parameters were set to constant values, taking the average value for the range of values reported for tropical rain forest tree species in the literature (Appendix 1). For the metamer dimensions a different approach was taken: the 20-cm-long segment and 20-cm-long leaf were chosen to simulate trees growing to considerable size, without the need to produce tens of thousands of leaves (Table 1). Excessive calculations (mainly for light interception) were thus avoided. Given that some tree-growth parameters are still estimated with considerable uncertainty, for example the respiration load of wood, one should of course be careful with taking the results in an absolute sense.

The model is therefore used in a comparative way in order to unravel the interaction effects of different constraints (flush probability, carbon economy, leaf longevity, and shading) and strategies (axillary inhibition, light response, correlated leaf traits) on tree growth. We compared ‘wide trees’ that had a low axillary inhibition (trees 1–5, Table 2) with ‘narrow trees’ that had a high axillary inhibition (trees 6–10, Table 2). For the wide trees 1–4 and the narrow trees 6–9 constraints were subsequently added to explore their effects one after the other. The trees 5 and 10 added a second development component, as these trees were able to respond to local light conditions (Table 2, Fig. 3c). The simulated trees 11–13 (Table 2) are wide trees that differ in correlated leaf traits, as has been observed for trees of a tropical forest tree community (Table 2 for calculations, Reich et al., 1991). These trees were ‘grown’ in different forest light environments to understand the effects of interactions between different constraints and development and leaf traits on growing trees. The time step was 10 d.

Table 2.  The settings of variable parameters for the simulated trees
ParametersUnitsSimulated trees Constraints and developmentLeaf traits
  1. Trees 1–10 are simulated in the absence of canopy and in a closed canopy, and trees 11–13 with no canopy, large gap (15 m radius), small gap (5 m radius), and a closed canopy. Leaf specifications were based on contrasting leaf life spans, and conform the broad interspecific pattern observed for a Venezuelan rain forest tree community (specific leaf area = 102.37 leaf−1 life span 0.33; max. assimilation = 102.67 leaf−1 life span 0·65, Reich et al., 1991). Leaf respiration was set to 5% of the maximum assimilation. Note that leaf life span is considered a constraint in simulations 1–10, and part of a leaf strategy in simulations 11–13.

Carbon economy +++++++++++
Leaf life spanmonths  242424  242424  3 1248
Self-shading +++++++
Axillary Inhibition  0.25 0.25 0.25 0.25 0.25 0.75 0.75 0.75 0.75 0.75  0.25  0.25 0.25
Light response ++
Leaf life spanmonths  242424  242424  3 1248
Specific leaf areacm2 g−
Max. assimilationnmol C g−1 s−159.359.359.359.359.359.359.359.359.359.3222.9 93.037.7
Leaf respirationnmol C g−1 s−1 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0  8.2  4.6 1.9

Results & Discussion

Four canopies and their light environments

Four different canopies and light environments were simulated, ranging from a closed canopy, a canopy with a 15-m-radius gap (large gap) and a canopy with a 5-m-radius gap (small gap), to no canopy (Figs 6, 7). The closed canopy and the gap borders were 40 m in height, had a homogeneous leaf distribution and a leaf area index of 3. For each canopy, the incident light was calculated at each hour of the day, and integrated to the daily incident light. From these calculations we obtained contrasting light levels at ground level (Fig. 6), and four vertical light-gradients (Fig. 7).

Figure 6.

Daily course of light intensity in different canopies and in Nouragues, a rain forest site in French Guiana. (a) Daily courses of light intensity in four different model-canopies on March 21, latitude 0°. Canopies were 40 m tall, had a LAI of 3 and had a 15-m radius gap (large), a 5-m radius gap (small), or no gap. Light values at the soil level are shown, in canopies with gaps in the gap center. Solid line, no canopy; dotted line, large gap; dashed line, small gap; dashed/dotted line, closed canopy. (b) The effect of the atmospheric humidity τ is shown for no canopy on March 21, latitude 0°, showing the difference between misty days (τ = 0.6) to extremely clear days (τ = 0.9). Solid line, τ= 0.6, dotted line, τ= 0.7; dashed line, τ= 0.8; dashed/dotted line, τ= 0.9. (c) Minute-averages of photosynthetic radiation (PAR) were measured with LI-COR SA-190 PAR sensors and logged on a LI-COR L-1000 data-logger for an extremely clear day for a sensor position that was not influenced by the forest canopy in Nouragues, a tropical rain forest in French Guiana.

Figure 7.

Vertical gradients of daily light intensity are shown for the four different canopies: a closed canopy, a 5-m radius gap, a 15-m radius gap, and no canopy. Canopy height was set at 40 m, LAI was set to 3, and leaves were homogeneously distributed. Solid line, no canopy; dotted line, large gap; dashed line, small gap; dashed/dotted line, closed forest.

Over the course of a day, light intensity followed the typical bell-shaped curve in the absence of a canopy (Fig. 6), as predicted (Campbell, 1977; Monteith & Unsworth, 1991). The parameter τ represents the gradient in atmospheric humidity, from mist (τ = 0.6) to very clear conditions (τ = 0.9). The data for one of the clearest days measured for a lowland tropical rainforest in French Guiana exhibited the light pattern with a τ-value of 0.8–0.9, but there were some low-value outliers due to small clouds. On such a clear day, daily irradiance passed 50 mol m−2 d−1, which is rather exceptional for rain forest. In further simulations, the τ-value was set at 0.7 to obtain a more realistic light intensity for sunny days (c. 40 mol m−2 d−1).

Light intensity also followed a diurnal bell-shaped pattern at the forest floor in the closed canopy, but at a much lower level than in the absence of a canopy (Fig. 6). The two gaps exhibited intermediate diurnal light patterns. In the centers of these gaps, light levels rapidly rose until the canopy did not intercept direct light anymore. From that moment onwards, the diurnal light curves were similar to that curve in the absence of a canopy, although it was slightly lower as the surrounding canopies intercepted some diffuse light. Later in the afternoon, light declined when the canopy started to intercept direct sunbeams again.

The structural canopy parameters were set at fixed values: the height was 40 m, the leaf distribution was homogeneous, and the leaf area index was 3. A height of 40 m approximates the heights of rain forests realistically (Lieberman & Lieberman, 1994; van der Meer, 1995; Richards, 1996). The homogeneous leaf distribution in the canopy contrasts with the clustered nature of leaves in actual canopies (Goudriaan, 1977; Chazdon, 1988; Russel et al., 1989), and with the variation in leaf density with height (Kira, 1978; Lieberman & Lieberman, 1994; Montgomery & Chazdon, 2001). The leaf area index of 3 in the canopy is lower than most index values reported for tropical forests (Kira et al., 1969; Richards, 1996). Since the low LAI was compensated for by a homogeneous leaf distribution, the parameters jointly resulted in first a realistic negative exponential vertical light gradient in a closed canopy (Yoda, 1974; Ellsworth & Reich, 1993; Koop & Sterck, 1994), from 100% light above the canopy to c. 2–4% on the forest floor (Chazdon & Fetcher, 1984; Valladares et al., 2000), and second in contrasting, realistic, light intensities at the forest floor, ranging from c. 1–2 mol m−2 d−1 in a closed forest, c. 4 mol m−2 d−1 in the small gap center, c. 17 mol m−2 d−1 in the large gap center, and c. 40 mol m−2 d−1 when there is no canopy (van der Meer, 1995). These four environments thus specified the range of canopy structures, and associated light environments that trees may encounter when they grow in tropical rain forest.

The effects of constraints and tree growth

Wide and narrow trees were subjected to a sequence of constraints: no constraints (Table 2, tree 1, 6), carbon economy (Table 2, tree 2, 7), leaf longevity (Table 2, tree 3, 8) and self-shading (Table 2, tree 4, 9). Narrow and wide trees are visualized in Fig. 8, and their growth in different canopies (no canopy vs closed canopy) in Fig. 9. The effects of the different constraints on tree growth are further discussed using Fig. 9(g,j). Note that the numbers in this graph correspond with the different trees of Table 2.

Figure 8.

The architecture of the simulated trees 1–10 (Table 2) in a closed canopy. Note that the line numbers correspond with the tree numbers in Table 2. Note that for wide trees (simulations 1–5) and narrow trees (6–10) the same constraints and strategic factors were added subsequently: no constraints (1, 6), carbon limits (2, 7), limited leaf life span (3, 8), self-shading effect (4, 9) and local light response (5, 10). Trees with fewer than 500 leaves (simulations 1, 2, 3, 6, 7 and 8) and 10-yr-old trees (4, 5, 9 and 10) are shown.

Figure 9.

The effects of constraints and development on growth-patterns in the wide trees 1–5 and the narrow trees 6–10 (Table 2). Note that the line numbers correspond with the tree numbers in Table 2. Note that for wide trees (1–5) and narrow trees (6–10) the same constraints and strategic factor were added subsequently: no constraints (1, 6), carbon limits (2, 7), limited leaf life span (3, 8), self-shading effect (4, 9) and local light response (5, 10). Trees with fewer than 500 leaves (simulations 1, 2, 3, 6, 7 and 8) and 10-yr-old trees (4, 5, 9 and 10) are shown.

Flush probability, carbon economy, leaf longevity and tree size

Trees 1 and 6 grew exponentially, as they were not limited by a carbon economy or leaf longevity (1 in Figs 9g and 6 in Fig. 9j). The wide tree grew most rapidly since its probability to produce new metamers at higher branch orders (Fig. 3b) was higher than in the narrow tree. The growth rates of these trees were thus only limited by the flush probability of the metamers.

Trees 2 and 7 were subjected to a carbon economy (tree 2 in Fig. 9g, and tree 7 in Fig. 9j). After a exponential phase, these trees grew logistically and, as was to be expected, they reached a steady state in biomass (not shown). The logistic pattern resulted from the increase in tree size. The new leaves are connected to longer woody pipes and, consequently, the growth and maintenance costs rise per new leaf and reduce the growth rate. The carbon economy and the fact that the simulated trees tended to grow taller and taller resulted in a size constraint. Trees 3 and 8 nicely show the result of the interaction between the carbon economy, tree size and leaf longevity. After these trees passed the exponential and logistic phase, they ran into a decay phase (tree 3 in Fig. 9g, tree 8 in Fig. 9j) in which they could not pay back their leaves any more due to the high costs of new leaves with ever long woody pipes. Trees gradually ran out of leaves and branches, and died when they dropped the last leaf and branch. The same results were obtained by an analytic version of the model (F. Schieving, unpublished results). These simulations thus show that the trees do not automatically reach a steady state and that they need additional strategic ‘choices’ to reach such a steady state, but such choices were not explored here.

The effects of shading

In the absence of a canopy, there is some effect of self-shading on tree growth (tree 4 in Fig. 9a, and tree 9 in Fig. 9d), but this effect is small in magnitude. In a closed canopy, the same trees 4 and 9 show dramatic decreases in growth rate compared with trees 3 and 8 (Fig. 9g,j), that is the latter trees having no self-shading effect. The mass production goes down for those self-shaded trees 4 and 9, and so does the leaf production (Fig. 9h,k) and height growth rate (Fig. 9i,l). In addition, trees 4 and 9 exhibited a dramatic change in architecture compared with trees 3 and 8 (Fig. 8). They had only few leaves and branches, and most leaves were located in the periphery of the upper crown half due to the slow growth rate and the loss of leaves that fell below the compensation point at the crown bottom (respiration exceeding photosynthesis). These results suggest that self-shading in a closed canopy resulted in mono-leaf-layered crowns. For temperate forest trees, Horn (1971) suggested that trees produce ‘adaptive’ mono-leaf-layered crowns in the shade, but did not relate crown architecture to underlying growth-mechanisms. His view has also been echoed as a working-hypothesis for tropical rain forest trees (Kohyama & Hotta, 1990; Sterck et al., 2001; Poorter et al., 2003). By contrast, we hypothesize that a mono-leaf-layered crown shape is not adaptive, but rather that it results from the strong self-shading effects on trees growing in a closed canopy. This hypothesis applies to rain forest trees as well as to temperate forest trees, but we cannot present any test at this stage (and see further complications, next sections). The model shows that adaptive or strategic effects should not be confused with constraint effects, and thus provides new insights in the cause-effect relationships underlying tree growth in forest light environments.

Interactions between constraints and strategies

After this discussion of interactions between constraints like carbon economy, leaf longevity, tree size and shading, we now introduce different strategies. A first strategic contrast is the wide trees (simulations 1–5, Table 2) vs narrow trees (simulations 6–10), that is those trees that had low vs high axillary inhibition (Fig. 3b, Eqn 1), respectively. Because wide trees had a higher metamer flush probability on higher branch orders, they were thicker, wider and shorter than narrow trees (see Fig. 8, compare Fig. 9c with Fig. 9f, and Fig. 9i with Fig. 9l). In a biological sense, wide trees may be associated with trees that had a lower apical control than narrow trees. A second strategic contrast was introduced with trees 5 and 10 (Table 2). The metamer flush probability in these trees is a function of local light levels, whereas there was no such response in trees 4 and 9 (Fig. 3c, Eqn 1). In a biological sense, trees 5 and 10 may be associated with trees that can ‘forage for light’ (Sorrensen-Cothern et al., 1993; Stoll & Schmid, 1998). To discuss the effects of these development traits, we restrict ourselves to the exponential and logistic growth phases.

When there was no canopy, trees 5 and 10 (with light response) hardly differed from trees 4 and 9 (Fig. 9a,d). In a closed canopy, it appears that wide tree 5 (with light response) had a higher growth and hence a higher ‘steady state’, that is maximum biomass, than wide tree 4 (no response, Fig. 9g). In terms of crown architecture (Fig. 8), tree 5 had a similar round crown shape to tree 4, but at the same time tree 5 had more leaves (Fig. 9h) and had these leaves more equally distributed over the crown volume. The narrow trees 9 (no light response) and 10 (with light response) showed the reverse pattern. Tree 10 grew slower than tree 9 (Fig. 9j), had dropped its bottom branches, and only supported leaves in the crown top (Fig. 8). Tree 10 is long and narrow compared with the wide tree 5 (Fig. 9l vs 9i), faced a stronger vertical light gradient (Fig. 7), and thus favored new metamer production in the upper crown at the expense of the crown bottom (Fig. 8). The light response thus lead to reduced growth rates, as new leaves involved greater costs (due to tree-size constraint) and bottom branches ran out of leaves and were dropped. These results show the subtle effects of the interaction between axillary inhibition (‘apical control’) and light response (‘light foraging’) on trees growing in a closed canopy.

Do constraints and development traits contribute to interspecific functional patterns?

How far do such interaction effects explain the broad interspecific patterns observed in tropical rain forests? The most consistent pattern has been observed for tree species differing in adult stature: taller species produce smaller crowns and more slender stems and grow more rapidly in height than shorter species (Kohyama & Hotta, 1990; King, 1991b; Sterck et al., 2001; Poorter et al., 2003). These patterns largely fit to trees differing in axillary inhibition. The narrow-tree produced narrower crowns, more slender stems and grew more rapidly in height than the wide-tree in both light environments, and the narrow trees thus resemble the taller stature species. In a closed canopy, wide trees grew more rapidly in biomass (but not in height), reached a higher ‘steady state’, that is maximum biomass (Fig. 9g vs 9j), and produced a bigger crown with more leaves, whereas narrow trees (Fig. 9h vs 9k, Fig. 8) rapidly ran into the decay phase due to a strong size-constraint. Self-shading resulted in the ‘mono-leaf-layer’-like crown of tree 4 (Fig. 8). Compared with tree 4 (no light response), tree 5 (with light response) had more leaves (Fig. 9h), distributed those leaves over a larger volume (Fig. 8), and grew more rapidly (Fig. 9g, tree 5 vs tree 4), and thus suggests that a light-response has adaptive value for wide-trees in a closed canopy. Positive effects of a light response have been reported by earlier studies (Sorrensen et al., 1993; Stoll & Schmid, 1998; Sprugel, 2002) but, as far as we know, never for short stature species of tropical rain forest. The resultant crown architecture of tree 5 fits nicely with field studies that show that short species have bigger crowns than taller species (Sterck et al., 2001; Poorter et al., 2003), but contrasts with the shade-adaptive crown shape (mono-layered) suggested by Horn (1971). We hypothesize that the interaction effects of weak apical control vs light response contribute to the crown shape, growth and survival of short stature species in a closed canopy.

Do leaf traits contribute to interspecific functional patterns?

In the trees 11–13 (Table 2, Figs 10 and 11) leaf life span correlated negatively with maximum assimilation and specific leaf area (Reich et al., 1991; Wright et al., 2004). Here we consider trees with leaf life span of 3 (tree 11), 12 (tree 12), and 48 months (tree 13) in four different canopies. As axillary inhibition did not affect the major results (Appendix 2, available online as supplementary material), we present only the results for wide trees. Note that the qualitative ontogenetic biomass patterns of trees 11–13 were similar to those of trees 1–10 discussed (for example, Fig. 11b vs Fig. 9h), but that we present height growth (Fig. 11a,d,g) instead of biomass.

Figure 10.

The architecture of the simulated trees 11–13 in the four contrasting canopies: no canopy, large gap (15-m radius), small gap (5-m radius), and a closed canopy. Trees 11–13 differed in leaf life span (LLS) and correlated leaf traits (Table 2). The illustrated individuals were 1 yr old.

Figure 11.

The effect of correlated leaf traits (Table 2) on tree-growth in the four contrasting canopies: no canopy, 15-m-radius gap, 5-m-radius gap, and closed canopy (see methodology section for details). For leaf trait specifications, see Table 2. Solid line, no canopy; dotted line, 15-m-radius gap; dashed line, 5-m-radius gap; dashed/dotted line, closed forest.

Tree 11 (3 months leaf life span) showed the strongest response to these canopies: in the absence of a canopy, it grew most rapidly in height (Fig. 11a,d,g), in biomass (for leaves, see Fig. 11b,e,h), and had the biggest crown (Fig. 10); in a large gap it grew slower, and it died rapidly in a small gap or in a closed canopy. The architectural variation for this short leaf life tree (tree 11) was fully attributed to differences in the carbon-economy and the resultant growth rate. With increasing shade, tree 11 was unable to replace leaves, ran out of leaves and branches, produced smaller crowns, or died rapidly. Vice versa, the longer-leaf life trees 12 and 13 were less sensitive in carbon economy, growth and architecture in response to the light environments of different canopies, and survived for years in every canopy. These patterns were not affected by the parameter settings (Appendix 2), suggesting that the performance differences among trees differing in leaf traits were robust. These results reflect the patterns observed along the shade-tolerance axis among tropical rainforest tree species: a short leaf life, high assimilation, high specific leaf area, high growth rate, and low survival rate in the shade, is typical for pioneers growing in open environments (Boojh & Ramakrishnan, 1982; Ackerly, 1996; Veneklaas & Poorter, 1998), whereas the reverse pattern holds for shade-tolerants (Coley, 1988; Williams et al., 1989; Reich et al., 2003). Our hypothesis is confirmed as the interaction effects of correlated leaf traits vs light environment resulted in the growth–survival trade-off that has been observed among different tree species (pioneer vs shade-tolerants) of tropical rainforests (Kitajima, 1994).

In open conditions (no canopy and large gap), the short leaf life tree 11 (3 months) turned from positive to negative growth at a shorter stature than trees with intermediate leaf life span tree 12 (12 months) (Fig. 11a,b vs Fig. 11d,e). The tree with the longest leaf-life span (48 months) did not yet reach that height during the simulation (Fig. 11g), owing to its slow growth. The short leaf life tree 11 was most sensitive to the size constraint: at greater size, it did not pay back its leaves due to the increasing growth and maintenance costs of ever longer woody pipes. The intermediate leaf life tree 12 maintained positive growth at the same size and grew to a greater stature than the short leaf life tree 11 (Fig. 11d vs 11a). Leaf economy compared with tree size interactions thus determined the differentiation in carbon economy, growth and maximum stature among these trees, as observed for species that successively dominate in the early phases of secondary forest succession. The simulations suggests that the joint effect of leaf economy and tree size results in short-lived pioneers that win the early height battle but die at a short stature, and in long-lived pioneers that grow slower and taller, and dominate in later phases of succession (Finegan, 1996).


The presented model integrated the growth responses at the metamer level to growing, 3-D tree structures. Given that some tree-growth parameters are still estimated with considerable uncertainty, for example the respiration load of wood, the model results are only interpreted in a comparative way. The results were robust for different parameter settings, and were therefore most useful to unravel the interaction effects of different constraints (carbon economy, tree size, leaf longevity, and shading) and strategies (apical control, light response, correlated leaf traits) on tree growth in different canopy light environments.

The results suggest: first that the constraints result in an exponential, logistic, and decay phase; second that trees need additional strategies to reach a biomass steady state; third that trees need additional strategies to prune lower branches and develop branch-free stems; fourth that a mono-leaf-layered tree crown results from self-shading in a closed canopy, rather than adaptating to such conditions; fifth how the interactions between apical control and light response contribute to differences in the crown architecture and growth between tall vs short stature species; sixth how correlated leaf traits result in the growth vs survival trade-off as observed between pioneer species vs shade-tolerant species; and seventh how correlated leaf traits result in the growth – adult stature trade-off as observed between short-lived pioneers vs long-lived pioneers. Results 4–7 are formulated as hypotheses as they need further validation from comparative field studies and experimental studies. With the rise of multiple species data sets over the last decade (Westoby et al., 2002), we think that testing some of these hypotheses is nowadays possible for tree communities of different forests.


We thank Lex Bijlsma for technical assistance; Niels Anten, Heinjo During and Marinus Werger for their helpful suggestions during MS writing; This study was supported by the grants W 01–53 of the Netherlands Foundation for the Advancement of Tropical Research (NWO-WOTRO).

Supplementary material

The following material is available as supplementary material at


Appendix 1

Explanation of the parameters codes used.

Appendix 2

A sensitivity analysis showing the effects of fixed parameter values (see Table 1) on tree height and survival at the age of 1 year.