Optimal leaf display and biomass partitioning for efficient light capture in an understorey palm, Licuala arbuscula

Authors


  • Author to whom correspondence should be addressed. E-mail:takenaka@nies.go.jp

    ¶Current address: Laboratory of Ecology and Evolution, Department of Biology, Faculty of Science, Shinshu University, Matsumoto 390-8621, Japan.

Summary

  • 1 The effects of leaf display and biomass partitioning on light capture efficiency were examined in a non-branching understorey palm, Licuala arbuscula, by using a three-dimensional geometric simulation model. This species has several fan-shaped laminae, attached on long petioles at a mostly constant deflection angle (DP). The petiole of the youngest leaf was almost vertical, and slanted downwards as it aged.
  • 2 The combination of large DP and small Zmax (zenith angle of the oldest leaf’s petiole) maximized light capture for a plant with few leaves; this combination kept the lamina facing in the approximate direction of the zenith with high light intensity. For a plant with many leaves, the combination of large Zmax and small DP increased light capture because it reduced self-shading.
  • 3 For a given total leaf biomass, the plant increased its total leaf area by producing many small leaves. This occurred because the leaf area per unit biomass decreased with increasing biomass per individual leaf. This effect was most pronounced in larger plants. However, an increasing number of leaves intensified self-shading among leaves. Allocation of biomass to the petioles reduced self-shading, but decreased leaf area.
  • 4 There was an optimal allocation of biomass to petioles and an optimal number of leaves that maximized the crown’s light capture. Greater investment in petioles as the number of leaves increased was the favoured strategy for larger plants.
  • 5 In most cases, the leaf geometry and biomass partitioning in the plants were close to the optima predicted by a simulation model developed in this study. There were noticeable differences in a few cases, but the reduction in the crown’s light capture due to these differences was small.

Introduction

The environment of the forest understorey is characterized by a shortage of light. This reduces the production of photosynthate and limits plant growth and survival; there must have been strong selection pressure on understorey plants to construct crowns that capture light efficiently with the least possible investment of carbon. Light capture by a plant depends on its total leaf area, which is determined by the allocation of photosynthate to leaf production. Whole-plant light capture also depends of the amount of light captured per unit of leaf area; it is determined by the orientation of the leaves with respect to incoming light (Chazdon 1985; Ackerly & Bazzaz 1995; Muraoka et al. 1998; Pearcy & Yang 1998) and the degree of self-shading among the leaves (Warren Wilson 1981; Niklas 1988; Takenaka 1994).

One possible strategy to minimize the amount of self-shading is to make fewer leaves. In the extreme case – the production of only one flat leaf – no self-shading would occur. However, the specific leaf area (leaf area per unit of dry biomass) would decrease with increasing lamina area, because correspondingly greater biomass must be invested to provide sufficient mechanical stability (Chazdon 1986; Niinemets 1996). Therefore, total leaf area is likely to be smaller in a plant that makes few large leaves than in one with many small leaves. Another way to reduce self-shading is to distribute leaves in a larger volume of canopy space. However, this requires more investment in support tissues such as stems, petioles and rachises. Again, there is a trade-off between the total leaf area and the degree of self-shading.

As a result of these trade-offs, it is reasonable to expect that an optimal crown architecture exists for any given light environment and amount of biomass available to construct the crown. Unfortunately, determination of this optimal structure is not straightforward, because light capture by a crown is a non-linear function of the many mutually dependent morphological features of the crown (Pearcy & Yang 1998). Moreover, the complex branching architecture of plants makes it hard to infer the optimal structure under a given set of conditions.

Takahashi & Kohyama (1997) studied the crown architecture of Licuala arbuscula Becc. (Palmae), a small, non-branching understorey palm that grows in lowland rain forests in south-east Asia (Whitmore 1985). The palm has a stagnant vertical stem less than 40 cm high with several fan-shaped, palmately compound laminae on long petioles. The simplicity of the crown structure of L. arbuscula facilitates analysis of the structural dependency of light capture by its crown. Takahashi & Kohyama (1997) discussed the possible advantages of the various morphological characteristics of this species in terms of receiving light efficiently. These characteristics include the deflection angle between the lamina and its petiole, the range of petiole zenith angles and the biomass investment in petioles.

The purpose of the present study was to examine quantitatively the effects of changes in leaf display pattern and in biomass partitioning on whole-plant light capture in L. arbuscula. We tested the hypotheses that L. arbuscula adjusts the maximum zenith angle of the petiole, the deflection angle between the lamina and its petiole, the size and number of leaves and the relative biomass allocation to the laminae and petioles so as to maximize whole-plant light capture.

Materials and methods

PLANT MATERIAL

Licuala arbuscula has palmately compound leaves that clump near the top of the stem (< 40 cm high). The leaf has a thin petiole and wedge-shaped leaflets (≈ 25 or more). The proximal leaflets are shorter than the distal ones, and the petioles do not elongate after the full expansion of laminae. The leaf size and number of leaves within a crown increase in the later stage of plant growth, while the ratio of petiole length to lamina diameter also increases as the plant grows. The leaf lamina deflects ≈ 40° from the horizontal to the inside of the crown when the petiole stands vertically (i.e. the deflection angle between a lamina and its petiole is ≈ 50°). The petiole of the youngest leaf is almost vertical, but slants increasingly downwards with age; consequently, the petiole of the oldest leaf within the crown has the maximum zenith angle. The deflection angle of the lamina to the petiole does not change with age, but the maximum zenith angle within a crown increases as the plant grows. In the juvenile stage, this angle amounts to ≈ 50° and all laminae cluster at the top of the crown. In older plants, the zenith angle increases to ≈ 110°, and the leaves form a hemispherical crown.

THE MODEL PLANT

A computer model of the crown of L. arbuscula was developed based on geometrical measurements of 24 plants growing on the forest floor of a primary lowland dipterocarp forest on the slope of Mount Berui (West Kalimantan, Indonesia). In addition to these non-destructive measurements, we harvested 38 leaves to measure the biomass of the leaf laminae and petioles. Unless otherwise stated, the values of the parameters reported below are based on these measurements. Details of the study area and the sampling procedures are described by Takahashi & Kohyama (1997).

An example of the plant simulated with the computer model is shown in Fig. 1. The petiole of the youngest leaf was vertical, and those of older leaves slanted downwards. In the present study, the zenith angle of the nth leaf [with the youngest leaf in the crown assigned a value of 1 in a plant with N leaves (N > 1)] is defined as Zn = Zmax(n – 1)/(N – 1), where Zmax is the maximum zenith angle of the petioles within the crown. The leaves were arranged in a spiral phyllotaxis, with a divergence angle between successive leaves of 135°.

Figure 1.

An illustration of the computer model of Licuala arbuscula. The dry biomass of each of the five leaves was set at 33·7 g, the relative allocation to the lamina was set at 0·583, Zmax (zenith angle of the oldest leaf) was set at 90° and DP (the angle between the lamina and petiole) was set at 49·6°. Leaves are numbered from 1 to 5 with the increasing zenith angle of the petioles. (a): Side view of the model plant; (b): vertical view. The inset in (b) shows the distribution of sampling points (●) on each lamina for the determination of photon flux density.

The lamina of the fan-shaped, palmately compound leaf was approximated as an ellipse. The ratio of lamina area to the area of the enclosing ellipse was 0·535. The major axis of the ellipse ran from the basal end to the tip of the leaf, and the ratio of the minor axis to the major axis was set at 0·84. From the leaves sampled, it was found that the specific leaf area (SLA, m2 g−1) of the lamina depended on leaf size. A regression equation for SLA was developed as a function of the area of the ellipse that enclosed the leaf’s lamina (E, m2) as follows:

image(eqn 1)

(R2 = 0·360, P < 0·01). The area of the ellipse represented by a given biomass of leaf lamina of the model plant was calculated from this relationship. Generally, SLA is affected by various factors other than leaf size, such as light and water conditions. In the present study, such environmental factors were not taken into account because they did not vary much under the closed forest canopy where the sample plants were located.

The shape of a petiole was approximated as a truncated cone. The diameter at the distal end of the cone was 66% of that at the basal end. The petiole was considered to attach to the major axis of the ellipse that represented the lamina at a point 40% of the distance along the axis from the proximal end of the lamina.

Given the biomass of a petiole and the attached leaf lamina, we determined the length of the petiole by assuming that it could be simulated as a cantilever fixed to the stem and loaded with its own weight plus that of its lamina. In the calculations, the cantilever was assumed horizontal because the bending moment working on a petiole is greatest in this case. The maximum bending stress of the leaves sampled did not show any significant correlation with leaf size. Thus, the maximum stress at the base of a petiole was assumed to be constant at 2·08 × 107 N m−2, which is the mean moment calculated from all the leaves sampled. Based on this assumption, the size of a petiole was calculated for a given biomass of leaf lamina plus petiole. The details of this calculation are shown in the Appendix. The petiole geometry was not determined using an empirical relationship between its mass and the length of leaves sampled because we intended to show explicitly that the mechanical stability is the constraint to which the petiole should conform.

BIOMASS ALLOCATION IN REAL PLANTS

The relationships between total leaf biomass and both the number of leaves and the relative allocation of biomass to the lamina were derived using data from the plants sampled. These relationships were used to compare the optimal crown structure predicted by the computer simulation with the structure of real plants.

Equations 2 and 3 were obtained from the 38 leaves harvested by using a reduced-major-axis regression (Niklas 1994):

image(eqn 2)

(r2 = 0·893, P < 0·01) and

image(eqn 3)

(r2 = 0·952, P < 0·01), where mp and ml are the dry biomasses (g) of a leaf’s petiole and lamina, respectively, L is the length (m) of the petiole and A is the area (m2) of the ellipse that encloses the leaf’s lamina. The total leaf biomass was calculated for each of the 24 non-destructively sampled palms with these equations. From the number of leaves and the estimated total leaf biomass for each of the 24 sample palms, equation 4 was obtained; this was used to estimate the number of leaves (N) as a function of the total leaf biomass (W):

image(eqn 4)

(r2 = 0·695, P < 0·01). The allometric relationship between lamina and petiole was derived from their respective estimated biomasses:

image(eqn 5)

(r2 = 0·914, P < 0·01), where Ml and Mp are the total dry biomasses of all laminae and petioles, respectively. The number of leaves per plant and the relative allocation of biomass to the laminae and petioles were calculated for a given total leaf biomass using equations 4 and 5.

CALCULATION OF LIGHT CAPTURE

Light capture by the model plant was calculated by following the method described by Takenaka et al. (1998) and Muraoka et al. (1998). Firstly, the three-dimensional structure of the model plant was simulated on the computer. Light capture was estimated by incorporating the directionality of the light source, the self-shading among the leaves of the crown and the incidence angles of light striking the leaves.

The sky hemisphere above the horizon was divided into a 200-cell ‘web’ (Anderson 1964). Unlike Anderson’s web, all the cells represented the same solid angle (Fig. 2). Ten sampling points, each representing 10% of the leaf area, were spread uniformly over the surface of each model leaf, as shown in Fig. 1. Obstruction of these sample points by other leaves was assessed for light from each of the 200 cells over the sky hemisphere by using three-dimensional analytical geometry. If leaves did not obstruct the light source at a point, the full quantity of light from that cell was considered to reach the point. Conversely, if the light was obstructed, the photon flux density (PFD) from that cell was reduced by the light absorptance of the obstructing leaf. Here, the light transmissivity was assumed to be 1 – (leaf area)/(ellipse area), i.e. 0·465. After calculating the quantity of incoming light from all cells at a sampling point on the leaf, the PFDs were cosine-corrected using the angle of incidence of the light, and the values from all cells were summed to give the total PFD at the sampling point.

Figure 2.

The web dividing the hemisphere above the horizon into 200 cells of identical solid angle. The web is drawn on a circular image of the polar projection of the hemisphere. The arrow indicates the direction of north.

Determining the angular distribution of the incoming light that reaches a plant on the forest floor requires estimates of the distribution of both irradiance across the sky and openings in the forest canopy. The distribution of irradiance was modelled using the standard overcast sky (SOC) model of diffuse light distribution (Moon & Spencer 1942). Here, irradiance at the zenith is considered to be three times greater than that at the horizon. In the present study, the light intensity at the zenith was chosen so that the PFD measured on a horizontal surface above the forest canopy equalled 1000 µmol m−2 s−1. The solar beam coming directly from the solar disk was not taken into account for the sake of simplicity: doing so would not greatly alter the angular altitude dependency of the relative light source intensity averaged over a year.

Information on the distribution of openings in the forest canopy where the palms were sampled was not available. Instead, hemispherical photographs of the canopy of a tropical forest in Peninsular Malaysia with a similar floristic composition were analysed. Sixty-nine photographs taken under the closed forest canopy were provided courtesy of Dr T. Okuda (National Institute of Environmental Studies, Japan). Details of the forest are provided by Okuda et al. (1997). Images of the forest canopy with distinct canopy gaps were not used, because the present study focused on the adaptive geometry of the palms under light-limited conditions, not those in open gaps with abundant illumination.

The photographs, which were taken with a fisheye lens, were scanned into a computer. The images of the hemisphere were divided into 200 cells as described above and the proportion of open canopy was determined for each cell, checking the proportion of bright pixels. Analysis of the photographs provided a measure of how the canopy openness depended on the zenith angle (Table 1). Superimposing the zenith angle dependency of the canopy openness on the SOC model of diffuse light generated the directionality of the diffuse light over the sky.

Table 1.  Relationship between zenith angle and canopy openness, determined from hemispherical photographs taken under the canopy of a tropical rain forest
Zenith angle (°)Canopy openness (%)
  0·0–11·58·5
−23·16·7
−35·05·5
−45·64·2
−55·92·4
−67·11·3
−77·90·34
−90·00·04

SIMULATION OF LIGHT CAPTURE

Two simulations of light capture by the model plant were performed. The first analysed the effect of the deflection angle between the leaf lamina and its petiole (DP) and the effect of the range of zenith angles of the petioles within the crown. In the simulations, DP varied from 0 to 90° at 10° intervals. The range of zenith angles for the petiole was represented by the maximum zenith angle (Zmax), i.e. the zenith angle of the oldest leaf. This angle varied from 30 to 135° at 15° intervals. These two ranges included the observed range for real plants. Among the sample plants, DP was constant at around 50° and Zmax ranged between 50 and 110° (Takahashi & Kohyama 1997). All combinations of DP and Zmax were tested to determine the interaction of their effects on light capture by the crown. The biomass of the lamina and petiole of an individual leaf were kept constant at 23·4 and 10·3 g, respectively (the mean values for the sampled leaves). The simulation was carried out for model plants with four different numbers of leaves (5, 10, 15 and 20). To estimate the effects of the angle of incident light and self-shading on whole-plant light capture separately, hypothetical values for light capture (Ins) were calculated without consideration of the effects of self-shading within a crown. As an index of the relative contribution of self-shading to reducing light capture, the ratio (Ins − I)/Ins was calculated, where I is the light capture when self-shading is considered (calculated as before).

The second simulation studied the effect of changes in the allocation pattern of leaf biomass. The number of individual leaves and their biomasses were varied, with the total leaf biomass per plant held constant. In the simulation, the number of leaves was varied from 1 to 30. The effects of the allocation of biomass on the lamina and petiole were also analysed. The relative allocation of biomass to the leaf lamina [lamina biomass/(lamina biomass + petiole biomass), Rl] was varied from 0·1 to 0·9 at intervals of 0·1. The total leaf dry biomass of the model plant was varied from 20 to 640 g (six values, each double the previous one). The ranges of these parameters included the observed ranges of the plants in the forest. The largest total dry biomass of leaves per plant among the 24 sample palms was 469 g, and the minimum and maximum numbers of leaves per plant were 2 and 13, respectively. The minimum and maximum Rl values for the sampled leaves were 0·44 and 0·73. In the simulations, the deflection angle between the petiole and the leaf lamina was kept constant at 49·6°, which is the mean value for the sample data. Zmax was fixed at 90° in all simulations. Thus, the most slanted leaf within a crown had a horizontal petiole. All combinations of total biomass, number of leaves and relative allocation of biomass to the lamina were tested.

Results

EFFECTS OF THE GEOMETRY OF LEAF DISPLAY ON LIGHT CAPTURE

For every Zmax and number of leaves tested, the curve of the relationship between DP and whole-plant light capture per unit of leaf area was concave, with an optimal value of DP that maximized light capture (Fig. 3). The optimal value of DP decreased as Zmax increased. The optimal combination of DP and Zmax depended on the number of leaves. As leaf number increased, the optimal value of DP decreased and that of Zmax increased (Table 2). The dependency of light capture on DP was weak near the optimal value of DP. Light capture at DP = 50°, the value that was observed in real plants irrespective of the number of leaves within a crown, was less than that at the optimal DP by 2% or less for each number of leaves.

Figure 3.

The relationship between whole-plant light capture per unit of leaf area (µmol m−2 s−1) and the deflection angle between the lamina and the petiole (DP). The maximum zenith angle of the leaf petiole (Zmax) was set at 30 (×), 60 (○), 90 (▴) and 135° (●). Results are shown for plants with 5 (a), 10 (b), 15 (c) and 20 (d) leaves. The dry biomass of individual leaves and Rl (relative allocation to lamina) were set at 33·7 g and 0·583, respectively. Upper arrows indicate the deflection angle at which light absorption is maximal for each plant. Lower arrows with a broken line indicate the mean DP observed in the field for L. arbuscula.

Table 2.  The optimal combinations of DP (angle between lamina and petiole) and Zmax (zenith angle of the oldest leaf) for plants that maximized light capture
Number of leavesDPZmax
 560 60
1055 75
1545 90
2040105

Figure 4 shows the ratio (Ins – I)/Ins, an index of the relative contribution of self-shading to reducing light capture. In a simulated plant with five leaves, this ratio was consistently < 0·1, except when Zmax was 30° (with leaves being packed into a very small space, this ratio ranged from 0·1 to 0·2) (Fig. 4a). The ratio was generally much larger in plants with 20 leaves, although the ratios for plants with high Zmax values were closer to those for the simulated plant with five leaves (Fig. 4b). This indicates that self-shading becomes more severe as the number of leaves increases, and that larger Zmax values reduce this effect.

Figure 4.

The relationship between the relative contribution of self-shading to reducing light capture and DP in plants with 5 (a) or 20 (b) leaves. The maximum zenith angle of the leaf petiole (Zmax) was set at 30 (×), 60 (○), 90 (▴) and 135° (●).

EFFECTS OF BIOMASS PARTITIONING ON LIGHT ABSORPTION

In the present model, leaf area does not increase proportionally with lamina biomass. This is because of the decrease in SLA for larger leaves, as described in equation 1. Thus, the total whole-plant leaf area increases if the available biomass is divided into many small leaves (Fig. 5). The increase in total leaf area with an increasing number of leaves was more pronounced for large total leaf biomasses than for small total biomasses. Decreasing the biomass allocated to each leaf’s lamina reduced the total lamina biomass, and diminished the effects of leaf number on total leaf area. When the total leaf biomass was small, increasing the allocation of biomass to the lamina resulted in a proportional increase in total leaf area (e.g. doubling the allocation to the lamina doubled the total leaf area), irrespective of the number of leaves. However, as total leaf biomass increased, increasing the allocation to the lamina had diminishing returns in terms of increasing the total leaf area, particularly with fewer leaves.

Figure 5.

The relationship between the total leaf area of a simulated plant and the number of leaves. The relative allocation to the lamina (Rl) was set at 0·3 (×), 0·5 (○) and 0·7 (▪). The total biomass of the leaves was set at 20 (a), 40 (b), 80 (c), 160 (d), 320 (e) and 640 (f) g .

The length of the petiole increased with increasing biomass, but at a slower rate than the increase in the lamina. This is because long petioles must become thicker to withstand the increased bending moment imposed by the greater leaf biomass and the increasing length of the petiole as a cantilever. With a fixed relative allocation to the lamina (Rl), increasing leaf biomass produced leaves with disproportionately short petioles (Fig. 6).

Figure 6.

The relationships between the leaf dry biomass and the length of the major axis of the ellipse that encloses the leaf’s lamina (solid line) and the length of the petiole (broken line). The relative allocation to the lamina (Rl) was 0·1 (a), 0·3 (b), 0·5 (c), 0·7 (d) and 0·9 (e).

For these reasons, large plants should improve their light capture by increasing the number of leaves, thereby keeping SLA high and lowering Rl so that petioles can grow longer and reduce self-shading. The results of our simulation supported this expectation. For each given total leaf biomass, there was a combination of leaf number and Rl that maximized light capture (Fig. 7). The optimal number of leaves increased and the optimal Rl decreased with increasing total leaf biomass.

Figure 7.

Relationship between the number of leaves and the whole-plant light capture for plants with total leaf biomasses of 20 (a), 40 (b), 80 (c), 160 (d), 320 (e) and 640 (f) g . Rl was set at 0·1 (×), 0·3 (○), 0·6 (▴) and 0·9 (▪).

The numbers of leaves and Rl values for L. arbuscula plants were estimated for each total leaf dry mass from 20 to 640 g by using equations 4 and 5 (Table 3). The estimated numbers of leaves and Rl increased with increasing biomass as in the optimal model plants. The estimated numbers of leaves and values of Rl compared well with the optima for plants with total leaf biomasses of 160 g or more. In smaller plants, however, the estimated numbers of leaves were larger and Rl was smaller than the predicted optima.

Table 3.  The number of leaves and relative allocation to the lamina (Rl) estimated for given total leaf biomasses in real plants. The optimal leaf number and Rl estimated for the model plant are also shown for each total leaf biomass
 Total leaf biomass (g)
 204080160320640
Number of leaves
Real plant5·15·56·27·710·616·4
Model plant34461216
Rl
Real plant0·690·660·630·60 0·57 0·54
Model plant0·90·80·80·7 0·6 0·6

Discussion

GEOMETRY OF LEAF DISPLAY

The results of the simulation supported Takahashi & Kohyama’s (1997) hypothesis that the geometry of leaf display in L. arbuscula contributes to efficient light capture. The increasing value of Zmax with increasing numbers of leaves in a crown was advantageous in terms of light capture under the following constraints: all leaves are identical in shape, DP is fixed and the zenith angle of the petiole increases gradually with increasing leaf age until it reaches Zmax. With few leaves, a small Zmax, combined with little variation in the zenith angle of the plant’s petioles, lets all laminae face the brightest part of the hemisphere near the zenith. Then, a DP larger than the observed value (50°) is favoured. However, the difference between the amount of light captured at the optimal and the observed DP was small. As the plant grows in size, the number of leaves increases. With many leaves, self-shading becomes severe. A large Zmax then reduces self-shading by distributing the leaf laminae throughout a larger volume. The negative effect of small Zmax on the light capture of plants with many leaves cannot be reduced much by adjusting DP (Fig. 3). When Zmax is large, greater variation in the orientations of the leaf laminae becomes unavoidable. We found that a DP of 50° represents a good compromise for plants with large Zmax values. These results support the conclusion that a DP of around 50°, as measured in L. arbuscula, is a good solution for the palms in which Zmax changes during growth.

Chazdon (1985) pointed out that angular efficiency (the ratio of a leaf’s projected area, as viewed from directly above, to leaf area) decreased with increasing plant size and increasing numbers of leaves in two understorey palm species [Geonoma cuneata (H. Wendl) ex Spruce and Asterogyne martiana (H. Wendl) H. Wendl]. This occurred because more of the leaf laminae deflected away from the horizontal. However, the magnitude of the reduction in light capture caused by the slanting of the leaf laminae is overestimated when the light is assumed to come exclusively from the zenith. Under such a light regime, a leaf lamina slanted by 60, 75 and 90° receives 50, 26 and 0%, respectively, of the light received by a horizontal one. In the present study, the assumed light regime took light from the entire hemisphere into consideration. Under these conditions, the leaf laminae slanted by 60, 75 and 90° receives 58, 46 and 42%, respectively, of light received by a horizontal one. By assuming the more realistic light regime, we showed that wide variations in petiole orientation contribute strongly to the avoidance of self-shading, especially in plants with many leaves. In a study of the adaptive geometry of plants, care should be taken in modelling the directionality of light source because it may influence the results substantially.

BIOMASS PARTITIONING

Many woody plants develop branching structures when they expand their crown. In these plants, self-shading can be reduced by distributing leaves throughout a large space by allocating biomass to branches (Kohyama 1987; Kohyama & Hotta 1990). Licuala arbuscula does not expand its crown by branching. All leaves sprout from the top of a single stem. In this case, increasing only the number of leaves results in severe self-shading. In contrast, increasing leaf size without increasing the number of leaves results in small SLA values because of the need to increase the mechanical stiffness of the laminae. More investment to the petioles required to support large laminae is also disadvantageous to light capture. The present study demonstrates that increasing both the size and number of leaves is a reasonable strategy for maintaining a high light capture efficiency as the single-stemmed palm grows, as Takahashi & Kohyama (1997) hypothesized.

Increasing the number of leaves inevitably intensifies self-shading, but elongation of the petiole can effectively reduce self-shading among the leaves on a vertical non-branching shoot (Takenaka 1994). Takahashi & Kohyama (1997) suggested that L. arbuscula reduces self-shading during crown development by investing more biomass in petioles. Our simulation study demonstrated quantitatively the advantage of increasing the investment in petioles as plant size increases. Optimization of petiole length was also reported in another understorey plant, Adenocaulon bicolor Hook (Pearcy & Yang 1998). The latter study showed that the petiole length of this species maximizes whole-plant light capture given the trade-off between petiole length and lamina area. The increase in allocation of biomass to the petiole has been reported in other understorey palm species (Chazdon 1986) and in non-branching single-stem trees (Yamada & Suzuki 1996).

In small L. arbuscula plants, the observed allocation of biomass to the petiole was considerably larger than the predicted optimum. One possible reason for this is that the palm has adapted to vertical gradients in light availability. More investment in long petioles enables a small palm with a stagnant stem to raise the lamina. If a vertical gradient of light exists near the ground in the natural habitat of this species, positioning leaf lamina higher above the ground is advantageous to light capture. If that is true, there must have been selection pressure in favour of more investment in petioles in L. arbuscula in the early stages of development.

Acknowledgements

We thank T. Okuda of Japan’s National Institute for Environmental Studies for providing the hemispherical photographs taken in the forest in Malaysia, which we used to determine the spatial distribution of canopy openings in our model.

Received 29 January 2001; revised 24 May 2001; accepted 24 May 2001

Appendix

In the simulation, the petiole was approximated as a truncated cone. The length and radii of the basal and distal ends of the truncated cone for a given dry biomass (mp) were calculated as follows.

The volume of a petiole V is given as:

image(eqn A1)

where ρ is the density of the petiole. Let the ratio of the radius at the distal end to that at the basal end be represented by α. Then, V can be calculated as:

image(eqn A2)

where r is the radius of the basal end and B is (1 + α + α2). Thus,

image(eqn A3)

where L is the length of the petiole. Consider the petiole as a horizontal cantilever, with the basal end fixed to the plant’s stem. The lamina attached to the petiole acts as a load at the distal end. The case of a horizontal petiole is considered. The bending moment at the basal end of the petiole was divided into two components. The first, Il, is that caused by gravity acting on the lamina and the other, Ip, is that caused by gravity acting on the petiole itself. Il is given by:

image(eqn A4)

where ml is the dry biomass of the lamina, kl is the ratio of fresh to dry biomass for the lamina, and g is the gravitational constant. The other component, Ip, is given by:

image(eqn A5)

where kp is the fresh mass to dry mass ratio of a petiole, and C is (1 + 2α + 3α2). The section modulus of the petiole at the base (Y) is a function of r:

image(eqn A6)

The maximum stress at the basal end of the petiole (σ) is calculated as

image(eqn A7)

Thus, for given a value of σ,

image(eqn A8)

Then, r is given as:

image(eqn A9)

We determined the values of these parameters from the sampled leaves of L. arbuscula as follows: α = 0·66, ρ = 1450 kg m−3, kg = 2·67, kl = 2·08 and σ = 20·8 MPa. From these parameters and equations A1, A3 and A9, the radii and length of a petiole were determined for a given petiole and lamina biomass.

Ancillary