T. Kohyama, Graduate School of Environmental Earth Science, Hokkaido University, Sapporo 060–0810, Japan (fax + 81 11 706 4954; e-mail email@example.com).
1Maximum attainable height varies greatly between tree species in tropical rain forests and covaries with demographic and allometric traits. We examined these relationships in 27 abundant tree species in a mixed dipterocarp forest. These species were monitored over 3 years in two 1-ha plots in western Borneo. A 95-percentile upper height limit was used to represent maximum height, to avoid sample size differences among populations.
2Average growth rate in trunk diameter was regressed against trunk diameter using a maximum likelihood model and assuming that growth rates were exponentially distributed around the average. Estimated average growth rate at small trunk diameters (up to 11 cm) was independent of maximum height among the 27 species, while the degree of growth reduction at larger diameters was larger for species with smaller maximum height.
3The recruitment rate efficiency of saplings was negatively correlated with maximum height, regardless of the measure used to assess species abundance. In particular, sapling recruitment per unit basal area declined greatly with increasing maximum height, consistent with model predictions of the traits required for the stable coexistence of species at different heights within the canopy.
4Allometric analyses showed that understorey species had shorter heights at the same trunk diameter, and deeper crowns at the same tree height, than canopy species. Therefore, understorey species showed adaptive morphology to deep shade.
5The regressed size-dependent pattern of average growth rate and an assumption that the population was in a steady state readily explained the observed trunk diameter distributions for 21 species among 27 examined. These species, for which the projected size distribution hardly changed when the natural increase or decrease of the population was set at γ = ±0.005 year−1, had mortality rates of more than four times the value of γ.
The coexistence of many tree species in tropical rain forests has been examined in various contexts. The differentiation between species along topographic and nutritional gradients has been investigated (Ashton 1964, 1977), as have density-dependent regulation (Janzen 1970; Connell et al. 1984), disturbance-mediated heterogeneity (Denslow 1987) and dispersal limitation (Hubbell et al. 1999; Hubbell 2001), all in a horizontal landscape. Tropical rain forests are characterized by their well-developed canopy architecture, with multiple crown layers, and vertical species partitioning may also contribute to tree species coexistence (Richards 1996; Kohyama 1996; Turner 2001).
Because each tree must develop by growing up through the forest profile from the forest floor, the light resource gradient does not provide the basis for a simple niche differentiation among species. Using a size-structured model of population dynamics, in which upper foliage density suppresses growth and recruitment, Kohyama (1992, 1993) showed that species with overlapping size distributions can coexist, even without the horizontal heterogeneity caused by tree fall gaps. The model suggested that shorter species, where relative growth rate is strongly negatively size-dependent, can coexist with taller species with less sensitive growth rates, providing shorter species have higher recruitment rates per unit species basal area than taller species. Species can then coexist along the gradient of trade-off constraints between vegetative growth and reproduction.
By examining 14 abundant species in a warm-temperate rain forest, Aiba & Kohyama (1996) showed that differences in demography between species were associated with variation in their maximum observed size. Thomas (1996a) and Davies & Ashton (1999) showed that the threshold size for reproduction is proportionally larger for taller species in a Malaysian rain forest. Allometric characteristics of trees are different in taller and shorter species (King 1990; Aiba & Kohyama 1996, 1997; Thomas 1996b; Davies et al. 1998) and morphological and physiological parameters of leaves from the same understorey stratum were also differentiated between canopy and understorey species (Thomas & Ickes 1995; Thomas & Bazzaz 1999). However, no direct examination has been made of differences in the trade-off between growth and recruitment among species with different maximum tree size in tropical rain forests, which would test the prediction of the forest architecture hypothesis (Kohyama 1993).
This paper examines the differentiation between co-occurring tree species in relation to their upper size limit, based on inventory data from trees in permanent plots over a span of 3 years, in a species-rich primary dipterocarp forest. We describe size-specific growth pattern, recruitment rate and allometries of these species. We use the size-specific growth pattern to calculate the expected size stricture, assuming that the population is in a stationary state, and compare it with the observed size structure, and thus examine the contribution of growth patterns.
The extreme tree species diversity in mixed dipterocarp forests makes it difficult to extract population-level traits of species, due to the relative rarity of each species. To overcome the small number of trees per species that can be examined, we employ procedures such as the maximum-likelihood estimate of the size growth function (Kubo & Ida 1998), and the Gf estimate of recruitment rate (Kohyama & Takada 1998). The former enables direct estimation of parameters of average tree growth rate as a function of tree size, under the assumption that growth rate is exponentially distributed around the average at any given size. The latter is used to estimate recruitment rate of each population as the product of the density of trees in the lowest size class and the average growth rate of that class. It gives therefore a reasonable estimate of recruitment rate even when few recruitment events are observed.
The site was at 0°45′ N 110°06′ E in a lowland mixed dipterocarp forest on Gunung (= Mount) Berui, Desa Serimbu in West Kalimantan, Indonesia. Two plots, each 100 × 100 m in size, and located about 500 m apart on the gentle plateau of a ridge, were established in November 1992. All trees larger than 15 cm in girth at breast height (130 cm above ground) were marked with aluminium tags. Trunk girth was measured at breast height or above buttresses using a steel measure, and used to calculate diameter at breast height (hereafter d.b.h. in cm). A re-census of trunk girth was made in November 1995. The overall height (H, m) and the height of the lowest living branch were measured for all trees in August 1994, by trigonometric calculation from the angular elevations of the crown top, the lowest branch and the trunk base, and the distance from the observer to the target tree. Crown depth (C, m) was defined to be the tree height minus the height of the lowest branch. Two perpendicular horizontal widths of the crown were measured for all trees in August 1994 and used to calculate crown projection area (S, m2) assuming an ellipsoid shape. Damaged trees with broken or leant trunks were recorded. Overall dynamics in tree size structure of these plots, ignoring species identity, and detailed description of the plots are provided in Kohyama et al. (2001).
Repeated sampling of voucher specimens enabled us to classify trees into clearly recognizable morphological species. We stored voucher specimens in the Herbarium Bogoriense (BO) and the Herbarium of the Faculty of Science, Kagoshima University (KAGS); voucher numbers are specified for the unidentified species. The entire list of tree species is available at URL http://hosho.ees.hokudai.ac.jp/kohyama/kspyk/serimbu.dat. We analysed the 27 species that had total population sizes of 20 or more trees in the 2-ha area (Table 1). All of these species were clear monomorphic populations. One Diospyros species, which had characteristic tri-nerved leaves, was clearly distinguished from other congeners in the plots and is likely to be a new species. Although the majority of Macaranga species are typical pioneers with large round leaves and occur in heavily disturbed, bright sites in the Indo-Malaysian subkingdom (Davies et al. 1998), Macaranga brevipetiolata is a shade-tolerant species with smaller oblong leaves. Similarly, Mallotus penangensis is a shade-tolerant, narrow-leaved species, unlike light-demanding round-leaved congeners. Specific gravity of wood was determined in one or a few tree trunks per species (except Scorodocarpus borneensis). Trees of basal diameter around 5 cm were sampled just outside the plots, cut into 20-cm long pieces from the base, and volume and oven-dry mass with intact bark were measured (Suzuki 1999).
Table 1. Species analysed (n = 27), wood density, abundance and upper limit size in two 1-ha plots on Gunung Berui, Serimbu, West Kalimantan
Sample population size varied from 20 to 180, and we therefore used the 95-percentile of the size distribution (above 5 cm d.b.h.) rather than maximum observed size, to characterize species by maximum attainable size. This represents the size of [0.95n]-th tree in a population of n trees. The upper values for height and diameter are presented in Table 1, but we report analyses only on height because this is closely correlated with the upper d.b.h. limit (Table 1). We carried out correlation analysis between maximum height and demographic or allometric parameters among the 27 species.
The expected rate of annual increment of d.b.h., G (cm year−1), for each species during the 3 years between 1992 and 1995 was expressed as a function of d.b.h., D (cm) by:
G = aDexp(−D/b)(eqn 1)
where a is the model parameter such that the relative growth rate (year−1) converges to a close to 0 cm d.b.h., and b (cm) is the model parameter expressing the degree of size-dependent reduction in growth rate. The growth rate of similar sized trees is highly variable, with strong positive skewness within the population, and the standard deviation of growth rate at a given size is roughly proportional to the average rate itself (Kohyama & Hara 1989). An ordinary least-squares procedure of regression, under the assumption of normal distribution around the average, is therefore not appropriate. We employed a procedure of maximizing logarithmic likelihood to estimate the two parameters of equation 1, assuming that the growth rate at each size was exponentially distributed around the average (Kubo & Ida 1998). The observed growth rate was the d.b.h. increment between 1992 and 1995 divided by three (the duration in years) for each survivor. Some trees showed negative increments, which do not fit within the exponential distribution and were therefore treated as zero for the purposes of regression. To examine the significance of among-species difference in equation 1 parameters, we compared logarithmic maximum likelihood of the separate regressions for each species with that of a regression of all species combined, with Akaike's Information Criterion (Akaike 1974). We calculated the distribution of normalized growth rate (observed growth rate divided by regressed average growth rate), and compared it with a normalized exponential distribution (where average = 1), to validate the assumption of exponential variation.
Recruitment rate per unit land area R ([no. trees] ha−1 year−1) is defined here as the number of recruits per year entering through the arbitrary minimum size of 6 cm d.b.h. between 1992 and 1995. The observed number of recruits was limited, and inevitably underestimated, because we could not record the number of ‘recruited-and-died’ trees. We thus applied the technique of Gf estimation (Kohyama & Takada 1998). The Gf estimate for the midpoint of the minimum size class examined is the tree density f ([no. trees] ha−1) in the minimum size class (here 5–7 cm d.b.h.) multiplied by the average absolute growth rate G (cm year−1) of survivors within the class, divided by the class width of 2 cm. We used equation 1 with species-specific parameters to obtain the average growth rate at 6 cm d.b.h. The recruitment rate efficiency is defined as R divided by an abundance measure for the species, and six separate measures were tested, i.e. (i) tree density, (ii) basal area, (iii) cumulative d.b.h.3, (iv) cumulative d.b.h.5, (v) leaf mass density, and (vi) increment rate of above-ground biomass, all defined for trees ≥ 6 cm d.b.h. It is assumed that reproduction is proportional to size-weighed densities (ii)–(v), or to the vegetative growth rate (vi), rather than conventional tree density (i), of each species. According to the pipe model theory (Shinozaki et al. 1964), per-species basal area, or the sum of D2, is expected to be proportional to leaf mass. Use of cumulative D3 is based on the allometric relationship between reproductive organ and stem diameter for various land plants by Niklas (1994) and cumulative D5 corresponds to the relation between number of reproductive organs and d.b.h. for trees of a mixed dipterocarp forest in the Malay Peninsula, determined by Thomas (1996c). Leaf mass, and the increment rate of above-ground biomass, were estimated using allometric equations (all species combined) by Yamakura et al. (1986) for a mixed dipterocarp forest in east Kalimantan, as in Kohyama et al. (2001).
We did not carry out mortality analysis, due to the limitation of sample size.
The species-specific allometric relationships between d.b.h. D, tree height H, crown depth C, and crown projection area S were analysed using ordinary allometric equations, or power functions between dimensions. To avoid biasing the allometries, damaged trees (11% of trees observed in 27 spp., Table 1) were excluded from the allometric analyses. For the overall range of dimensions, some allometric relationships, particularly that between d.b.h. and tree height, show non-linear relationships with an upper bound for one dimension (i.e. tree height) on log-log co-ordinates (Aiba & Kohyama 1996; Thomas 1996b), requiring that non-linear regression is applied. There is, however, no straightforward statistic to test species difference in such non-linear models, and as it is not appropriate to compare species with huge size differences, we compared trees with 5–20 cm d.b.h., which included individuals of all the species. Within this range, the log-linear, or power function model was appropriate. Considering 5–10 and 5–30 cm trees gave very similar results. We applied analysis of covariance to test whether species differ in the slope as well as the intercepts of the allometry (cf. Kohyama 1987).
The shape of the size distribution of a species’ population reflects rates of both size growth and mortality, as well as the transient status of populations. With time-invariant demographic parameters, populations converge towards a time-independent stable (not stationary) size distribution with time, when population size changes at relative rate of γ (year−1) (VanSickle 1977; Caswell 2001). When the population is at a dynamic equilibrium with γ= 0, the observed distribution will agree with the (stable and) stationary size distribution projected from observed demographic rates.
We tested the hypothesis that a species population is close to equilibrium using equation 1. The dynamics of size-structured populations can be formulated by a one-dimensional drift equation or ‘continuity equation’ of fluid dynamics (VanSickle 1977; Vance et al. 1988):
where f (ha−1cm−1) is the d.b.h. distribution, G is growth rate of d.b.h., and M is mortality, all defined as functions of both time t and d.b.h. D. In a population with stable size distribution, the intrinsic rate of natural increase γ is defined by
and equation 2 is rewritten in terms of stable size distribution f as
at the instantaneous moment t, where R=R(t) (ha−1 year−1) is the recruitment rate per unit area at the boundary d.b.h. D=D0 set at 6 cm. For the stationary state population with f/t = 0 and γ= 0, R is constant irrespective of time t. The frequency distribution of d.b.h. f can be numerically solved, using equation 1 for G, provided the mortality function M is known. We assumed that the mortality rate, M, of each species population was a constant independent of d.b.h., and was equal to the recruitment rate R divided by the whole population density F above 6 cm d.b.h. (ha−1), or M=R/F (year−1). We calculated size distribution f according to equation 3, as well as performing 2-cm interval integration of f, applying the trapezoidal formula for numerical integration. We then compared calculated and observed size distributions at 2-cm d.b.h. classes using the Kolmogorov-Smirnov one-sample test, to examine whether populations were at equilibrium, stable, or transient.
The total number of species recorded among the 2636 trees above 5 cm d.b.h. in 1992 was 410. The most abundant 27 species were analysed (Table 1) and these accounted for 45% of tree number and 41% of the basal area, even though they contributed only 7% to the species count. Most of these species were abundant in both plots (percentage similarity between the two plots was 57% in tree number), except that Fordia splendidissima occurred only in one plot, and that Archidendron cf. ellipticum and Scorodocarpus borneensis were found only in the other.
Size-dependent growth pattern regressed by equation 1 was significantly different in 27 species. The distribution of observed tree growth rates relative to average growth rate calculated by equation 1 almost fit the expected exponential distribution (Fig. 1), thus validating the primary assumption of the parameter estimation for equation 1. The only disagreement was the observed tail with 6% of trees showing negative growth.
The parameter a of equation 1, representing the initial growth rate, was independent of maximum height U (Fig. 2a; P = 0.2 for a vs. ln U correlation). Except for Neoscortechinia kingii, all species had positive parameter b, and decreased relative growth rate with d.b.h. The parameter b showed positive correlation to maximum height (Fig. 2b; r2 = 0.56, P = 10−5 for ln b vs. ln U plot, and r2 = 0.26, P = 0.007 for b vs. ln U plot including value for N. kingii), i.e. larger statured species showed a smaller decline in relative growth rate with increasing d.b.h. When d.b.h. was analysed separately for classes at 1-cm intervals, the expected growth rate showed no correlation against maximum height up to 11 cm d.b.h. (P > 0.05), and it was positively correlated to maximum height above 12 cm (P < 0.05).
The various measures of recruitment rate efficiency all showed significant negative correlations with maximum height (Table 2). These apparent relationships might, however, be affected by autocorrelation because measures of species abundance, excluding tree density, were also positively correlated with maximum height in these 27 abundant species (Table 2). To examine the possibility of autocorrelation, we carried out multiple regression of recruitment rate efficiency by maximum height and abundance measures. Logarithmic recruitment rate efficiency was explained by logarithmic maximum height, with no significant additional contribution from logarithmic abundance, except when tree density and D5 were used as measures of abundance (Table 2). To test the possible effect of collinearity in the multiple regression analysis, we divided species into abundance classes, using basal area as the index of abundance (see Fig. 3), within each of which there was no correlation between recruitment rate efficiency and abundance (Fig. 3b). We still found a significant negative correlation of recruitment rate efficiency with maximum height in each class (Fig. 3a). Therefore, we conclude that the negative correlation between recruitment rate efficiency and maximum size characterizes the differences between the 27 species examined.
Table 2. Correlation analysis between upper height limit (U) and recruitment rate efficiency R/A; i.e. recruitment rate per land area R (ha−1 year−1) divided by several measures of species abundance A
Either parameter i, j or k with P > 0.05 is removed.
Tree density (ha−1)
−0.61 ln U + 0.54 ln A − 3.71
Basal area (m2 ha−1)
−2.94 ln U + 9.06
Cumulative D3 (m3 ha−1)
−4.36 ln U + 14.8
Cumulative D5 (m5 ha−1)
−3.02 ln U − 0.66 ln A+ 9.78
Leaf mass (t ha−1)
−2.83 ln U + 10.6
Biomass increment (t ha−1 year−1)
−3.34 ln U + 12.6
Allometries between six sets of dimensions all showed significant species differences (Table 3a). The difference appeared only in the intercept, except for tree height vs. crown depth and crown depth vs. crown area, where slope was also significantly different (Table 3a). By comparing species-specific intercepts among the six allometric relationships, there was a tendency for species with larger tree height at the same d.b.h. to have a wider crown relative to height, and for species with deeper crown at the same d.b.h. and at the same height to have a larger crown area at the same d.b.h., height and crown depth (Table 3b). Species with larger maximum height had larger tree height and shallower crown depth at a given trunk diameter, and they had shallower crown depth and marginally significantly (P = 0.06) smaller crown projection area at a given tree height (Table 3b, Fig. 4). Wood density was marginally negatively correlated with upper height limit (P = 0.07, data in Table 1) and was not correlated with species-specific intercept of any allometric relationship (P > 0.05).
Table 3. Species difference in allometric characteristics, expressed by logarithmic linear regression between independent variable x and dependent variable y, for dimensions d.b.h. D (cm), tree height H (m), crown depth C (m) and crown projection area S (m2), for undamaged trees with 5–20 cm d.b.h.; sample size varies due to missing data for each of dimension
(a) Interspecific difference in allometries between tree size dimensions
ln x − ln y
Sample size n
F ratio for species difference in
Range of intercept
ln D − ln H
ln D − ln C
ln D − ln S
ln H − ln C
ln H − ln S
ln C − ln S
(b) Interspecific correlation coefficients between allometry intercepts with common slope, and with logarithmic upper height limit U
The first degree of freedom of F ratio is 52 for total, and 26 for intercept and slope, and the second d.f. is n−54 for total and slope and n−28 for intercept. Bold values are significant at P < 0.05 (with Bonferroni correction applied for among-allometry correlations).
ln x − ln y
ln D − ln C
ln D − ln S
ln H − ln C
ln H − ln S
ln C − ln S
ln D − ln H
ln D − ln C
ln D − ln S
ln H − ln C
ln H − ln S
ln C − ln S
We calculated, using equation 3, the stationary d.b.h. distribution under the assumption of equilibrium populations (γ = 0) and size-independent mortality, and the stable d.b.h. distribution when γ =+0.005 or −0.005 year−1. Such 0.5% annual changes would bring about substantial changes in population size, totalling a 65% increase or a 39% decrease, respectively, over 100 years.
Figure 5 compares projected stationary distribution (full line), stable distributions (broken lines) and the observed d.b.h. distribution in 1992 (circles). Generally, the calculated stationary distribution agreed with the observed distribution (P > 0.05 for 21 of 27 species, Kolmogorov-Smirnov test). Dryobalanops beccarii and Strombosia ceylanica, as well as the less common Neoscortechinia kingii, Syzygium cf. acutangulum, Palaquium cf. oxleyanum and Shorea parvifolia, not shown in Fig. 5, showed a significant separation of the observed distribution from the projected stationary distribution (P < 0.05). For the two species shown in Fig. 5, the projected stationary distribution was an underestimate at large size classes and an overestimate at small size classes and thus may suggest that the populations are transient. The deviation was, however, not simply explained by changing population growth rate.
For four species (Pimerodendron griffithianum, Amyxa pluricornis, Mallotus penangensis and Dysoxylum cauriflorum), increasing or decreasing population growth rate γ produced a remarkable change in projected d.b.h. distribution (Fig. 5). Calculated mortality rates M for their populations were no more than 0.01 year−1 (twice the value of γ used). M-values for Dryobalanops beccarii, Macaranga brevipetiolata and Strombosia ceylanica, which showed moderate changes in d.b.h. distribution with γ, were somewhat higher (but still less than 0.02 year−1). Other species with M > 4 γ showed robust projected d.b.h. distribution, and a large change in population growth rate only caused slight modifications.
The assumption that tree growth rate was exponentially distributed at any size successfully described the large observed variation in growth rate (Fig. 1), confirming our procedure for estimating parameters of the averaged growth rate function, even when the population size is small. The only disagreement between the observed variation and that expected from the exponential distribution is the small proportion of trees showing negative growth. As trunks of dicotyledonous trees grow cumulatively, records of negative increment are likely to reflect measurement error and/or temporary variation in hydration of cambial tissue. Although our conversion of such negative growth to zero growth may bias the estimation, because other possible error or positive fluctuations are omitted, this bias is small as only 6% of trees showed negative growth, and to a limited extent (Fig. 1).
We detected the demographic trade-off among species with different maximum sizes, as predicted by the forest architecture hypothesis (Kohyama 1993). This hypothesis, which is based on a model of multispecies size-structure dynamics, showed that species with overlapping size distributions can coexist along a vertical gradient of light as a result of differences in maximum attainable size. This predicted trade-off between maximum size, size increment and recruitment capacity has also been observed in a warm-temperate rain forest (Aiba & Kohyama 1996). In our forest, sapling recruitment rate per unit basal area increased more than 100-fold over a fivefold decrease in maximum height. Although we examined only 27 of the 410 co-occurring species, these species comprised nearly half of the stems on the study plots, and we conclude that the trade-off detected is a dominant factor in species coexistence.
Differentiation between tall species and short species was also found in allometric characteristics, suggesting an adaptive morphological basis of their demographic differentiation. For a given trunk diameter, short species tended to have shorter heights, but similar crown area than did tall species. This suggests that the higher size-dependent depression of trunk diameter growth rate in short species (Fig. 2b) is not due to preferential allocation to vertical or branch growth in place of radial expansion of trunk. Short species are likely to allocate more to radial growth, thereby acquiring the physical stability needed to survive in the lower strata, at the expense of vertical growth. Having deeper and wider crowns at the same height is an adaptation to shade of short species via retention of lower and older branches. This is consistent with a physiological analysis of understorey leaves of species with different maximum size in a Malaysian rain forest, carried out by Thomas & Bazzaz (1999). They found that leaves of tall species, even in understorey deep shade, showed the characteristics of leaves adapted to exposed conditions rather than shaded conditions, unlike understorey leaves of short species. Tall species exhibit physiological and allometric adaptation to exposed conditions in upper canopy in sacrifice of their capacity to survive under deep shade.
Kohyama (1987), Kohyama & Hotta (1990) and Welden et al. (1991) pointed out that differences between co-occurring shade-tolerant species in allometry of above-ground architecture at the sapling stage were independent of species’ upper size limit. Differentiation at the sapling stage (less than 3 m height) is primarily explained by another trade-off, i.e. whether or not they can persist for a long time in dark conditions. King (1990) suggested that there was allometric differentiation between similarly sized saplings of tall and short species in a Panamanian rain forest. This is in accordance with the present results, as short species are near to their reproductive size in his analysis. Thomas (1996b) found that the initial slope of diameter–height allometric functions, for trees ≥ 1 cm d.b.h., was negatively correlated with asymptotic height in 38 tree species of a Peninsular Malaysian forest. Davies et al. (1998) indicated for Bornean Macaranga that small statured species had larger slopes and intercepts of diameter–height allometry for saplings (≤ 4 cm d.b.h.) than large statured congeners, suggesting slender trunk shapes at the sapling stage. This study detected species differences in intercept of the diameter–height allometry for trees with 5–20 cm d.b.h., suggesting that small statured species were characterized by less slender trunks and by deeper and wider crowns. Allometric differentiation among species in relation to maximum size therefore depends on the stage, or size range, of trees, which will, in turn, be related to differentiation among species in size at onset of reproduction (Thomas 1996a; Davies & Ashton 1999). Thomas (1996b) showed that species with larger asymptotic size tended to have less dense wood and faster growth rate, particularly at the adult stage, though in our study, wood density was only marginally negatively correlated to maximum size, and there was no growth rate difference for trees < 11 cm d.b.h. Suzuki (1999) reported that pioneer species with less dense wood grew faster than heavy-wooded, shade-tolerant species at our research site. Our species were from a range of phylogenetic groups, unlike those of Thomas (1996b), who selected species from only a few genera. Evolutionary relationships between wood density, initial growth rate and maximum size may therefore have been hidden in our study by variation reflecting different phylogenetic constraints.
A simplified model of the dynamics of tree size structure (equations 2 and 3) together with equation 1 for tree size growth explained, to a fair extent, the observed species-specific patterns of size distribution (Fig. 5). Size-dependent changes in average growth rate may therefore be the primary determinant of the observed size distribution of tree species. Agreement between projected stationary and observed distributions does not necessarily indicate that a population is near equilibrium, because populations with fast turnover rates (in terms of tree mortality) were robust to changes in the intrinsic rate of natural increase. Kohyama et al. (2001) performed detailed modelling of size structure dynamics using data from the same plots, but ignoring species differences, and showed that the resulting steady state distribution due to light-competition among trees was close to that observed. This provides complementary evidence for the stability of this forest system. Two species, Dryobalanops beccarii and Strombosia ceylanica, showed a significant separation between projected and observed distributions, which could not be explained by changes in population growth rate (Fig. 5). This may be due to tree mortality being size dependent and higher in smaller size classes (Kohyama et al. 2001).
The continuity equation for the dynamics of size-structured populations appears effective, even for small populations. Condit et al. (1998) examined a 50-ha plot in Panama, and concluded that the observed species-specific size structure was mostly explained by species differences in demographic rates at around the steady state, rather than the transient status of species populations. They assumed neither a particular growth equation nor the continuous equation of dynamics, but applied the size-class divided projection matrices to describe the demographic performance of species. Whilst this technique is superior where large samples are available, the present approach usefully describes performance in terms of a small number of demographic parameters.
We thank Soedarsono Riswan, Herwint Simbolon, Naohiko Noma and Koichi Takahashi for field collaboration, and Peter Ashton, Fakhri Bazzaz, David Burslem, Stuart Davies, Matthew Potts, Tristram Seidler and anonymous reviewers for valuable comments on earlier manuscripts. The Indonesian Institute of Sciences (LIPI) sponsored the research, and the Research Centre for Biology of LIPI offered full facilities during research. Financial support came from the Ministry of Education, Science, Sports and Culture of Japan in 1992–93, and from the Japan Society for the Promotion of Science in 1995. T. Kohyama acknowledges the Bullard Fellowship of Harvard University for providing the opportunity to accomplish this study.