study sites and field sampling
The research was conducted within the moist cold subzone of the Interior Cedar-Hemlock (ICHmc) biogeoclimatic zone of British Columbia (Banner et al. 1993). Forests of this region represent a transition between the interior and coastal forests of Northwestern British Columbia (Pojar et al. 1987). Mature natural forests in the ICH zone are dominated by western hemlock (Tsuga heterophylla (Raf.) Sarg.), but have a diverse tree species mix with western redcedar (Thuja plicata (Donn ex D. Don in Lamb), subalpine fir (Abies lasiocarpa (Hook.) Nutt.), lodgepole pine (Pinus contorta var. latifolia Engelm.), hybrid spruce [the complex of white spruce (Picea glauca [Moench] Voss), Sitka spruce (P. sitchensis [Bong.] Carr.) and occasionally Engelmann spruce (P. engelmannii Parry ex Engelm.)], paper birch (Betula papyrifera Marsh.), trembling aspen (Populus tremuloides Michx.), and black cottonwood (Populus balsamifera ssp. trichocarpa Torr. & Gray). Subalpine fir is typically replaced by amabilis fir (Abies amabilis Dougl. ex Forbes) at higher elevations. Morainal parent materials dominate the area, ranging in texture from loamy sand to clay loam. Eluviated Dystric Brunisols, Orthic Dystric Brunisols and Orthic Humo-Ferric Podzols are the most common soils.
We sampled across a wide range of stand ages, disturbance histories, tree species composition and competitive neighbourhoods to obtain a data set that allowed analysis of all dominant tree species (nine) found in the ICH zone (Table 1). A core set of measurements were taken at the Date Creek Silvicultural Systems Study (Coates et al. 1997), located near Hazelton, British Columbia, Canada (55°22′ N, 127°50′ W; 370–665 m elevation), but sampling also occurred at sites throughout the ICH zone. The Date Creek sample sites are fully described in Canham et al. (2004). Briefly, they were in experimental plots in mature and old-growth stands that were either undisturbed or subject to two levels of partial cutting (30% or 60% basal area removal). Our sampling took advantage of the spatial variation in canopy structure created by the treatments to sample the wide range of local competitive environments created by logging, but only in the two dominant age-classes found at Date Creek. Because of the small sample sizes for species other than western hemlock and western redcedar, only those two species were reported on in our earlier study (Canham et al. 2004).
Table 1. Samples sizes and mean, minimum and maximum d.b.h. (cm) for the study species
|Tsuga heterophylla||Western hemlock||245||29.9||6.1||103.9|
|Thuja plicata||Western redcedar||192||25.1||6.0|| 69.5|
|Abies amabilis||Amabilis fir|| 91||28.2||6.5|| 76.7|
|Abies lasiocarpa||Subalpine fir|| 95||15.4||4.0|| 45.4|
|Picea hybrid*||Hybrid spruce||196||22.0||4.3|| 59.1|
|Pinus contorta||Lodgepole pine|| 93||18.7||3.7|| 43.2|
|Populus tremuloides||Trembling aspen||101||20.1||4.0|| 47.3|
|Betula papyrifera||Paper birch||149||17.5||1.8|| 42.1|
|Populus balsamifera ssp. trichocarpa||Cottonwood|| 39||21.1||8.0|| 63.0|
We added 19 additional stem-mapped plots to the eight original sites used by Canham et al. (2004). The location of all trees > 2 m tall were mapped. Mapped areas varied from 0.1 to 1 ha in size. Species, (d.b.h., 1.3 m) and condition (live or dead) were recorded for each mapped tree. Species composition of each mapped area varied widely depending on stand age and disturbance history (Appendix S1). Sample trees were selected to have a minimum 15 m radius of mapped neighbours on all sides. A total of 1201 target growth trees were cored, with sample sizes of the nine study species varying between 39 and 245 (Table 1). Increment cores were taken at 1.3 m height. The average radial growth (mm year−1) over the last 5 years was used as the response variable.
Our analysis is an extension of the spatially explicit models of tree competition outlined in Canham et al. (2004). As in our previous studies, the observed radial growth (RG) of a target tree is analyzed as a function of: (i) the potential growth of a hypothetical ‘free growing’ tree (PRG), (ii) the size (d.b.h.) of the tree, (iii) the degree of shading, and (iv) crowding of trees by neighbours:
RG = PRG × Size Effect × Shading Effect × Crowding Effect(1)
RG and PRG are in units of mm year−1, and the remaining three terms on the right hand side of Eqn 1 are scalars ranging from 0 to 1 that act to reduce potential growth. As in other recent studies (Canham et al. 2004, 2006), we use a lognormal function for the shape of the size effect:
where δ is the d.b.h. (of the target tree) at which PRG occurs, and σ determines the breadth of the function. This function is flexible enough that for the effective range of adult trees the shape can be monotonically increasing (i.e. when δ is very large), decreasing (i.e. when δ is very small), or have a single ‘hump’ and a skew to the left when δ is within the normal range of d.b.h. (Canham et al. 2004).
Canham et al. (2004) presented an empirical method of calculating the shading of a target tree by neighbours, and we use the same method here. For our purposes, ‘shading’ is the percent of incident, seasonal total photosynthetic photon flux density blocked by neighbours. The calculations are based on a spatially explicit model of light transmission, parameterized specifically for the species at our study sites (Canham et al. 1999). The calculations use tree allometry, grown geometry and light extinction characteristics of each of the tree species (reported in Canham et al. 1999) to determine the areas of the sky around each target tree that are blocked by neighbours, and then weight those areas of the sky by a sky brightness distribution calculated for our study sites (see Canham et al. 2004 for details). The shading effect is then assumed to reduce potential growth following a negative exponential function:
Shading Effect = e−S×Shading(3)
The parameter S measures the sensitivity of the target tree to shading: at S equals zero, the target tree is insensitive to shading.
Our analysis of the effects of crowding follows from the long tradition of distance-dependent analyses of competition, in which tree growth is analyzed as a function of the sizes and distances to neighbouring trees (e.g. Bella 1971; Hegyi 1974; Lorimer 1983; Wimberly & Bare 1996; Vettenranta 1999; Berger & Hildenbrandt 2000; Canham et al. 2004, 2006; Uriarte et al. 2004a,b; Stadt et al. 2007). The net effect of a neighbouring tree on the growth of a target tree of a given species is assumed to vary as a direct function of the size of the neighbour, and as an inverse function of the distance to the neighbour. Most previous studies have assumed that all species of competitors are equivalent. In our analysis, the net effect of an individual neighbour is multiplied by a species-specific competition index (λs) that ranges from 0 to 1 and allows for differences among species in their competitive effect on the target tree. Then, for i = 1 ... s species and j = 1 ... n neighbours of species s within a maximum radius (R) of the target tree, a Neighbourhood Competition Index (NCI) specifying the net crowding effect of the neighbours on the target tree is given by (Canham et al. 2004):
where α and β are estimated by the analyses (rather than set arbitrarily as in previous studies), and determine the shape of the effect of the d.b.h. and the distance to the neighbour, respectively, on NCI. Because of our ability to explicitly incorporate estimates of shading in our analyses, we interpret our ‘crowding’ term (Eqn 1) as primarily a measure of below-ground competition. Our analysis also estimates R, as a fraction of the maximum neighbourhood radius of 15 m (the limit allowed by the size of our mapped plots and transects). To keep the number of parameters in the model manageable, α, β and R were assumed to be identical for all species of neighbours.
We have an a priori interest in quantifying interspecific variation in per capita effects of crowding by different species of neighbours. This is motivated by our interest in understanding the consequences for timber yield of designing silvicultural systems that manage for specific mixtures of species in local neighbourhoods. This issue is also relevant to recent debates about neutral theory in ecology in contrast to niche differentiation as a means to explain coexistence among species. In recent studies from species-rich tropical forests (Uriarte et al. 2004a,b, 2005), the most parsimonious models typically lump many species of neighbours as equivalent competitors. This appears to be largely a function of the very small samples of any given species of neighbour as a result of very high tree species diversity in those forests. For comparison with those studies, we also tested three different groupings of neighbouring species in Eqn 4: (i) a model in which all species were considered equivalent (i.e. fixing λ = 1 regardless of the species of neighbour); (ii) a model that calculated a separate λ for conspecifics and a single, separate λ for all heterospecific neighbours; and (iii) a model that calculated one λ for all neighbours of conifer species and a separate λ for all neighbours of deciduous species.
We assume that growth declines as a negative exponential function of NCI:
Crowding Effect = e−C×NCI(5)
Canham et al. (2004) used a simpler, linear reduction in growth with increasing crowding, but that formulation requires truncating the function at some level of NCI to prevent predictions of negative growth. Other recent studies have tested for a sigmoidal reduction in growth with increasing crowding (Uriarte et al. 2004a,b; Canham et al. 2006), but have consistently found that the simple exponential decline provides the best fit to the data, so we did not test a sigmoidal model here.
We also tested a variant of Eqn 5 in which the effect of crowding on target tree growth varied as a function of target tree d.b.h. This effect is independent of the underlying effect of target tree size on potential growth (Size Effect, in the absence of competition). This allowed us to test whether a given level of crowding had a greater effect on smaller (or larger) target trees (Canham et al. 2004). To test this, we allowed the exponential decay term (C) in Eqn 5 to vary as a function of target tree size (d.b.h.):
C = C′ × d.b.h.γ(6)
If γ = 0 there is no variation in sensitivity to crowding as a function of target tree size. If γ < 0, then sensitivity to crowding declines as target tree d.b.h. increases (i.e. smaller trees suffer a greater reduction in growth from a given level of crowding than do larger trees). If γ > 0, then larger trees are more sensitive to a given level of crowding than smaller trees.
parameter estimation and comparison of alternate models
Growth of each of the nine species was analyzed separately. For each analysis, the regression models described by Eqns 1–6 require estimation of n + 10 parameters for n species or groups of competitors. We solved for the coefficients of the regression models using maximum likelihood estimation and simulated annealing (Goffe et al. 1994), a global optimization procedure. The parameter estimation was done using software written specifically for this study using Delphi for Windows (Borland Software Corp.). Residuals were normally distributed, and unlike our earlier studies (Canham et al. 2004, 2006), the variances were uniform across the range of predicted values. We used asymptotic, two-unit support intervals (Edwards 1992) to assess the strength of evidence for individual maximum likelihood parameter estimates. A two-unit support interval is roughly equivalent to a 95% support limit defined using a likelihood ratio test. The slope of the regression (with a zero intercept) of observed radial growth on predicted radial growth was used to measure bias (with an unbiased model having a slope of 1) and the R2 of the regression was used as a measure of goodness-of-fit.
Our likelihood approach uses two different methods to assess the strength of evidence for (and magnitude of) different processes incorporated in our models. In many cases, the parameter estimates themselves provide the basis for determining the magnitude of the effect of a given process. For example, if the maximum likelihood estimate for parameter S in Eqn 3 is effectively zero, then there is no effect of shading by neighbours on the growth of a target tree species. We have also used formal model comparison methods, parameterizing alternate models with and without specific terms, and have then used the Akaike Information Criterion corrected for small sample size (AICcorr) to incorporate both parsimony and likelihood in comparing alternate models. In addition to the different groupings of species of neighbours described above to simplify the number of distinct λ parameters, we used formal model comparison in three additional cases. First, we tested whether there was evidence that sensitivity to crowding varied as a function of the target tree d.b.h., by comparing models including γ in Eqn 6 versus models omitting this term. Then we tested three simpler models in which we dropped: (i) shading, (ii) crowding, or (iii) shading plus crowding, leaving size as the only term modifying predicted potential growth.
Since both shading and crowding are functions of the distributions of neighbours, it is unavoidable that there is some degree of collinearity in the above- and below-ground competition experienced by individuals within a given target tree species. The correlation among calculated levels of shading and the crowding term ranged from very low (r = 0.05 for spruce) to relatively high (0.91 for amabilis fir), but even when high, there was considerable scatter among individuals, and the models converged without difficulty. Our method of model comparison, in which simpler models were fit using only shading or crowding alone, further helps guard against spurious inclusion of both above- and below-ground effects.