field sampling and growth measurements
We developed and evaluated field-parameterized models of probability of mortality as a function of recent radial growth for seven species important in boreal and near boreal forests. The species evaluated were trembling aspen, paper birch, yellow birch, white spruce, mountain maple, balsam fir and eastern white cedar. For each species, we attempted to sample an equal number of individuals across different size classes. Size classes varied depending on species growth rate, with fast growing species sampled in larger size classes (Table 1). For example, aspen, a clonal species reproducing primarily from root suckers, exhibits rapid early growth, often attaining a height of up to 1 m in the first year. Aspen clonal connections, although maintained in smaller saplings, may disappear in larger individuals (DesRochers & Lieffers 2001). When species were not abundant, we sampled two size classes, whereas for abundant species we used three size classes (Table 1).
Table 1. Characteristics of live and dead trees by size class and composition. Species are ordered by shade tolerance ranking as suggested in the literature (Burns & Honkala 1990 for the tree species and Rook 2002 for mountain maple)
|Balsam fir||0–1|| 8.0|| 4.6|| 54.4|| 16.6|| 9.97|| 3.6|| 61.7|| 20.32|
|1–2||21.9|| 8.6||146.7|| 59.2||22|| 7.1||143.6|| 38.7|
|2–4||33.1|| 9.9||281.1|| 81.6||34.68|| 9.5||271.4|| 98.8|
|White cedar||0–1.5||11.2|| 5.7|| 81.8|| 29.6||12.66|| 7.6|| 86.7|| 49.1|
|1.5+||23.2|| 8.1||186.9|| 61.0||31.5|| 9.4||178.2|| 43.2|
|Mountain maple||0–2||13.1|| 6.8||136.1|| 46.6||31.75||13.4||134.3||109.7|
|White spruce||0–1|| 7.6|| 6.0|| 62.8|| 22.1|| 8.95|| 2.5|| 64.7|| 23.7|
|1–3||14.1|| 4.8||138.5|| 35.1||16.67|| 4.5||143.3|| 37.2|
|Yellow birch||0–1|| 6.1|| 4.6|| 64.1|| 3.6|| 6.1|| 4.6|| 94.0|| 36.1|
|1–2||10.2|| 6.4||108.1|| 6.0||10.2|| 6.4||178.1|| 78.0|
|2–4||25.9||11.6||242.2|| 11.9||25.9||11.6||342.2|| 99.3|
|Paper birch||0–2||13.6||14.7||118.4|| 45.2||11.0|| 4.3||107.3|| 45.5|
|2–4||20.1|| 6.3||281.0|| 78.6||23.6|| 7.8||401.6||389.0|
|Aspen||0–2||13.2||10.3||145.6|| 44.7||17.2||32.4||184.6|| 89.7|
|2–4||24.7|| 8.7||272.8|| 81.0||27.7||13.6||369.3||160.8|
Two criteria were employed in choosing appropriate field sites for our statistical methods: adequate sample sizes to estimate parameters for the mortality model and sufficient variation in the predictor variable (see below). Our sampling protocol requires finding sufficient numbers of live and recently dead individuals of a focal species at the same sampling site. Finding adequate sample sizes for live individuals was rarely a problem, but finding adequate numbers of recently dead individuals was often difficult. Hence, the minimum size of a sample site had to be large enough to include a target of 30 recently dead individuals of the focal species. This resulted in variably sized sample sites ranging from approximately 40 to 8000 m2. The smaller sample areas were for the more common species (e.g. balsam fir and trembling aspen), for which high densities of individuals occurred in the smaller size classes. Larger individuals, as well as species with lower densities (e.g. cedar and the largest size class of aspen) required the largest sample areas. We sampled three replicate sites of each species size-class combination for a total of 45 sites. Sampling occurred in 1998 for balsam fir, white spruce and trembling aspen, in 1999 for mountain maple, cedar and paper birch and in 2000 for yellow birch. Sample sites were located where we expected variation in the recent growth of individuals (the predictor variable), i.e. across light gradients found between gap and non-gap environments. To span variation in growth rates, we randomly sampled live individuals, stratified across the heterogeneous growth environments at each site.
Three sets of field data were collected at each sample site: (i) the numbers of live and dead individuals of the focal species at the site were used to estimate mortality rate; (ii) a random sample of live individuals was selected for growth measurements; and (iii) a random sample of dead individuals allowed growth leading up to death to be measured. The total numbers of live and dead individuals of a focal species were estimated by either sampling the entire population or subsampling with randomly placed rectangular quadrats. Depending on the sample site, a different number (between three and eight) and different size (5 m2, 10 m2, 20 m2, 50 m2 and 100 m2) of quadrats was used.
Stem cross-sections at 10 cm above the root collar were obtained for a random sample of live individuals (25 ≤ N ≤ 55) and a random sample or the entire population of recently dead individuals (21 ≤ N ≤ 57). For our approach to work, it is important that our sample of recently dead trees contains only individuals whose likely cause of death was growth-related suppression. We carefully excluded any recently dead trees that showed signs of disease, herbivory, insect-infestation or mechanical damage. Annual growth rings were measured along a representative radius (the radius bisecting the angle formed by the longest and shortest radii of the cross section). Growth rings were measured with a VELMEX digital ring analyser (0.025 mm resolution) connected to a 40X stereo microscope. In young individuals (< 10 years) all rings were measured, whereas in older individuals only the 10 most recent rings were measured.
Dead saplings used for growth measurements were estimated to have died within the last 3 years (i.e. ‘recently dead’ individuals were defined using methods developed by Kobe et al. (1995) and Kobe & Coates (1997) and validated for our study sites). By selecting only individuals that show no external causes of death we focus on growth-related mortality and by focusing on the three most recent years of growth we are able to evaluate whether the risk of mortality for an individual sapling increases as growth rate decreases.
Time since death was estimated using the best discriminating features of buds (presence, intactness, colour), bark (coverage, intactness), stem suppleness and leaves (estimate of leaves/needles remaining, how easily the leaves are removed, colour, brittleness, intactness), as determined from characteristics sampled from trees that were known to have been dead (killed in spacing and weeding operations) for 1, 2, 3, 4 and 5 years. Two additional sources of information were used to establish time since death criteria. Spruce and fir saplings (n = 10 for each species) were transplanted into buckets and killed through lack of water and characteristics were observed over 4 years (1999–2003). In addition, criteria were validated, similarly to Newberry et al. (2004), against known death times from a long-term seedling monitoring project (L. Mathias et al., unpublished data). For all species, we found the highest discriminating power of time of death at 3 years, as manifested by bud, bark and leaf features (Kobe & Coates 1997). Careful application of these criteria minimizes potential error in estimating time since death.
Recently dead saplings at all sites showed variation in estimated time of death, supporting the theory that mortality events occurred continuously over the 3-year window and did not occur as a response to a single anomalous event (e.g. pest outbreak, extreme drought).
parameter estimation using maximum likelihood
We used maximum likelihood methods to estimate parameters and 95% support for species- and site-specific models characterizing the probability of mortality as a function of recent growth (g).
The statistical method that we used incorporates information from three sources in a likelihood function in order to estimate the most likely functional relationship between probability of mortality and recent growth, i.e. site-specific mortality rate based on a sampling of the population of live and dead individuals at a site, growth rates of ‘recently dead’ individuals (representing the last 3 years growth and thus the growth rates prior to death) and growth rates of live individuals.
We did not use the methods proposed by Wyckoff & Clark (2000) because assuming an overall mortality rate for a species and ignoring among-site variation can lead to biased estimates (R. K. Kobe, unpublished). The likelihood function that we used was
( eqn 1 )
where the first term, based on the counts of live and recently dead individuals within each study site, is the probability that D dead saplings are encountered in a total population of N individuals (D and N estimated from the sampled quadrats) where represents the mean probability or expectation of mortality (and is estimated as the denominator in the second term), m(g) is the mortality function (probability of mortality as a function of growth for an individual sapling) and h(g) is the probability density function of growth rates of all saplings at a given site.
The second term in equation 1 is the conditional probability density function of growth given that a sapling will die, based on measured growth rates leading up to the mortality of the ‘recently dead’ saplings. The third term of the likelihood function is the probability density function of growth, conditioned on the status of being live. A search algorithm is used to test different sets of parameter values in the likelihood function to obtain those that result in the highest likelihood of replicating the data set. The conditional density function of growth given that a sapling will die is detailed in Kobe et al. (1995), Kobe & Coates (1997) and Caspersen & Kobe (2001).
As in Kobe et al. (1995), we used a gamma density function to specify h(g) because the two-parameter gamma is flexible in shape and, by definition, g≥ 0. We used the Metropolis algorithm (Szymura & Barton 1986) to search for parameter values and functional forms of m(g) and parameter values of h(g) that yielded the highest likelihoods. Ninety-five per cent support intervals for all estimated parameters were estimated by inverting the likelihood ratio test (LRT) (Edwards 1992; Pacala et al. 1996).
We used the arithmetic average of the four most recent years of growth (excluding the last ring) based on LRT comparisons of 2, 3, 4 and 5-year averages of recent growth. Previous work (Kobe et al. 1995) has also shown this to provide the highest likelihoods. We excluded the most recent growth ring to ensure that growth measurements from both live and recently dead individuals were from complete growing seasons.
We specified m(g) as the cumulative distribution function of an exponential random variable
( eqn 2 )
where A and B are parameters to be estimated from the data. Equation 2 assumes that mortality increases with lower growth rates. Although m(g) in the likelihood function could be derived analytically from density functions of growth for the live and dead individuals and mortality rate based on stem counts (Wyckoff & Clark 2000), we chose to use equation 2 instead because it is much simpler and has a more straightforward biological interpretation than the complex form of m(g) derived from growth densities. In addition, the flexibility of equation 2 makes it ideal for approximating complex relationships that are analytically derived (Wyckoff & Clark 2000), as equation 2 accommodates a wide range of functional relationships between mortality and growth.
We also tested a negative exponential model
( eqn 3 )
and simplified forms of the negative exponential and equation 2 where A or a is set equal to one. Equation 2 with A = 1 resulted in the best fits weighted by the number of parameters (LRT, P < 0.05) and thus we report results only for this model.
variation in mortality functions within and among size classes
The above methods were used to estimate parameters for species, site and size class specific mortality models. To parameterize generalized models for a species in a given size class, all three data sets for a particular species-size class combination were evaluated simultaneously. The generalized likelihood function was the product of site-specific variants of equation 2, with a mortality model [m(g)] common to all three sites. That is, each site retained its site-specific N, D, Ū and h(g), but one generalized m(g) was estimated for all three sites (Kobe 1996). We assessed empirical support for specifying one generalized mortality model per species–size class combination vs. site-specific mortality models with Akaike's information criteria corrected for small sample sizes (AICc). In general, the model with the lowest AICc has the greatest empirical support, with a difference of > 2 AICc units representing stronger empirical support for the model with the lower AICc (Burnham & Anderson 2002).
Similarly, we estimated a single mortality function for a given species across all sites and size classes (i.e. m(g) common to all sites and size classes for a given species). The generalized species models were used as a basis for comparison with size class specific models to assess empirical support for effects of size class on the functional relationship between probability of mortality and recent growth. We compared size class specific vs. generalized species models using AICc. All results are expressed in terms of ΔAICc, which is defined as the AICc of a given model minus the minimum AICc (Burnham & Anderson 2002); thus, the best supported model will have ΔAICc = 0.