2.1. Forest Inventory Data
Trees identified as black spruce (
Picea mariana (Mill.) BSP), interior Douglas-fir (
Pseudotsuga menziesii var.
glauca (Beissn.) Franco), or western hemlock (
Tsuga heterophylla (Raf.) Sarg.) were compiled from databases provided by agencies in British Columbia, Alberta, Saskatchewan, Manitoba, Ontario, New Brunswick, Nova Scotia, Newfoundland and Labrador, and the United States (
Figure 1).
Interior Douglas-fir was differentiated from the coastal variety by only including trees east of the Cascade/Coastal Mountain range. At each field plot, a census was taken for trees exceeding a threshold breast-height diameter. The lowest observations of breast-height diameter were below 5.0 cm with the exceptions of Saskatchewan and Newfoundland and Labrador, which only considered trees with a breast-height diameter greater than 7.1 and 9.1 cm, respectively. Between 1% to 5% of the samples were affected by some form of management and these observations were excluded from the samples. The biomass of bark, branches, foliage, and stemwood was calculated from dimension analysis based on diameter and height [
41,
42]. Total aboveground biomass was then calculated from the summation of foliage, branches, bark, and stemwood. All values of biomass were converted to the mass of carbon assuming a carbon-dry wood ratio of 0.5 [
43].
The final samples included between 5779 and 8206 field plots (
Table 1). Sampling was conducted between the 1960’s and 2010’s and targeted stands that were between 10 and 190 years old. On average, black spruce, interior Douglas-fir, and western hemlock were representative of continental, cold-temperate, and maritime-temperate climates, respectively. Referencing estimates of warm-season mean soil water content from a monthly surface water balance model suggested that all species covered large ranges in hydrological conditions, but that the sample of interior Douglas-fir was, on average, substantially drier than the other two species.
Statistical modelling was conducted for three dependent variables. The first dependent variable was the periodic mean annual aboveground biomass growth rate of individual trees, Gag (kg C tree−1 year−1). Growth was calculated from the annualized difference between estimates of tree biomass at sequential census dates. By measuring a wide range of stand ages, the observations of growth included trees that eventually succumb to density-dependent mortality, yet the growth of trees during a period in which trees died was not measured due to uncertainty in the year that the tree died. The second dependent variable was the annual probability of recruitment, Pr (% year−1), which was inferred from a binary variable describing whether the tree was a recruit (a value of “1”) or not (a value of “0”) during the measurement interval. The third dependent variable was the annual probability of mortality, Pm (% year−1), which was inferred from a binary variable describing whether the tree died (a value of “1”) or not (a value of “0”) during the measurement interval.
Independent variables included the aboveground biomass of individual trees at the first measurement date, Bag (kg C tree−1); stand age at the first measurement date, SA (years); biomass of all trees in the plot at the first measurement date, SB (Mg C ha−1); and biomass of larger trees in the plot at the first measurement date, SBLT (Mg C ha−1). Stand age was either supplied by the data source, or estimated from the subsample of trees that were measured at each plot. The biomass of larger trees was calculated by summing the product of aboveground biomass of individual trees and area expansion factors for individuals with a greater biomass than the focal tree.
2.2. Modelling
A function,
Sawtooth, was written in Matlab (MATLAB R2016a, The Mathworks Inc., Natick, MA, USA) to calculate the mass balance of trees for
i = 1 …
m annual time-intervals and
j = 1 …
n homogeneous spatial units (i.e., stands), each with an area of 10000 m
2. The biomass dynamics of each stand were ultimately driven by equations describing the recruitment, growth, and mortality of trees. The two defining features of each equation are: (1) the sample of field observations that the equations were calibrated against (e.g., typically representing a specific combination of tree species and region); and (2) the specification of the equations. For this study, we focused on calibrating and applying relatively simple equations for each species, herein called “Default 1” equations, with the intention of representing basic patterns of ontogeny, aging, and competition among trees. The Default 1 equations intentionally avoided additional complexity and, thus, provide a high-bias/low variance alternative to equations that attempt to represent additional system complexity [
26,
30].
Stand-level variables were initialized as m × n (stand-level) matrices. For each stand, the model tracked the vital status and growth rate of k = 1 … SDmax trees by initializing a second set of intermediate m × SDmax matrices for each variable. The value, SDmax, specified the maximum possible number of live trees per spatial unit, or stand density (SD). As such, there were SDmax spaces (with an area of SDmax/10,000 m2) available where a live tree could exist. If SDmax is set high enough, the balance between predicted recruitment and mortality rates will yield a steady-state value of stand density that never reaches SDmax. Each tree in the tree-level matrix was randomly assigned a species according to prescribed fractions of each initial species.
During the first time-interval, all SDmax spaces were initially classified as “empty” and then l = 1 … Nseed were randomly selected to germinate. The status of these germinating trees was converted to “live tree”, and the aboveground biomass of the germinating trees was converted to a low value, Bag,seed. The values of Bag,seed were drawn randomly from a normal distribution with the mean, Bag,seed,mu, and standard deviation, Bag,seed,sig, and were constrained by Bag,seed > 0. Hence, a degree of within-stand variation in germination rates was assumed to lead to an initial cohort of trees with a biomass that follows a normal distribution at the end of the first time-interval.
For each stand, the model looped through i = 2 … m time-intervals. First, trees that were “live” at the end of the previous time-interval were identified and used to calculate the stand-level explanatory variables required to predict recruitment, growth, and mortality for that time-interval. This included stand age (SA), aboveground biomass density (SBag), and aboveground biomass of larger trees (SBLTag). Stochastic processes during each time-interval were controlled by vectors of random numbers drawn from a uniform distribution between 0 and 1 for all spaces inhabited by live trees (rlive) and empty spaces (rempty).
Models of tree recruitment predict the number of trees that surpass a specified tree size threshold, and range from simply applying a constant probability, to dynamic models that predict responses to varying stand conditions [
44,
45,
46,
47]. Overall, modelling recruitment generally requires separate predictions of occurrence and frequency. Here, the strategy was to treat species occurrence as a boundary condition—an assumption made by the user that a seed source of the focal species exists, or was predicted to occur using established models within stand,
j [
48,
49,
50,
51]. In instances where the focal species occurs, logistic regression was used to predict the annual probability of recruitment of the
kth tree:
We assumed that the most simplistic version of a dynamic recruitment model would consider dependence on the intensity of stand competition. To represent this type of dynamic model, the Default 1 equation was defined as:
where
a0 and
a1 are fixed effects. (See the previous paragraph for a definition of explanatory variables.) The equations were calibrated against ungrouped samples of each species. The logistic regression equations were fitted against observations of ingrowth; the sample consisted of trees that were counted at both the first and second measurement dates (“0” or “not a recruit”) and trees that were not counted at the first measurement date, but surpassed the threshold tree size for inclusion between the first and second measurement dates (“1” or “recruit”). As SD is constrained by SD
Max, the probability of recruitment is only calculated and applied to the available empty spaces, by comparing
Pr(empty) with the
rempty. For any trees with
Pr(empty) >
rempty, the status of the empty space was converted from “empty” to “live tree”, the age of the recruited tree was converted to one year, and the aboveground biomass of the recruited tree was converted to
Bag,seed.
Whereas many comparable models predict the annual increments of diameter and height before deriving the growth rate of stemwood volume or biomass, Sawtooth assumes aboveground biomass growth as the dependent variable [
30]. For this study, estimates of aboveground biomass growth were derived from dimension analysis of the tree diameter and height using Canada’s national biomass equations [
42] and a dry weight-to-carbon ratio of 0.5 [
43]. The Default 1 equation predicted the natural logarithm of the aboveground biomass growth of the
kth tree as:
where
b0 …
b5 are fixed effects. By including ln(
Bag,k) and
Bag,k, the equation has the flexibility to represent an optima in the size-dependence of growth [
30,
31,
52,
53]. By including SBLT
ag and SB
ag, the equation has the flexibility to represent relative and absolute expressions of the intensity of stand competition. Predictions of aboveground biomass growth are then added to the existing aboveground biomass of live trees.
The annual probability of tree mortality (
Pm) was calculated as the sum of simulated and prescribed mortality. Simulated mortality was inferred from statistical models (e.g., logistic regression). Prescribed mortality provided the flexibility to also utilize observations of discrete disturbance events derived from forest inventory and monitoring systems. The detection strength and prediction accuracy of tree mortality observation systems are often highly specialized and, therefore, unevenly distributed across the complete spectrum of landscape-scale tree mortality. To simultaneously accommodate data sources of varying specialization, simulated and prescribed types of mortality were further partitioned into non-stand-replacing and stand-replacing components:
where
Pm,SimNSR is simulated non-stand-replacing mortality,
Pm,SimSR is simulated stand-replacing mortality,
Pm,PreNSR is prescribed non-stand-replacing mortality, and
Pm,PreSR is prescribed stand-replacing mortality. As the landscapes in the present study were hypothetical, we relied on a combination of
Pm,SimNSR and
Pm,SimSR to represent the total rates of tree mortality, while prescribed disturbance events,
Pm,PreSR and
Pm,PreNSR, were set to zero.
Pm,SimNSR represents lethal processes in the high-frequency/low-magnitude part of the landscape mortality spectrum; this is the frequent occurrence of low rates of mortality within intact stands, commonly referred to as ‘background’ or ‘regular’ mortality. The processes that are represented by
Pm,SimNSR will depend on the specification of the equations and quality of the sample that it was calibrated against. Technically, there is nothing stopping
Pm,SimNSR from predicting 100% mortality in rare cases. As with recruitment, we used logistic regression to predict
Pm,SimNSR of the
kth tree, defining the Default 1 equation as:
where
c0 …
c5 are fixed effects. By design, the Default 1 equation of mortality focuses on representing effects of ontogeny, aging, and competition on mortality rates. The equation is underspecified because it does not include many other factors that contribute to overall rates of mortality observed at field plots, including wind, insects, pathogens, weather, animals, and pollutants [
27,
54]. To some unknown extent, these factors partially contribute to the average of predictions, but failure to specify them in the model (due to difficulties in measuring them) likely introduces bias in the predictions. Nevertheless, the Default 1 equation tests fundamental constraints on the longevity of trees that are central to biomass dynamics of intact stands. Upon the calculation of Equation (5), mortality counts were tallied according to the number of trees with
Pm,SimNSR(live) >
rlive. The status of the trees that died were converted from “live” to “empty”, and the age and aboveground biomass of the trees that died were converted to zero.
The model also accounted for lethal processes that operate in the low-frequency/high magnitude part of the landscape mortality spectrum by simulating stand-replacing mortality as:
where
Pm,SimSR,H is mortality caused by harvesting,
Pm,SimSR,F is mortality caused by fire,
Pm,Sim SR,I is mortality caused by insects, and
Pm,SimSR,P is mortality caused by pathogens. To represent landscape-scale forest biomass dynamics, it was essential to impose stand-replacing disturbance in this study, yet the study did not strongly depend on the specific nature of the disturbance regime. Across all time-intervals and all stands, stand-replacing disturbance was represented by assuming a constant fire return interval of RI
Fire = 200 years, corresponding to
Pm,SimSR,F = 0.005. Other stand-replacing disturbances were not considered, setting
Pm,SimSR,H,
Pm,SimSR,I, and
Pm,SimSR,P to 0.0.
Upon completing the time-intervals of each stand, the aboveground biomass of each tree was partitioned into foliage, bark, branches, and stemwood based on species-specific allometric equations [
42]. The belowground biomass for each tree was derived from aboveground biomass based on allometric equations for coniferous trees [
55]. Belowground biomass was partitioned into coarse and fine roots based on allometric relationships [
55,
56]. Total tree biomass was then calculated as the sum of foliage, branches, bark, stemwood, coarse roots, and fine roots. Biomass turnover of foliage, bark, branches, coarse roots, and fine roots,
τF,
τBk,
τBr,
τRc, and
τRf, respectively, were calculated based on constant rate coefficients for the tissues of coniferous trees [
55].
Corresponding columns of the stand-level matrix were then populated with summaries of the tree-level information from that stand, including stand age, SA (years); stand density, SD (trees ha−1); stand biomass growth (of survivors and recruits), SG (Mg C ha−1 year−1); stand biomass turnover due to litterfall, SLF (Mg C ha−1 year−1); stand biomass loss due to mortality, SM (Mg C ha−1 year−1); and stand biomass density, SB (Mg C ha−1). Stand net primary production, SNPP (Mg C ha−1 year−1), is defined as the difference between the photosynthetic fixation of carbon and autotrophic respiration of carbon, and was approximated from the sum of stand biomass growth and biomass turnover due to litterfall.
The subsequent description of forest biomass dynamics was based on the mean value of each variable across
n stands of the simulated landscape. To facilitate a comparison with field observations, we reported predictions of net ecosystem biomass production (NEBP), defined as the net change in biomass for stands during time-intervals in which there were no stand-replacing disturbances:
Net biome biomass disturbance (NBBD) was defined as the net change in biomass for stands during time-intervals where stand-replacing disturbances occurred:
Net biome biomass production (NBBP) was calculated as:
which is equal to net change in mean stand biomass density. Consistent with the differentiation between NEBP and NBBP, we also differentiated between the biomass residence time of intact ecosystems:
and biomass residence time of the forest biome, including disturbances: