We present a simple, generic model of annual tree growth, called “
Forests cover about 30 % of the land surface (Bonan, 2008) and are
estimated to contain 861
Climate factors, such as temperature and moisture availability during the
growing season, are important drivers of tree growth (Harrison et al.,
2010). This forms the basis for reconstructing historical climate changes
from tree-ring records of annual growth (Fritts, 2012). However,
photosynthetically active radiation (PAR) is the principal driver of
photosynthesis. Models for primary production that use temperature, not PAR,
implicitly rely on the far-from-perfect correlation between temperature and
PAR (Wang et al., 2014). PAR can change independently from temperature
(through changes in cloudiness affecting PAR or atmospheric circulation
changes affecting temperature) and this may help to explain why statistical
relationships between tree growth and temperature at some high-latitude and
high-elevation sites appear to break down in recent decades (D'Arrigo et al.,
2008). CO
Modelling is needed for this purpose. Empirical models of annual tree-growth
and climate variables (temperature and precipitation) have been used to
simulate tree radial growth (Fritts, 2012). Process-based bioclimatic models
might be preferable, however, because this allows other factors to be taken
into account (e.g. the direct impact of CO
Simple equations representing functional and geometric relationships can describe carbon allocation by trees and make it possible to model individual tree growth (Yokozawa and Hara, 1995; Givnish, 1988; Falster et al., 2011; King, 2011). Such models are built on measurable relationships, such as that between stem diameter and height (Thomas, 1996; Ishii et al., 2000; Falster and Westoby, 2005), and crown area and diameter or height (Duursma et al., 2010) that arise because of functional constraints on growth. The pipe model represents the relationship between sapwood area and leaf area (Shinozaki et al., 1964; Yokozawa and Hara, 1995; Mäkelä et al., 2000). The ratio of fine-root mass to foliage area provides the linkage between above- and below-ground tissues (Falster et al., 2011). These functional relationships are expected to be stable through ontogeny, which implies that the fraction of new carbon allocated to different compartments is variable (Lloyd, 1999). In this paper, we combine the two modelling approaches previously developed in the global carbon-cycle (ecophysiology) and forest-science (geometric and carbon allocation) literature to simulate individual tree growth.
We use a light-use efficiency model (the
The
Potential annual GPP is the product of the PAR incident on vegetation
canopies during the growing season (PAR
Leaf-internal [CO
Model application flow. We combined the simple light-use efficiency
and photosynthesis model (
The temperature dependence of
The
We assume that potential GPP is the first-order driver of tree growth both at
stand and individual level. The
Carbon is allocated to different tissues within the constraint of the basic functional or geometric relationships between different dimensions of the tree.
Asymptotic height–diameter trajectories (Thomas, 1996; Ishii et al., 2000;
Falster and Westoby, 2005) are modelled as
The model also requires the derivative of this relationship:
The relationship of diameter increment to stem increment is then given by:
Crown fraction (
Actual GPP (
NPP is derived from annual GPP, corrected for
foliage respiration (which is set at 10 % of total GPP, an approximation
based on the theory developed by Prentice et al., 2014 and Wang et al.,
2014) by further deducting growth respiration and the maintenance
respiration of sapwood and fine roots. Growth respiration is assumed to be
proportional to NPP, following:
NPP is allocated to stem increment (d
Parameter description and the derivation of parameter values.
The season over which GPP is accumulated (i.e. the effective growing season) is defined as running from July in the previous year through to the end of June in the current year. This definition is consistent with the fact that photosynthesis peaks around the time of the summer solstice (Bauerle et al., 2012) and that maximum leaf area occurs shortly after this (Rautiainen et al., 2012). Carbon fixed during the later half of the year (July to December) is therefore either stored or allocated for purposes other than foliage expansion. Observations of tree radial growth show that it can occur before leaf-out (in broadleaved trees) or leaf expansion (in needle-leaved trees), thus confirming that some part of this growth is based on starch reserves from the previous year (Michelot et al., 2012). This definition of the effective growing season is also supported by analyses of our data, which showed that correlations between simulated and observed tree-ring widths are poorer when the model is driven by GPP from the current calendar year rather than an effective growing season from July through to June.
Estimation of parameter values for the application of the
We use site-specific information on climate and tree growth from a relatively
low-elevation site (ca. 128
We applied the model to simulate the growth of 46 individual
For statistical analyses and comparison with observations, the individual
trees were grouped into three cohorts, based on their age in 1958: young
(0–49 years); mature (50–99 years); old (
Comparison between simulations and observations for the three
age cohorts (young: 0–49 years; mature: 50–99 years; old
There are only small differences between different age cohorts in the mean simulated ring width, with a mean value of 1.43 mm for young trees, 1.31 mm for mature trees, and 1.37 for older trees. These values are comparable to the mean value obtained from the observations (1.48, 1.29, and 1.34 mm, respectively). However, the general impact of ageing is evident in the decreasing trend in ring widths between 1958 and 2007 within any one cohort (Fig. 3). The slope is stronger in the observations than in the simulations, indicating that the model somewhat underestimates the effects of ontogeny.
There is considerable year-to-year variability in tree growth. The simulated
interannual variability (standard deviation) in simulated ring width is
similar in all the age cohorts (0.265 mm in the young, 0.265 mm in the
mature, and 0.264 mm in the old trees). This variability is somewhat less
than shown by the observations, where interannual variability is 0.274,
0.367, and 0.245 mm, respectively in the young, mature, and old cohorts. The RMSE is
0.263, 0.332, and 0.284 mm, respectively for young, mature, and old age
cohorts. The correlation between the observed and simulated sequence in each
cohort is statistically significant (
Parameter sensitivity analyses for the
Despite the fact that the model reproduces both the mean ring width and the interannual variability in tree growth reasonably well, the range of ring widths simulated for individual trees within any one cohort is much less than the range seen in the observations. This is to be expected, given that individual tree growth is affected by local factors (e.g. spatial variability in soil moisture) and may also be influenced by ecosystem dynamics (e.g. opening up of the canopy through the death of adjacent trees). These effects are not taken into account in the model.
GLM analysis of tree-growth response to the climatic factors and
age, based on simulations and observations. The dependent variable is mean
ring width series (1958–2006) for each age cohort (young, mature, and old).
The independent variables are the growing-season total annual
photosynthetically active radiation (PAR
To evaluate the sensitivity of the model to specification of individual
parameters, we ran a series of simulations in which individual parameter
values were increased or decreased by 50 % of their reference value. For
each of these simulations, the
Tree-growth response to climate and tree age: partial residual plots
based on the GLM analysis (Table 2), obtained using the visreg package in
CO
The model simulates a rapid initial increase in ring width, with peak ring
widths occurring after ca. 10 years, followed by a gradual and continuous
decrease with age (Fig. 4). The model is comparatively insensitive to
uncertainties in the specification of fine-root specific respiration rate
(
The GLM analysis revealed a strong positive relationship between PAR
The model reproduces these observed relationships between climate factors and
tree growth. The slope of the observed positive relationship with
The GLM analysis also showed that age, as represented by the three age cohorts, has an impact on ring width: young trees have greater ring widths than mature trees, while old trees have somewhat greater ring widths than mature trees. This pattern is seen in both the observations and simulations, although the differences between the young and mature cohorts are slightly greater in the observations.
The overall similarity in the observed and simulated relationships between
growth rates and environmental factors confirms that the
Elevated levels of CO
We have shown that radial growth (ring width) can be realistically simulated
by coupling a simple generic model of GPP with a model of carbon allocation
and functional geometric tree growth with species-specific values. The model
is responsive to changes in climate variables, and can account for the impact
of changing CO
Our simulations suggest that, after a brief but rapid increase for young plants, there is a general and continuous decrease in radial growth with age (Fig. 4). This pattern is apparent in individual tree-ring series, and is evident in the decreasing trend in ring widths shown when the series are grouped into age cohorts (Fig. 3). It is a necessary consequence of the geometric relationship between the stem diameter increment and cross-sectional area; more biomass is required to produce the same increase in diameter in thicker, taller trees than thinner, shorter ones. However, we find that ring widths in old trees in our study region are consistently wider than those in mature trees, and this property is reproduced in the simulations (Fig. 5). This situation arises because the old trees are, on average, smaller than the mature trees at the start of the simulation (in 1958). Thus, while the difference between average ring widths in the mature and old cohorts conforms to the geometric relationship between stem diameter increment and cross-sectional area, it is a response that also reflects differences in the history of tree growth at this site, which determined the initial size of the trees in 1958. Lack of climate data prior to 1958 or detailed information about stand dynamics precludes diagnosis of the cause of the growth history differences between mature and old trees.
Studies attempting to isolate the impact of climate variability on tree growth, including attempts to reconstruct historical climate changes using tree-ring series, often describe the impact of ageing as a negative exponential curve (Fritts, 2012). However, our analyses suggest that this is not a good representation of the actual effect of ageing on tree growth, and would result in the masking of the impact of climate-induced variability in mature and old trees. The simulated NPP of individual trees always increases with size (or age). This is consistent with the observation that carbon sequestration increases continuously with individual tree size (Stephenson et al., 2014).
We have shown that total PAR during the growing season is positively correlated with tree growth at this site. This is not surprising given that PAR is the primary driver of photosynthetic carbon fixation. However, none of the empirical or semi-empirical models of tree growth uses PAR directly as a predictor variable; most use some measure of seasonal or annual temperature as a surrogate. PAR is determined by latitude and cloudiness. Although temperature varies with latitude and cloudiness, it is also influenced by other factors, including heat advection. Temperature changes can impact the length of the growing season, and hence have an impact on total growing-season PAR, but this is a trivial effect over recent decades. In fact, we show that mean annual temperature is negatively correlated with tree growth at this site. Given this decoupling, and the potential that longer-term changes in cloudiness will not necessarily be correlated with changes in temperature (Charman et al., 2013), we strongly advocate the use of growing-season PAR for empirical modelling, as well as in process-based modelling.
We found no age-related sensitivity to interannual variability in climate; the interannual variability in ring width is virtually identical between age cohorts. The strength of the relationship with individual climate variables is also similar between the three age cohorts. It is generally assumed that juvenile and old trees are at greater risk of mortality from environmental stress than mature trees are (e.g. Lines et al., 2010; McDowell et al., 2008). This may be true in the case of extreme events, such as wildfires, windthrow, or pest attacks. Our results suggest that, although climate variability has an important effect on tree growth, it is not an important influence on mortality.
We have assumed that the period contributing to growth (i.e. the effective growing season) in any year includes carbon stores generated during the second half of the previous year. The total foliage area determines the radial area of the stem, and, once this is achieved, NPP is allocated either to fine-root production or stored as carbohydrate for use in stem growth in the early part of the subsequent year. This is consistent with observations that radial growth begins before leaf-out (Michelot et al., 2012) and that maximum leaf area is generally achieved by mid-summer (Rautiainen et al., 2012). The MAIDEN model also allows tree growth to be influenced by a fixed contribution from the previous year's growth (Misson, 2004). Defining the effective growing season as being only the current growth year had no impact on the influence of climate on ring widths, or the shape of the ageing curve. It did, however, produce a considerably lower correlation between simulated and observed interannual variability in growth. Since tree-ring width reflects the integrated climate over the “effective growing season”, reconstructions of climate variables reflect conditions during that season, not only during the current calendar year.
The high degree of autocorrelation present in tree-ring series is often seen as a problem requiring pretreatment of the series in order to derive realistic reconstructions of climate variables (e.g. Cook et al., 2012; Anchukaitis et al., 2013; Wiles et al., 2014). However, spatial or temporal autocorrelation is a reflection of the causal mechanism underpinning the observed patterning. Here, we postulate that the mechanism that gives rise to the temporal autocorrelation in tree-ring series is the existence of carbon reserves that are created in one year and fuel early growth in the next. If a large reserve of carbon is created in the second half of the growing season, because of favourable conditions, this will offset poor conditions in the following year. However, large reserves may not be necessary if conditions during the subsequent growing year are very favourable. The fact that the relative influence of one year on the next can vary explains why the measured autocorrelation strength in a given tree-ring series varies through time.
The
Our modelling approach integrates the influence of climate, [CO
We thank Wang Han for assistance with the