Materials and Methods
2.1 Classical growth equations for the
rising growth
trend
The
Gompertz and logistic equations are two important types of sigmoidal
trajectories. In order to better describe the unimodal trajectory, here,
we discretized these two equations. For the Gompertz model, we first
introduced a time parameter T to re-describe growth rate, whereT refer to the formation time of unit tissue. Although still
considered conceptual, T is not biologically arbitrary, but
describes the time during which tissue development is controlled by
genes and physiological activity, independent of size. Assumingf (M ) is the total biomass of new tissues during timeT , then the expression term of growth rate isf (M )/T in this period of time. In mathematics,T can also be infinitely small so thatf (M )/T → dM /dt. So the introduction ofT has a broader significance. Applying this concept and the
Bertalanffy paradigm, we not only can obtain the discrete Gompertz
equation, but also extend it further. The discrete logistic equation can
be directly derived from the population model based on the intrinsic
growth rate and reproductive generations. See supplementary information
for the detailed derivation steps of all equations. The resulting
discrete growth expressions can be written as:
(1a)
(2)
Eq. 1a is the extension of the discrete Gompertz equation, and can be
simplified to
(1b)
Where f (M )/T represent the growth rate; (ignoreo ) and parameter o ensures that the growth equation can
move along the transverse axis, reflecting the modular strategy of tree
growth. The vertex of Eq. 1b is (, ).
Since T , gr and b are relatively
stable, M max is mainly determined bycM /mr . By definition,cM ∝ resource uptake, 1/mrand 1/gr . Obviously, Eqs. 1a and 2 belong to the
Bertalanffy family.
The effects of parameterscM , b ,T ×mr /gr,M max and λ on growth curves are shown in
Figs. 1a and b. Note that the parameter cMreflects the average levels of resource uptake and respiration
consumption. The change ofT ×mr /gr mainly
depends on mr , because mris more sensitive to the environment (Van Iersel 2003). Mathematically,
unimodal curves connected in series can be divided into types αand β, as shown in Fig. 1c. Type α indicates that unimodal
curves in series have the same starting point. In type β , the
starting point of the new curve can be anywhere on the old curve,
reflecting modular growth. Biologically, the difference between αand β types lies in the involvement of the old tissues in the new
growth pattern. Considering Eq. 1 as an example, continuous changes in
parameters (cM andT ×gr /m r ) may
result in a mixed trajectory, as shown in Fig. 1d. We referred to this
type as type γ.
We speculated that changes in the relationship between functional traits
and tree size may affect cascading growth, with both positive and
negative effects (PE and NE) on growth. The effects of PE and NE on Eqs.
1 and 2 are related to changes in functional traits such as module and
overall performance, as shown in Table. 2. On the module scale, we
considered both leaf and stem economics (Table. 2), which are related to
photosynthesis, hydraulic transport, and respiration consumption. These
traits are important for tree ontogeny (Westoby 1998; Weiher et al.
1999; Poorter et al. 2008; Héraul et
al., 2011). On a more comprehensive scale, we considered the number of
trait modules relative to size, where some morphological traits that are
related to the total amount of photosynthesis and respiration deserve
special attention. The product of average crown width/(DBH+crown width)
(abbreviated as CW/(D+CW)) and crown height/tree height (i.e., crown
ratio) (abbreviated as CH/H) can be considered a proxy of canopy/size.
Note that the size in this indicator is not biomass but
volume. Higher canopy/size means a
resource intake strategy, then increase cM andM max. The attenuation of light within the canopy
will cause a decrease in light utilization after the canopy closes, so
the uptake rate of trees to resources is not directly proportional to
their canopy.
Sapwood is the living, outermost portion of a woody stem or branch, and
heartwood is the dead, inner wood, which often comprises the majority of
a stem’s cross-section. Sapwood, not heartwood, serves as a sink tissue
that consumes sugars for daily metabolism through respiration (Lehnebach
et al., 2016; Bamber et al., 1976).
Because heartwood is not active, themr of the trunk could decrease with a decreasing
sapwood/heartwood ratio. Given that the tree trunk accounts for most of
the tree biomass, we calculated the sapwood area to heartwood area ratio
to assess the change of tree mr . Obviously, the
change of mr can affect maintenance respiration
(= mr × biomass) from the scaling relationship
with biomass. In fact, sapwood maintenance respiration may also be a
significant cause of growth decline with increasing tree age (Tatuo and
Shidei, 1967; Ryan and Yoder, 1997; Sillett et al., 2010). Overall,
these two ratios have great effects on plant growth.
2.2 Site description and experimental
design
We studied four subalpine primevalAbies fabri forests located in the Gongga Mountain, in the
southeastern Tibetan Plateau, and at altitudes of 2,900 m, 3,000 m,
3,300 m, and 3,600 m (treeline). The mean annual temperature was 4 °C,
and the mean annual precipitation was 1,938 mm, with roughly 50% of the
annual total rainfall occurring from June to September in 2015. The
characteristics of these survey plots are described in Wang et al
(2017).
Abies fabri trees have large stature and long life history, and
we expected to observe significant biomass changes on the century scale.
First, we estimated the ideal and average growth trajectories (with
respect to size) of this species by the DBH sequence of all sampled
trees and biomass equations. These results can be used to test H1 and
H2. Note that the ideal growth trajectory is composed of the maximum
growth increment of different trees along the size gradient. Then, we
focused on changes in some important functional traits with tree size.
This can be achieved by analyzing the functional traits of different
trees. Some variable functional traits may be the key to cascading
growth. Finally, we tried to quantify the effects of these variable
functional traits on unimodal growth to test H3. Sampled functional
traits is current, so they are only valid for recent growth dynamics.
Thus, some individual trees that follow the unimodal trajectory only in
recent growth would be selected, all located at an altitude of 3,100 m.
We expect that the height and length of these unimodal trajectories
(related to parameters cM ,mr , λ and M max) are
determined by some current functional traits (Tab. 1).
2.3 Data collection, processing and
analysis
In September 2015, we measured the tree ring sequence and DBH
(> 10cm) of each tree in four large plots along elevation
gradients. For details of the method, see Wang et al. (2017). In June
2019, we sampled and measured the relevant functional traits of trees
belonging to different DBH classes. DBH classes are in the 5 cm
interval. These randomly sampled trees covered more than half of the
sample areas surveyed in 2015.
Meanwhile, the functional traits of the selected trees were sampled
separately. For leaf and stem economics (listed in Tab. 1), we sampled
three times at the south and north sides of trees, and at different
heights (e.g., 5m, 10m,15m). The number of leaves sampled each time
ranged between 20 and 60. Some leaves were used for element analysis.
Leaf N concentrations were analyzed with a Vario MAX CN element analyzer
(Elementar, Hanau, Germany). The vanadium molybdate yellow colorimetric
method was used to measure Leaf P, and flame photometry was used to
measure Leaf K. We used foliar dry matter content to locate different
trees on a resource use axis rather than specific leaf area (SLA)
(Wilson et al., 1999) due to the large vertical span of the fir canopy
and sampling difficulty. Details of trait measurement are described in
Wilson et al. (1999) for foliar dry matter content. For stem economics,
trunk xylem density was calculated as the ratio of mass of samples dried
for 72 h at 105℃ after the removal of bark, to their fresh volume as
calculated by the displacement of liquid volume (water). Trunk bark
thickness was measured by vernier caliper. Trunk wood moisture was
determined by the fresh mass and dry mass of the wood, and was expressed
as fresh mass/dry mass-1. We determined the sapwood width using diameter
increment borers. Sapwood and heartwood areas are equal to
π×(DBH/2)2-π×(DBH/2-sapwood width)2and π×(DBH/2-sapwood width)2, respectively. Other
morphological traits were obtained by direct forestry survey.
We used the actual tree ring sequence to determine the aboveground
biomass dynamic of individual trees. To reduce error, the minimum time
interval for describing growth dynamic was set to 10 years. The increase
in DBH for each decade can be derived from the current DBH and tree-ring
data. Based on previous results (Zhou et al., 2013), we established the
aboveground biomass equation of individual trees dependent on DBH, which
allows us to obtain the growth trajectory of individual trees (see
supplementary information). From these results, we can determine the
best and average DBH or biomass increment within different DBH classes
(> the maximum diameter increments) to reveal the growth
dynamic along along the DBH or size gradient. The emphasis of maximum
increment is on species upper-quantile growth change (95% percentile)
(Wright et al., 2010), reflecting a
relatively ideal growth state.
For Eq. 1b, f (M )/T can represent the annual average
growth rate over a decade. Thus, the 10-year aboveground biomass
increment can be approximately equal to 10f (M )/T .
(3)
Where b = 0.75. Assuming a constant ratio of aboveground to
underground biomass (Shu et al., 2019), individual growth equations can
also be used directly to describe the change of aboveground biomass. We
used Eqs. 3 and 2 and their extended forms (see supplementary
information) to fit the growth dynamics of aboveground biomass and DBH
to directly test the validity of classical growth
equations. If these equations are
valid for the selected trees, we can get the length and height of two
type of unimodal curves (Figs. 1a and b), which are closely related to
parameter λ , M amax10mr /gr andcM . According to Tab. 1, we can directly test the
effect of selected variable traits on the formation of unimodal curve.