Species-specific allometric scaling under self-thinning: evidence from long-term plots in forest stands
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DOI: 10.1007/s00442-005-0126-0
- Cite this article as:
- Pretzsch, H. Oecologia (2006) 146: 572. doi:10.1007/s00442-005-0126-0
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Abstract
Experimental plots covering a 120 years’ observation period in unthinned, even-aged pure stands of common beech (Fagus sylvatica), Norway spruce (Picea abies), Scots pine (Pinus sylvestris), and common oak (Quercus Petraea) are used to scrutinize Reineke’s (1933) empirically derived stand density rule ( \(N \propto \bar d^{-1.605} \), N = tree number per unit area, \(\bar{d}\) = mean stem diameter), Yoda’s (1963) self-thinning law based on Euclidian geometry (\(\bar w \propto N^{- 3/2}, \)\(\bar w\) = mean biomass per tree), and basic assumptions of West, Brown and Enquist’s (1997, 1999) fractal scaling rules (\(w \propto d^{8/3}, \)\(\bar w \propto N^{-4/3}, \)w = biomass per tree, d = stem diameter). RMA and OLS regression provides observed allometric exponents, which are tested against the exponents, expected by the considered rules. Hope for a consistent scaling law fades away, as observed exponents significantly correspond with the considered rules only in a minority of cases: (1) exponent r of \(N \propto \bar d^r \) varies around Reineke’s constant −1.605, but is significantly different from r=−2, supposed by Euclidian or fractal scaling, (2) Exponent c of the self-thinning line \(\bar w \propto N^c \) roams roughly about the Euclidian scaling constant −3/2, (3) Exponent a of \(w \propto d^a \) tends to follow fractal scaling 8/3. The unique dataset’s evaluation displays that (4) scaling exponents and their oscillation are species-specific, (5) Euclidian scaling of one relation and fractal scaling of another are coupled, depending on species. Ecological implications of the results in respect to self-tolerance (common oak > Norway spruce > Scots pine > common beech) and efficiency of space occupation (common beech > Scots pine > Norway spruce > common oak) are stressed and severe consequences for assessing, regulating and scheduling stand density are discussed.
Keywords
AllometrySelf-thinningStand density rule−3/2-Power lawEuclidian geometrical scalingFractal scalingIntroduction
Allometric scaling laws generalize the size-dependent structural relationships, partitioning and trade-offs between different organs’ or ecosystem elements’ growth. The stand density rule postulated by Reineke (1933) for woody plants, is an early empirically based species invariant scaling law with considerable importance in forest practice and forest science. The −3/2 power law of self-thinning formulated by Yoda et al. (1963) for herbaceous plants is the most prominent example for a scaling law based on Euclidian geometry. West et al. (1997, 1999) and Enquist et al. (1998) posit a scaling law for plants and animals, based on fractal geometry. The allometric coefficients and exponents of such laws give shape to underlying processes and although not elucidated in detail, they make the processes’ results operational for linkages between organismal and ecosystem level, for prognosis and scenario analyses. Simple and general rules still our innate propensity to reduce complexity, however they engender the risk of neglecting individual species peculiarities, which are essential for assessment and understanding the dynamics of organisms, populations or ecosystems.
Always based on the 3/4 scaling of metabolic rate, West, Brown and Enquist extend their considerations on plants, animals, and even on cells and mitochondria. They apply it on individual, community and ecosystem level and provoke Whitefield’s (2001, p. 343) question whether their approach is a “...theory of everything...”. Kozlowski and Konarzewski (2004) see the models’ positive influence in reviving interest in allometric scaling as a link between process and structure. However, after numerical and empirical scrutiny, they criticize West, Brown and Enquist’s model as neither mathematically correct nor biological relevant or universal. They claim more biological realism and analysis why scaling exponents differ between taxonomic groups.
My contribution to the ongoing debate about “the ultimative scaling law” is not more theory, but more empirical evidence. My database is a unique set of fully stocked, untreated long-term experimental plots in pure common beech, Norway spruce, Scots pine, and common oak stands in central Europe and it covers a 120 years’ observation period. The study is stimulated by the critical attitude towards general and species-invariant scaling rules of Gadow (1986), Niklas et al. (2003), Stoll et al. (2002), Weller (1987, 1990) and Zeide (1987). I focused on species-specific structural and temporal peculiarities under self-thinning.
Hypotheses
H1.1 claims for unthinned, fully stocked, even-aged pure stands a constancy of r of Reineke’s rule (exponent r in Formula 9) within stand development. H1.2 assumes that exponent r is equal for all m considered species, i.e. r_{1}=...=r_{m}. H1.3 claims r_{1}=r_{2}=... =r_{m}=−1.605, i.e. the validity of Reineke’s rule.
H2.1 postulates constancy of c [exponent c in Formula 10] within stand development. H2.2 postulates that slope c is equal for all m considered species, i.e. c_{1}=...=c_{m}. H2.3 scrutinizes whether the self-thinning line of the species follows Euclidian geometry, i.e. c_{1}=...=c_{m}=− 3/2 or fractal scaling, i.e. c_{1}=...=c_{m}=−4/3.
H3.1 postulates, that the coefficients of variation of \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} \) are equal for all species, i.e. \({\text{v}}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} _1 = \ldots = {\text{v}}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} _4 \) H3.2 uses the Pearson correlation \(r_{{\text{v}}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} ,\,c} \) between v \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} \) and slope c to detect connections between the spatial and temporal dynamic of the self-thinning process.
Data ^{1}
Growth and yield characteristics for the first/last survey of 28 non-thinned, fully stocked experimental plots of common beech, Norway spruce, Scots pine and common oak
Species | Experiment/ plot | n | First-last survey | Age span (years) | Stem number (trees/ha) | Mean height (m) | Mean diameter (cm) |
---|---|---|---|---|---|---|---|
Common beech/ Fagus sylvatica L. | ELM 20/1 | 13 | 1871–1967 | 49–145 | 5,844–400 | 13.2–35.0 | 7.7–36.0 |
FAB 15/1 | 12 | 1870–1958 | 48–136 | 6,220–477 | 12.5–32.2 | 7.6–34.1 | |
HG 333/3 | 11 | 1951–1999 | 56–104 | 2,368–840 | 12.7–26.3 | 13.6–26.9 | |
HAI 27/1 | 16 | 1870–1994 | 38–162 | 6,533–269 | 12.2–36.5 | 6.9–43.6 | |
KIR 11/1 | 10 | 1871–1936 | 49–114 | 5,146–755 | 11.1–26.5 | 8.6–27.8 | |
LOH 24/1 | 13 | 1871–1967 | 66–162 | 7,081–292 | 13.5–32.3 | 8.2–39.6 | |
MIS 25/1 | 15 | 1870–1981 | 42–153 | 1,1242–439 | 8.7–29.1 | 5.7–35.8 | |
ROT 26/1 | 14 | 1871–1967 | 48–144 | 5,458–425 | 13.2–34.0 | 8.2–37.0 | |
WAB 14/1 | 15 | 1870–1967 | 48–145 | 6,206–650 | 10.4–28.8 | 7.8–29.3 | |
Norway spruce Picea abies (L.) Karst. | DEN 05/1 | 18 | 1882–1990 | 35–143 | 3,528–508 | 14.4–40.6 | 13.5–47.3 |
EGL 72/1 | 13 | 1906–1990 | 36–120 | 6,256–712 | 10.5–32.5 | 8.8–35.9 | |
EGL 73/1 | 12 | 1906–1983 | 42–119 | 2,240–672 | 14.4–33.2 | 15.2–36.9 | |
OTT 08/1 | 14 | 1882–1963 | 32–113 | 4,232–632 | 14.0–38.8 | 12.8–40.6 | |
OB 697/2 | 17 | 1928–1999 | 42–113 | 3,623–548 | 14.5–36.8 | 12.2–40.9 | |
SAC 02/1 | 15 | 1882–1972 | 32–122 | 4,100–492 | 14.2–38.8 | 12.8–44.7 | |
SAC 03/1 | 14 | 1882–1965 | 33–116 | 7,428–596 | 10.6–38.2 | 8.6–42.0 | |
SAC 67/1 | 14 | 1902–1990 | 43–131 | 3,496–443 | 15.9–41.4 | 13.7–50.7 | |
SAC 68/1 | 14 | 1902–1990 | 42–130 | 2,952–544 | 16.5–40.4 | 15.2–45.3 | |
Scots pine/ Pinus sylvestris L. | SLU 50/1 | 13 | 1899–1991 | 26–118 | 4,900–550 | 8.9–29.3 | 8.3–32.1 |
SNA 57/1 | 13 | 1901–1995 | 44–138 | 5,104–456 | 9.1–23.8 | 7.8–28.3 | |
BOD 229/9 | 8 | 1961–1999 | 36–74 | 4,650–850 | 8.0–19.0 | 6.7–19.5 | |
WAS 234/1 | 9 | 1962–1999 | 86–122 | 1,117–358 | 14.0–23.6 | 15.3–28.2 | |
BUL 240/1 | 7 | 1965–1999 | 59–93 | 1,080–620 | 12.5–19.4 | 15.5–25.4 | |
HED 243/6 | 6 | 1971–1996 | 72–97 | 2,067–1,056 | 16.9–22.4 | 14.6–21.8 | |
Common oak/ Quercus petraea (Matt.) Leibl. | WAL 88/2 | 9 | 1934–1989 | 48–103 | 1,676–514 | 16.5–30.3 | 13.4–30.9 |
WAL 88/5 | 11 | 1934–1999 | 48–113 | 1,643–457 | 16.3–31.6 | 13.4–33.2 | |
ROH 90/1 | 8 | 1934–1996 | 70–132 | 1,205–487 | 17.4–27.4 | 15.4–32.5 | |
ROH 620/1 | 5 | 1980–2001 | 54–75 | 1,569–1,308 | 18.7–24.2 | 15.4–21.9 |
Estimates ± standard error for k′ and a of the model \({\text{ln }}w{\text{ }} = k' + a\,{\text{ln }}d\) or adequately \(w = k\,d^a, \) with k’=ln k [w= above ground shoot weight (kg), d= diameter in brest height (cm), k, k′, a= regression coefficients, n= number of harvested sample trees, d_{min} to d_{max} minimum to maximum diameter of sample trees (cm)]
Tree species | n | d_{min} to d_{max} (cm) | Estimate k′_{OLS} | SE k′_{OLS} | Estimate a_{OLS} | SE_{OLS} | r^{2} | P |
---|---|---|---|---|---|---|---|---|
c. beech | 48 | 6.0 to 61.8 | −2.169 | ±0.089 | 2.503 | ±0.031 | 0.99 | 0.000 |
N. spruce | 18 | 7.1 to 41.2 | −3.119 | ±0.254 | 2.659 | ±0.090 | 0.98 | 0.000 |
S. pine | 14 | 9.9 to 30.3 | −2.200 | ±0.323 | 2.297 | ±0.107 | 0.97 | 0.000 |
c. oak | 14 | 9.8 to 33.0 | −2.353 | ±0.302 | 2.633 | ±0.098 | 0.98 | 0.000 |
Methods
Regression algorithm
After ln–ln-transformation, most of the datasets fulfilled the assumptions of linear regression analysis such as normal distributions of residuals, homoscedasticity and independence of residuals. Except RMA regression analysis, for all subsequent calculations, I used SPSS (Version 11.5). Those 10 out of 28 plots with nonlinear self-thinning lines were omitted from the subsequent analysis (cf. Sect. “Results”).
Scaling exponents and scaling coefficients were computed by both, Model I (ordinary least square regression, OLS) and Model II (reduced major axis regression, RMA). Zeide (1987) and Niklas (1994) argue that RMA regression (Model 2) represents the “true relationship” between the variables, as RMA slopes of x on y are exactly the inverse of those of y on x. I applied a program for reduced major axis regression from Bohonak (2002), who suggests that RMA should be used, if error variance of x exceeds one-third of the error variance in y. I also applied ordinary least square regression, to keep comparable with the original work of Yoda et al. (1963) and other studies, based on OLS-slopes. Sackville Hamilton et al. (1995) showed that slope estimates of all different algorithm converge with increasing r^{2}. As the fitting of Eqs. 9, 10, and 12 yielded mostly r^{2}>0.9, the differences between OLS and RMA slopes (2.3% in average) had no decisive effect on the final results of this study.
Above-ground shoot biomass’ estimation
Mathematical representation of the self-thinning line
Weller (1987) and Niklas (1994) argue that most empirical studies about self-thinning rule are methodologically flawed, as they base on ln \(\bar w\) versus ln N instead of ln W versus ln N. They argue that the relation ln \(\bar w\) versus ln N is equivalent to plotting total plant biomass/total number of plants ( \(\bar w\)) versus total number of plants/total area (N), so that total number of plants is shared by x and y. As x and y are not independently measured and total number of plants is mostly estimated by sampling, slopes obtained from the first formulation of the rule may be spurious. However, these arguments apply to studies about herbaceous plants rather than to forests. In forests, biased plant numbers play a minor role as biomass is not measured by harvesting all plants, but each individual tree is measured and counted (Prairie and Bird 1989). Thus stem number is the most precisely recorded variable; each tree is permanently marked and any flaws at a survey are corrected at the next survey at the latest. In order to anticipate criticism on account of the used model and to keep comparable with studies applying ln \(\bar w\) versus ln N (e.g. White 1981; Yoda et al. 1963) and ln W versus ln N (e.g. Weller 1987; Zeide 1987), both relations were analyzed.
Regression with centered data
To obtain individual species’ slopes for the relations \({\text{ln}}\,N\;{\text{versus}}\;{\text{ln}}\,\bar d,\)\({\text{ln}}\,\bar w\;{\text{versus}}\;{\text{ln}}\,N,\) and \({\text{ln}}\,W\;{\text{versus}}\;{\text{ln}}\,N\) all plots per species were integrated in an overall RMA and OLS analysis. Prior to RMA and OLS analysis, the mean values \(\overline {{\text{ln}}\,\bar w} \) and \(\overline {{\text{ln}}\,N} \) were calculated for each stand and used for standardisation \({\text{ln}}\,\bar w_i^\prime {\text{ = ln}}\,\bar w_i - \overline {{\text{ln}}\,\bar w} \) and \({\text{ln}}\,N_i^\prime {\text{ = ln}}\,N_i - \overline {{\text{ln}}\,N}. \) The standardisation has the effect of focussing all straight lines on the mean value ( \(\overline {{\text{ln}}\,\bar w} ,{{\text{ln}}\,N} \)) and eliminating of plot specific intercepts. The regression of \({\text{ln}}\,N\;{\text{versus}}\;{\text{ln}}\,\bar d,\)\({\text{ln}}\,\bar w^\prime \) versus \({\text{ln}}\,N^\prime, \) and ln W versus ln N yielded individual species’ scaling exponents. The slopes resulting from centered data are labeled by apostrophe (e.g. c′).
Results
Relationship w ∝ d^{a} between diameter and biomass
Table 2 displays the species-specific parameters k′ and a of the model ln w = k′ + a ln d [cf. Eq. 12]. Parameter a serves in this study for both, scrutiny whether slope a tends to follow a general scaling law and upscaling from tree diameter d to stand biomass W.
Parameter a is equivalent to scaling exponent a in w ∝ d^{a} (generalized form of Eq. 8) and represents the allometric relationship between diameter and biomass. The observed values for common beech, Norway spruce, Scots pine, and common oak differ considerably from a=3.0, assumed by Yoda et al. (1963), but vary around a=8/3, postulated by West et al. (1997) and Enquist et al. (1998).
Biomass of adult trees or even whole stands can hardly be measured completely. For the subsequent evaluation, individual tree biomass w was estimated in dependence on individual stem diameter d by model ln w = k′ + a ln d [Eq. 12]. Estimates of all individual trees’ biomass w enables calculation of \(\bar w\) and W. Slopes c and d, needed for scrutiny of H2, were derived by regression analysis on the basis of all value pairs (N, \(\bar w\)) and (N, W), respectively.
H1: scrutiny of Reineke’s stand density rule
Reineke’s relationship \(N \propto \bar d^r \) is another representation of \(\bar s \propto \bar d^b, \) as \(\bar s{\text{ = 1/}}N.\) Thus, scrutiny of slope r of \({\text{ln}}\,N{\text{ versus ln}}\,\bar d\) also exposes slope b of \({\text{ln}}\,\bar s{\text{ versus ln}}\,\bar d,\) as both slopes just differ in the sign (r=−b).
H1.1: each of the 28 \({\text{ln}}\,N{\text{ versus ln}}\,\bar d\)-trajectories was analyzed by OLS-regression with regard to quadratic effects by model \(\ln N = o_{\text{1}} + o_{\text{2}} \ln \bar d + o_{\text{3}} \,\ln ^{\text{2}} \bar d\). Negative o_{3}-values indicate a concave curve, as seen from below, while positive o_{3}-values produce a convex curve. Overall, significant quadratic terms (P<0.05) were obtained for 29% of the plots (8 out of 28 plots). Significantly negative quadratic terms were obtained in six cases, significantly positive ones in only two cases. For common beech there were three concave curves (o_{3} = −0.02 to −0.001, P<0.05) and one convex curve (o_{3} = +0.004, P<0.05). Norway spruce had two concave (o_{3}=−0.002 to −0.001, P<0.05) and one convex curve (o_{3}= +0.011, P<0.05). Scots pine had one concave curve (o_{3}= −0.001, P<0.001). Common oak was consistently linear (o_{3} ≅ 0).
Plotwise scaling exponents r_{RMA} and c_{RMA} for the relations \({\text{ln}}\,N\;{\text{versus}}\;{\text{ln}}\,\bar d\) and \({\text{ln}}\,\bar w\;{\text{versus}}\;{\text{ln}}\,N\), respectively)
Experiment/plot | n | ln N versus ln \(\bar d\) | In \(\bar w\) versus ln N | ||||
---|---|---|---|---|---|---|---|
r_{RMA}(SE) | 95% CI | r^{2} | c_{RMA}(SE) | 95% CI | r^{2} | ||
Common beech | |||||||
ELM 20/1 | 13 | −1.747(0.022) | −1.795 to −1.699 | 0.998 | −1.475(0.019) | −1.517 to −1.433 | 0.998 |
FAB 15/1 | 12 | −1735(0.022) | −1.785 to −1.686 | 0.998 | −1.485(0.020) | −1.529 to −1.441 | 0.998 |
HAI 27/1 | 16 | −1.747(0.019) | −1.787 to −1.706 | 0.998 | −1.472(0.015) | −1.505 to −1.440 | 0.998 |
MIS 26/1 | 15 | −1.873(0.048) | −1.976 to −1.770 | 0.992 | −1.374(0.034) | −1.448 to −1.300 | 0.992 |
ROT 26/1 | 14 | −1.723(0.025) | −1.778 to −1.668 | 0.997 | −1.490(0.022) | −1.538 to −1.442 | 0.997 |
Norway spruce | |||||||
EGL 72/1 | 13 | −1.669(0.046) | −1.769 to −1.568 | 0.992 | −1.595(0.044) | −1.691 to −1.498 | 0.992 |
EGL 73/1 | 12 | −1.607(0.081) | −1.787 to −1.427 | 0.975 | −1.648(0.083) | −1.833 to −1.463 | 0.975 |
PB 697/2 | 17 | −1.652(0.024) | −1.703 to −1.601 | 0.997 | −1.614(0.022) | −1.661 to −1.566 | 0.997 |
SAC 03/1 | 14 | −1.664(0.026) | −1.721 to −1.606 | 0.997 | −1.594(0.025) | −1.649 to −1.539 | 0.997 |
SAC 67/1 | 14 | −1.633(0.035) | −1.709 to −1.556 | 0.994 | −1.633(0.036) | −1.712 to −1.554 | 0.994 |
SAC 68/1 | 14 | −1.641(0.047) | −1.743 to −1.538 | 0.990 | −1.624(0.048) | −1.728 to −1.519 | 0.990 |
Scots pine | |||||||
SLU 50/1 | 13 | −1.561(0.037) | −1.642 to −1.480 | 0.994 | −1.467(0.034) | −1.542 to −1.391 | 0.994 |
SNA 57/1 | 13 | −1.679(0.063) | −1.817 to −1.541 | 0.985 | −1.369(0.052) | −1.482 to −1.255 | 0.984 |
WAS 234/1 | 8 | −1.764(0.100) | −2.020 to −1.508 | 0.984 | −1.667(0.094) | −2.361 to −0.972 | 0.910 |
BUL 240/1 | 6 | −1.651(0.104) | −1.906 to −1.396 | 0.976 | −1.392(0.250) | −1.623 to −1.161 | 0.972 |
HED 243/6 | 6 | −1.620(0.163) | −2.072 to −1.168 | 0.960 | – | – | – |
Common oak | |||||||
WAL 88/2 | 10 | −1.628(0.130) | −1.928 to −1.328 | 0.949 | −1.617(0.129) | −1.914 to −1.320 | 0.949 |
WAL 88/5 | 10 | −1.417(0.072) | −1.582 to −1.252 | 0.980 | −1.847(0.093) | −2.062 to −1.633 | 0.980 |
ROH 90/1 | 8 | −1.222(0.116) | −1.506 to −0.939 | 0.946 | −2.139(0.193) | −2.612 to −1.667 | 0.951 |
ROH 620/1 | 4 | −1.327(0.143) | −1.941 to −0.714 | 0.977 | – | – | – |
H1.3: Comparison of r with Reineke’s −1.605. In 60% of the cases (12 of 20 plots) −1.605 lies within the 95% confidence intervall of r_{RMA} (cf. Table 3). Bold letters denote that Norway spruce and Scots pine in majority correspond with Reineke’s slope, whereas the slope of common beech is steeper and the one of common oak is shallower than −1.605. Same evaluation on basis of r_{OLS} yielded 50% of the plots with −1.605 included in the 95% CI. The regression over all centered data \({\text{ln}}\,N^\prime {\text{ versus ln}}\,\bar d^\prime \) yielded slopes r_{RMA}′ between −1.778 and −1.457 (common beech<Norway spruce<Scots pine<common oak) and r_{OLS}′ between −1.773 and −1.423 with the same ranking of the species. Only the 95% confidence intervals of Scots pine’s slopes r_{RMA}′ and r_{OLS}′ include −1.605. Slopes r′ of common beech and Norway spruce are significantly steeper than −1.605; r′ of common oak is shallower than −1.605 (cf. S5).
H2: slopes of the dynamic self-thinning lines \(\bar w \propto N^c \) and \(W \propto N^d \), respectively
Weller (1987, 1990), Zeide (1987) and Niklas (1994) prefered the relation \({\text{ln }}W{\text{ versus ln }}N\) for slope estimation, while Yoda et al. (1963) used \({\text{ln }}\bar w{\text{ versus ln }}N\). To avoid unjustified rejection of my results on account of methodological disagreement, I applied RMA and OLS regression to both relationships. However, as \(d \cong c + {\text{1}}\) the evaluation for c and d yielded always analogous results and I concentrated my report on c.
H2.1: The OLS-regression of the quadratic model \({\text{ln}}\,\bar w = p_{\text{1}} + p_{\text{2}} \,\ln N + p_{\text{3}} \,{\text{ln}}^{\text{2}} N\) resulted in significantly (P<0.05) negative p_{3}-coefficients in three out of nine common beech plots (p_{3}=−0.187 to −0.075), two out of nine Norway spruce plots (p_{3}=−0.148 and −0.125), two out of six Scots pine plots (p_{3}=−1.323 and −0.473) and one out of four common oak plots (p_{3}=−0.135). Thus, in 29% of the cases, the slope is concave from below and becomes shallower within stand development. In one out of nine common beech plots (p_{3}=+0.456) and one out of nine Norway spruce plots (p_{3}=+0.119), i.e. in 7% of all cases a significant (P<0.05) convex curve was detected. Comparison between the straight self-thinning lines (Fig. 2, solid lines) and those detected as nonlinear (broken lines) indicates mainly a slight curvature. Altogether in 10 out of 28 cases, the relation \({\text{ln }}\bar w{\text{ versus ln }}N\) deviated significantly (P<0.05) from linearity, i.e. on 36%. The analysis on basis of the relation \({\text{ln }}W{\text{ versus ln }}N\) yielded the same percentages of nonlinear, concave and convex self-thinning lines.
H2.2: For each of the 18 plots with a straight self-thinning line, we estimated slopes c and d, for the relations \({\text{ln }}\bar w{\text{ versus ln }}N\) and \({\text{ln }}W{\text{ versus ln }}N\) by both, RMA and OLS regression. The regression \({\text{ln }}\bar w{\text{ versus ln }}N\) yielded r^{2} -values from 0.91 to 0.99, which were highly significant (P<0.001) in all cases. On average (min to max), the RMA-slopes were c_{RMA} =−1.459 (−1.490 to −1.374) for common beech, c_{RMA} =−1.618 (−1.648 to −1.594) for Norway spruce, c_{RMA} =−1.474 (−1.667 to −1.369) for Scots pine, and c_{RMA} =−1.868 (−2.139 to −1.617) for common oak (Table 3). The OLS slopes are in average (min to max) c_{OLS} =−1.457 (−1.489 to −1.366) for common beech, c_{OLS} =−1.610 (−1.630 to −1.586) for Norway spruce, c_{OLS} =−1.449 (−1.600 to −1.358) for Scots pine, and c_{OLS} =−1.830 (−2.087 to −1.575) for common oak.
ANOVA for detection of individual species’ slope c: any interspecific differences of the scaling exponent c were analyzed by ANOVA. Variance analysis included all 18 plots with linear self-thinning lines and was carried out for slopes, estimated by RMA and OLS. Levene’s statistic proved homogeneity of variances for the four species (P<0.05). The hypothesis that the slope c_{RMA} of the relation \({\text{ln }}\bar w{\text{ versus ln }}N\) is equal for all four considered species can be rejected (P<0.01). The mean slopes (± standard error) were c_{RMA}=−1.459 (±0.022), c_{RMA} =−1.618 (±0.009), c_{RMA} =−1.474 (±0.068) and c_{RMA} =−1.868 (±0.151) for common beech, Norway spruce, Scots pine and common oak, respectively. Multiple comparisons of group means by Scheffé’s procedure revealed significant differences between c_{RMA}-values of common beech and common oak (P<0.01) as well as between Scots pine and common oak (P<0.01). Variance analysis on basis of OLS-slopes underlines the differences between the species: Global hypothesis of equality was rejected (P<0.001), group means differed significantly between common beech and common oak (P<0.001), Norway spruce and Scots pine (P<0.05), Scots pine and common oak (P<0.001).
In passing, I emphasize that also the intercept of the self-thinning lines differ considerably between the species (cf. Fig. 2). This fact was recently revealed by Stoll et al. (2002) and will be analyzed in a subsequent report.
H2.3: Table 3 presents slopes, standard errors and 95% confidence intervals for each plot. Those slopes, corresponding with Yoda’s law are printed in bold letters. Yoda’s constant of −3/2 is in 10 out of 18 cases (56%) within the 95% confidence interval of the RMA-slope c_{RMA}. Fractal scaling constant −4/3 is in merely 5 out of 18 cases (28%) within the respective CI. In the majority slopes of common beech (four out of five) and Scots pine (three out of four) approximated −3/2; whereas, Norway spruce (two out of six) and common oak (one out of three) differed significantly. The same evaluation on the basis of OLS-slopes yielded similar results. The relation \({\text{ln}}\,{\ifmmode\expandafter\bar\else\expandafter\=\fi{w}}\ifmmode{'}\else$'$\fi\,{\text{versus}}\,{\text{ln}}\,{N}\ifmmode{'}\else$'$\fi\) was fitted by both, OLS- and RMA-regression analysis. The slopes c_{OLS}′ and c_{RMA}′ (cf. Fig. 4 and S5) all differ significantly from −3/2 as well as from −4/3.
H3: species’ oscillation around the self-thinning line
Mean, standard deviation and coefficient of variation v \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} \) of quotient \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} = \ln \,\left( {{{\bar w_{i + 1} } \mathord{\left/ {\vphantom {{\bar w_{i + 1} } {\bar w_i }}} \right. \kern-\nulldelimiterspace} {\bar w_i }}} \right)/\ln \,\left( {{{N_{i + 1} } \mathord{\left/ {\vphantom {{N_{i + 1} } {N_i }}} \right. \kern-\nulldelimiterspace} {N_i }}} \right)\)
Species | Plots | Survey periods | Mean survey period (years) | \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} \;{\text{Mean}}\) | \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} \;{\text{SD}}\) | \(\text{v}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} \,(\% )\) |
---|---|---|---|---|---|---|
c. beech | 5 | 69 | 4.5 | −1.62 | 0.49 | 30.2 |
N. spruce | 6 | 74 | 6.7 | −2.02 | 0.77 | 37.2 |
S. pine | 4 | 36 | 6.5 | −2.02 | 0.77 | 37.2 |
c. oak | 3 | 24 | 8.8 | −2.67 | 1.90 | 69.3 |
H3.2: Pearson’s correlation between slope c_{RMA} and \({\text{v}}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} \) resulted in \(r_{c_{RMA} {\text{,}}\,{\text{v}}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} } \) = −0.614 (P<0.01). With other words, the steeper the slope of \({\text{ln }}\bar w\,{\text{versus}}\,{\text{ln }}N\), the higher the variation around the self-thinning line. In common beech stands, e.g., self-thinning is more rigorous (c_{RMA}′ = −1.409) but more consistent ( \({\text{v}}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} \) = 30.2%) than in oak stands, where self-thinning is slower (c_{RMA}′=−1.794) but tree losses due to self-thinning come up in batches \(\left( {{\text{v}}\hat c = 69.3\% } \right).\)
Discussion
The partially nonlinear curvature of the relation between ln (stem number) and ln (plant dimension) was mainly caused by storm damage and ice breakage, which opened up crown space and lowered stand density. I applied a rather conservative but objective criterion for telling linear from nonlinear relationships. Nevertheless, mean plant biomass \(\bar w\) and biomass per unit area W have to be interpreted with due care, as biomass of forest stands can hardly be measured completely, but is estimated by scaling functions again (e.g., \(w \propto d^a \)). To avoid artefact due to two-stage biomass sampling, slopes on the basis of the primary variables \(\bar d\) and N were analysed as well. The plus of the used database lies in the length of time series. It displays the “dynamic self-thinning line” for a restricted number of sites. But it is not sufficiently scattered over the whole site spectrum of the considered species to yield a “species boundary line”, which would represent the upper boundary of yield–density relation for a species (Weller 1987, 1990).
Generalization
The exponent a of \(w \propto d^a \) expresses biomass allocation of a tree with a given diameter, while exponent b of \(\bar s \propto \bar d^b \) expresses the lateral crown expansion. The scaling exponents for common beech account for its high efficiency of space occupation. Compared with Norway spruce and common oak, common beech invests rather less in biomass, but the invested biomass is used more efficiently to occupy additional space. A given diameter growth is coupled with a relatively low biomass growth (Fig. 4a), however, with an increase of growing space \(\bar s\), topped by none of the other considered species (Fig. 4b). The opposite applies to common oak. Despite a comparably high investment in biomass, oak achieves a rather low lateral expansion. Another pattern of shape and biomass allocation shows Norway spruce and Scots pine, where a and b are counteracting. The results confirm Weller (1987, 1990) and Zeide (1987) in their view, that the individual species’ scaling exponents are a key for understanding the species’ ability to cope with crowding and should not be cast away, although generalization across species is tempting.
Scaling exponents for woody plants might be biased because of the progressive accumulation of dead inner xylem, which impairs the relation between average biomass and plant number. In contrast to herbaceous plants, for which the −3/2 law was initially developed, dead tissue in the stem’s core is negligible in the juvenile phase but amounts to 15–20% for common beech, 50% for Norway spruce, 35–40% for Scots pine and 65–70% for common oak in age 100 (Trendelenburg and Mayer-Wegelin, 1955). As the steepness of the slopes, revealed in this study, rank in the same way as the percentage of dead xylem wood (common oak>Norway spruce>Scots pine>common beech), it seems that the percentage of dead wood is behind the species-specific slopes, or at least influences them.
Since slopes c and d of common beech and Scots pine are even flatter than −3/2 and −1/2, respectively, although they should be steeper if we consider the dead core wood accumulation, Yoda’s law appears questionable. Enquist et al. (1998) state without reasons that their −4/3 self-thinning law applies across populations of herbaceous and woody plants of very different size but that it does not explain self-thinning within populations. Fractal scaling slopes −4/3 and −1/3, expected by West et al. (1997, 1999), Enquist et al. (1999, 2001) and Niklas (1994), are flatter than all observed slopes. In view of my results, fractal scaling slopes −4/3 and −1/3 appear in a new light: they might apply to ln \({ws}\) versus ln N and ln WS versus ln N, where \({{\text{ws}}}\) and WS describes sapwood biomass. In order to judge, if this hypothesis is reasonable, I estimated whole stem biomass using c_{RMA}-slopes and compared them with stem biomass estimated via slope −4/3. The difference [(w−ws)/w 100] of biomass in advanced stand age (300 trees per ha) amount to 20% for common beech, 56% for Norway spruce, 23% for Scots pine and 75% for common oak. These portions of dead xylem correspond to remarkable extend with empirical findings and justify the assumption, that −4/3 is not at all generalizable for ln \(\bar w\) versus ln N, but applies better to ln \(\overline {ws}\) versus ln N.
Ecological implication
Allometry under self-thinning reveals the species-specific critical demand on resources of trees of given size. If the number N of trees per area approximates maximum stand density, average growing space \(\bar s\) falls below a critical limit and induces the mortality process especially of trees with growing space s< \(\bar s\). By rearrangement \(\bar w \propto N^c \), \(\bar w \propto \bar s^{ - c} \), \(\bar s \propto \bar w^{ - {{\text{1}} \mathord{\left/ {\vphantom {{\text{1}} c}} \right. \kern-\nulldelimiterspace} c}} \) and differentiation we get \(g = {{({\text{d}}\bar s/\bar s)} \mathord{\left/ {\vphantom {{({\text{d}}\bar s/\bar s)} {({\text{d}}\bar w/\bar w)}}} \right. \kern-\nulldelimiterspace} {({\text{d}}\bar w/\bar w)}}\), where g is reciprocal of Yoda’s exponent taken with the opposite sign (g = −c^{−1}). Rate g reflects the relative gain of growing space \({\text{d}}\bar s{\text{/}}\bar s\) by a given biomass investment \({\text{d}}\bar w{\text{/}}\bar w\). Yoda’s slope c = −3/2 would yield g=0.667. In other words, under self-thinning conditions, regardless of species and site, 1% of biomass investment would always effect 0.667% of space occupation. My evaluation yielded individual species’ c_{RMA} ′-values of −1.409, −1.611, −1.421, and −1.794, so that g = 0.7097, 0.6207, 0.7037, and 0.5574 for common beech, Norway spruce, Scots pine, and common oak, respectively. Thus, common beech and Scots pine prove to be more efficient in space occupation than predicted by Yoda’s constant, Norway spruce and common oak are less efficient. If we define the efficiency of space occupation as the fraction of sequestered space \({\text{d}}\bar s{\text{/}}\bar s\) per fraction of biomass investment \({\text{d}}\bar w{\text{/}}\bar w\) and set common beech as reference, the species’ ranking of efficiency will be: common beech (100%) Scots pine (99%)>Norway spruce (88%)>common oak (79%). If we follow Zeide (1985) who revealed c = \({{{\text{(d}}\bar w{\text{/}}\bar w{\text{)}}} \mathord{\left/ {\vphantom {{{\text{(d}}\bar w{\text{/}}\bar w{\text{)}}} {{\text{(d}}N{\text{/}}N}}} \right. \kern-\nulldelimiterspace} {{\text{(d}}N{\text{/}}N}})\) as a measure for a species’ self-tolerance, we get a ranking of self-tolerance: common beech (100%)<Scots pine (101%)<Norway spruce (114%)<common oak (127%). Thus, a high efficiency of space sequestration is coupled with a low self-tolerance and a rigorous self-thinning process, and vice versa.
Conclusions
In view of the individual species’ slopes, stand density estimation algorithms, founded on generalized allometric relations, appear unsuitable. Questionable is, e.g. Reineke’s stand density index (Reineke 1933), founded on species invariante slope r=−1.605. It is frequently used to quantify stand density (Sterba 1981, 1987; Kramer and Helms 1985). Stand density management diagrams (SDMD), which are applied for many species as a tool for regulating stand density, use the self-thinning line with generalized scaling exponents as upper boundary and are the most prominent silvicultural application of the self-thinning rule (Oliver and Larson 1990). Bégin et al. (2001) list for a considerable number of tree species available SDMDs as guides for stand management. As long as those SDMDs ignore individual species allometry, flawed density control and contraoptimal thinning will result. Equivalent shortcomings apply for prognoses by growth models, which ignore individual species’ scaling exponents. Models, which base thinning and mortality algorithms on generalized scaling exponents (Eid and Tuhus 2001; Xue and Hagihara 2002; Yang and Titus 2002) should be replaced by more flexible approaches (Pittman and Turnblom 2003; Roderick and Barnes 2004; Zeide 2001).
Allometry and peculiarities of space sequestration are a benchmark for a species competitiveness in pure and mixed stands (Bazzaz and Grace 1997). In order to get a better understanding of competitive mechanisms in forest stands, further research should clarify individual species scaling rules rather than to continue search for “the ultimative law”, that appears like hunting for a phantom. In comparison with ecophysiological and biochemical processes, which are not thoroughly understood, size and structure of plants are much easier to measure. Since there is a close feedback between structure and process, organisms’ size and structure can become the key for revelation and prognosis of stand dynamic. Allometric slopes can serve as an interface between process and structure. If the numerous falsification trials concerning the rules from Reineke, Yoda and West, Brown and Enquist lead to a refined understanding of individual species allometry, allpervasive scaling exponents would appear as a stimulating myth.
This and all subsequent sections repeatedly refer to electronic supplementary material available on Springer Verlag’s server. References to this material are numbered as S1, S2 etc. See URL on title page.
Acknowledgements
The author wishes to thank the Deutsche Forschungsgemeinschaft for providing funds for forest growth and yield research as part of the Sonderforschungsbereich 607 “Growth and Parasite Defense” and the Bavarian State Ministry for Agriculture and Forestry for permanent support of the Forest Yield Science Project W 07. Prof. Dr. Hermann Spellmann of the Lower Saxony Forest Research Station in Göttingen complemented the Bavarian dataset with two experimental plots from the former Prussian Forest Research Station. Thanks are also due to Prof. Dr. Boris Zeide for helpful discussion, Hans Herling for preparation of graphs and anonymous reviewers, for constructive criticism.