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Testing predictions of the energetic equivalence rule in forest communities
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Testing predictions of the energetic equivalence rule in forest communities Wei-Ping Zhanga,∗ , E. Charles Morrisb , Xin Jiac , Sha Pand , Gen-Xuan Wange a Beijing Key Laboratory of Biodiversity and Organic Farming, College of Resources and Environmental Sciences, China Agricultural University, Beijing 100193, China b School of Science and Health, University of Western Sydney, Richmond 2753, NSW, Australia c Key Laboratory of Soil and Water Conservation and Desertification Combating, Ministry of Education, Beijing Forestry University, Beijing 100083, China d Department of Environmental Hygiene, School of Public Health, Guiyang Medical University, Guiyang 550004, China e Institute of Ecology, College of Life Sciences, Zhejiang University, Hangzhou 310058, China
Received 13 May 2014; received in revised form 24 March 2015; accepted 6 April 2015
Abstract As growth rate is a reasonable proxy measure of the rate of resource use per plant individual, the ‘energetic equivalence rule’ predicts that net primary productivity (the rate of biomass production per unit area, NPP) will be independent of plant biomass and maximum population density in plant communities. However, only a few studies have tested these relationships in plant communities. In this study, we investigated allometric scaling of net primary productivity (NPP) to tree biomass (M) and density (N) across a range of tree-dominated communities in China. The aim was to test the universality of the ‘energetic equivalence rule’ (i.e. whether the exponents of these relationships take a universal value of 0) in forest communities. We used both ordinary least square (OLS) and standardized major axis (SMA) regression for selected boundary points, and quantile regression (QR) to estimate the slopes of regression lines. QR, OLS and SMA regression all showed that four NPP–M and two NPP–N exponents were different from 0 across the 8 forest types. In addition, when we combined all the data to determine a larger pattern that typifies Chinese forests, five out of the six exponents of NPP–M and NPP–N relationships deviated strongly from 0. Therefore the universality of the ‘energetic equivalence rule’ does not hold for forest communities at both the regional and the national scale of China. However, the “zero” exponent seems to be a central tendency for NPP–M and NPP–N relationships in 7 out of 8 forest types. Deviation from the energetic equivalence possibly reflects multiple, unsound assumptions for “an average idealized forest” by metabolic scaling theory, as well as unaccounted-for variations of site factors (e.g. stand age and stand conditions) within forest communities. In addition, our study suggested that statistical methods should be subject to strict scrutiny in testing the ‘energetic equivalence rule’.
Zusammenfassung Da die Wachstumsrate näherungsweise ein gutes Maß für die Rate der Ressourcennutzung pro Pflanze ist, sagt die Regel von der energetischen Äquivalenz voraus, dass die Nettoprimärproduktion (die Rate der Biomasseproduktion je Flächeneinheit: NPP) von der Pflanzenbiomasse und der maximalen Populationsdichte von Pflanzengemeinschaften unabhängig sein ∗ Corresponding
author. Tel.: +86 10 62734684; fax: +86 10 62731016. E-mail address: [email protected] (W.-P. Zhang).
http://dx.doi.org/10.1016/j.baae.2015.04.005 1439-1791/© 2015 Gesellschaft für Ökologie. Published by Elsevier GmbH. All rights reserved.
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wird. Indessen haben nur wenige Studien diese Beziehungen in Pflanzengemeinschaften untersucht. Wir untersuchten die allometrische Beziehung zwischen der Nettoprimärproduktion (NPP) und der Baum-Biomasse (M) und -Dichte (N) über eine Reihe von baumdominierten Gemeinschaften in China. Unser Ziel war, die Allgemeingültigkeit der Regel von der energetischen Äquivalenz zu überprüfen, d.h., ob die Exponenten dieser Beziehungen einen universellen Wert von Null annehmen. Wir benutzten die Methode der kleinsten Quadrate (OLS), standardisierte Hauptachsen-Regression (SMA) für ausgewählte Grenzpunkte sowie Quantil-Regression (QR), um die Steigungen von Regressionsgeraden zu bestimmen. QR, OLS und SMA zeigten, dass über die acht untersuchten Waldtypen vier NPP-M- und zwei NPP-N-Exponenten von Null verschieden waren. Als wir alle Daten kombinierten, um das größere, für chinesische Wälder charakteristische Muster zu bestimmen, unterschieden sich fünf der sechs NPP-M- und NPP-N-Beziehungen stark von Null. Damit ist die Regel von der energetischen Äquivalenz für Waldgemeinschaften in China weder auf der regionalen noch auf der nationalen Ebene allgemein gültig. Indessen scheint der Null-Exponent eine zentrale Tendenz für die NPP-M und NPP-N-Beziehungen in sieben von acht Waldtypen zu sein. Die Abweichung von der energetischen Äquivalenz spiegelt möglicherweise mehrere unbegründete Annahmen der Theorie der metabolischen Skalierung hinsichtlich eines durchschnittlichen, idealisierten Waldes wider, sowie die unberücksichtigt gebliebenen Schwankungen von Standortfaktoren in Waldgemeinschaften (z.B.: Bestandsalter, Bestandsbedingungen). Darüberhinaus legt unsere Untersuchung nahe, dass bei der Prüfung der Regel von der energetischen Äquivalenz die statistischen Methoden einer äußerst eingehenden Überprüfung unterzogen werden sollten. © 2015 Gesellschaft für Ökologie. Published by Elsevier GmbH. All rights reserved.
Keywords: Scaling; Net primary productivity; Biomass; Density; Forest communities; Self-thinning; Upper boundary; Quantile regression; Standardized major axis regression; Metabolic scaling theory
Introduction Metabolic scaling theory (MST) argued that metabolic rate (or resource use rate, Q) per individual would scale approximately as the 3/4 power of body mass (M): (i.e. Q ∝ M3/4 ) (West, Brown, & Enquist 1997; Niklas & Enquist 2001). In addition, maximum population density (Nmax ) should scale with body mass M−3/4 (i.e. Nmax ∝ M−3/4 ) (Damuth 1981, 1987; Enquist, Brown, & West 1998; Niklas & Enquist 2001; Niklas, Midgley, & Enquist 2003). Since the maximum rate of resource use per unit area Rmax = Q × Nmax ∝ M 3/4 × 0 , the maximum rate of resource use per M −3/4 ∝ M 0 ∝ Nmax unit area will be independent of average plant biomass and density. The relationship is termed as the ‘energetic equivalence rule’ (Damuth 1981, 1987), and is considered as one of the fundamental laws of nature (Deng et al. 2008). The ‘energetic equivalence rule’ has been tested using different kinds of organisms and across a range of biological scales, but with mixed results (Enquist et al. 1998; White, Ernest, Kerkhoff, & Enquist 2007; King 2010). In addition, both the 3/4 scaling of metabolic theory (Q–M) and the −4/3 scaling of biomass–density relationship (M–Nmax ) are challenged on theoretical and empirical grounds (Russo, Wiser, & Coomes 2007; Dai et al. 2009; Mori et al. 2010; Bai et al. 2010, 2011). Therefore, controversy continues over the validity and universality of the ‘energetic equivalence rule’ (White et al. 2007; Deng et al. 2008; King 2010). It is mathematically difficult to examine the applicability of the ‘energetic equivalence rule’ directly in plant communities (Enquist et al. 1998; Deng et al. 2008). As growth rate is a reasonable proxy measure of the rate of resource use per plant individual (Q) (Niklas & Enquist 2001), net primary productivity (the rate of biomass production per unit area,
NPP) is predicted to scale proportionally to total energy use per unit area in plant communities. It follows from the ‘ener0 . getic equivalence rule’ that NPPmax ∝ Rmax ∝ M 0 ∝ Nmax More importantly, these relationships provide an alternative indicator for ‘energetic equivalence’. However, only a few studies have tested these relationships in plant communities (Niklas & Enquist 2001; Niklas et al. 2003). Recent literature indicates that both scaling of metabolic theory and biomass–density relationship are not a constant, but rather vary with environmental conditions or particular taxonomic groups across a range of forest communities in China (Li, Han, & Wu 2005; Li, Han, & Wu 2006; Zhang, Jia, Bai, & Wang 2011; Zhang, Jia, Morris, Bai, & Wang 2012). Based on the same dataset, we further investigated allometric scaling of net primary productivity (NPP) to tree biomass (M) and maximum density (Nmax ) across a range of tree-dominated communities in China. The aim of was to test the universality of the ‘energetic equivalence rule’ (i.e. whether the exponents of these relationships take a universal value of 0) in forest communities.
Materials and methods Data source We examined the allometric scaling of net primary productivity (NPP) to tree biomass (M) and density (N) by using the compilation of Luo (1996) for standing biomass and production. These data were compiled from the Chinese literature for continuous forest-inventory plots of the forestry departments between 1970 and 1994. The dataset includes
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biomass, production and associated climatic variables for 1266 selected forest plots over China (Luo 1996). In most plots, all biomass components of trees (i.e. stem, branch, leaf and root) were estimated by species-specific allometric equations based on tree height (H, m) and diameter at breast height (D, cm) (Luo, Li, & Zhu 2002). In addition, the tree biomass in a few plots was estimated by the “clearcutting method” (all trees in a plot were cut down) or “mean tree method”. For “mean tree method”: several “mean trees” in a plot were felled and standing biomass was then estimated by the product of these mean tree biomass and stand density (Luo 1996; Ni, Zhang, & Scurlock 2001). This method was mainly used for plantations and even-aged stands, in which the variations of tree size (height, diameter and biomass, etc,) were small. Therefore, the “mean trees” were usually accessed according to mean height, mean diameter and mean crown width empirically (Zhang et al. 2011). Net primary productivity (NPP, kg/ha/yr) for all tree components (i.e. stem, branch, leaf and root) were estimated from the sum of annual growth rate per tree (kg/yr). Due to the careful estimation of biomass and net primary productivity, the data of Luo (1996) has been used for analysis in many studies (e.g. Ni et al. 2001; Luo et al. 2002; Li et al. 2005, 2006; Zhang et al. 2011). More detailed information about the dataset can be found in Luo (1996), Luo et al. (2002) and Zhang et al. (2011). Luo (1996) stratified the 1266 plots into 17 major forest types. Among the 17 forest types, seven have only 9 to 22 plots, the sample size is thus too small to analyze these scaling relationships. In addition, self-thinning is a phenomenon found in dense monocultures, indicated by increasing stand biomass with decreasing stand density (Yoda, Kira, Ogawa, & Hozumi 1963; Zhang et al. 2011, 2012). There is a basic agreement that stands with different individuals (but with same species or life-form, and with similar conditions) and densities should converge to a single self-thinning line. If the stand conditions and species composition are not equivalent, there are no guarantees that all the stands will be on a single self-thinning line, even if they are fully stocked. Considering the possibility of different conditions of the sample stands, it is suggested to stratify the stands, or classify stands into groups depending on their conditions, and obtain a boundary line in each group of the stands (Zhang et al. 2011, 2012). Therefore, although another two forest types, i.e. temperate typical deciduous broadleaved forest and subtropical evergreen broadleaved forest, have more plots, they were not used in this study due to the variation in species composition. We selected sites which had at least 40 stands, represented to 3 or less species. Therefore, the remaining eight out of 17 forest types were further selected from the dataset as in Zhang et al. (2011). The forest types were: boreal/temperate Larix forest (BTLF), boreal/alpine Picea-Abies forest (BAPF), temperate Pinus tabulaeformis forest (TPTF), temperate/subtropical montane Populus-Betula deciduous forest (TSPF), subtropical montane Pinus yunnanensis and Pinus khasya forest (SPPF), subtropical Pinus massoniana
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forest (SPMF), subtropical montane Pinus armandii, Pinus taiwanensis and Pinus densada forest (SMPF), subtropical Cunninghamia lanceolata forest (SCLF) (Table 1, see Appendix A: Fig. S1). Mean tree biomass (kg/individual) was calculated as standing tree biomass divided by density (N, trees/ha). Net primary productivity (NPP, kg/ha/yr), mean tree biomass (M, kg/individual) and density (N, trees/ha) were then used to estimate net primary productivity–biomass (NPP–M) and net primary productivity–density (NPP–N) relationships, respectively. We also combined all the data to determine a larger pattern that typifies Chinese forests.
Fitting of regression lines It is suggested that the upper boundary of the log–log relationship between plant size (or growth) and density provides a stronger test of theory than fitting lines through the center of data (Coomes, Holdaway, Kobe, Lines, & Allen 2012). In addition, an upper boundary ultimately represents mature or near-mature forests, and mature forests have gone through an initial natural thinning due to species interactions (Zhang et al. 2011, 2013), with the biomass close to dynamic equilibrium (Luo 1996; Zhang et al. 2011). According to the traditional methods, the scaling of maximum NPP to biomass and density was estimated by selecting data points that lie close to the apparent upper boundary of bivariate plots (Osawa & Allen 1993). These selected data points are then used to estimate the slopes and intercepts by ordinary least square (OLS) or standardized major axis (SMA) regression (Osawa & Allen 1993; Enquist et al. 1998). However, some stands lay far above the cluster scatter and boundary lines, which indicated unusually large NPP for their biomass and densities. These stands were judged as outliers due to sampling problems, and they were excluded from the following analysis (Osawa & Allen 1993; Zhang et al. 2011). The traditional methods are sensitive to the data selected (Zhang, Bi, Gove, & Heath 2005), while this problem could be solved by taking quantile regression (QR), which uses all the data points, involves no subjective data selection and contains all information (Zhang et al. 2005). Therefore, upper boundary is also often estimated from quantile regression (QR) (Zhang et al. 2005; Coomes et al. 2012). Quantile regression can provide estimation for linear fit to any conditional quantile of a response distribution, including near the upper boundaries, and is free of stringent assumptions of the error distribution (Cade, Terrell, & Schroeder 1999; Zhang et al. 2005). The τth quantile (0 ≤ τ ≤ 1) of a random variable Y is inverse to the cumulative distribution function, F−1 (τ), which is defined as the smallest real value y such that the probability of obtaining any smaller values is greater than or equal to τ (Cade et al. 1999). The τth regression quantile for the linear model y = X + v(X) is defined as QY (τ|X) = X(τ) and (τ) =  + v(·)F −1 (τ), where y is an n × 1 vector of dependent responses, X is an n × p matrix of predictors,
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Table 1. Vegetation and climatic characteristics of eight forest types in Luo (1996). Variable
BTLF
BAPF
TPTF
Alt (m) Lon (◦ ) Lat (◦ ) T (◦ C) P (mm) Age (yr) Age (yr) Age of NMF Age of MF N (trees/ha) M (kg) MT (t/ha) NPP (t/ha/yr) NPP (t/ha/yr)
1409 (900) 114 (14) 46 (6) –1 (3) 601 (176) 93 (53) 30–195 81–100 101–140 2922 (4472) 200 (221) 152 (81) 9 (3) 3–16
3063 (765) 102 (9) 34 (6) 4 (3) 820 (257) 130 (46) 46–350 81–100 101–140 728 (567) 576 (805) 250 (145) 7 (2) 3–14
1457 (396) 113 (3) 37 (1) 6 (2) 611 (101) 48 (13) 15–95 51–60 61–80 1113 (1004) 94 (97) 68 (34) 5 (2) 2–12
TSPF 1567 (843) 115 (10) 39 (6) 4 (3) 661 (179) 50 (23) 25–222 16–20 21–30 1326 (1117) 145 (119) 120 (50) 11 (4) 3–24
SPPF
SPMF
SMPF
SCLF
2463 (626) 101 (2) 27 (1) 12 (3) 1049 (210) 61 (35) 20–150 51–60 61–80 995 (1444) 564 (684) 154 (71) 9 (3) 4–15
536 (307) 111 (4) 27 (3) 18 (3) 1441 (228) 29 (13) 15–101 51–60 61–80 1358 (766) 153 (128) 146 (72) 15 (4) 7–25
2060 (856) 105 (8) 29 (3) 10 (2) 1128 (460) 43 (29) 16–160 51–60 61–80 1622 (1259) 179 (248) 124 (66) 11 (3) 4–22
602 (377) 112 (5) 27 (3) 17 (3) 1526 (361) 23 (7) 16–55 21–25 26–35 2074 (729) 72 (58) 136 (93) 14 (6) 6–34
T is mean annual temperature. P is mean annual precipitation. Age and Age are forest age and its range, respectively. Age class was divided according to the national forest resource inventory conducted by China’s State Forestry Administration (SFA) (Luo 1996), NMF: near-mature forest, MF: mature forest. N is density. M is mean tree biomass. MT is standing biomass. NPP and NPP are net primary production and its range, respectively. Values with parentheses are means (standard deviation). BTLF: boreal/temperate Larix forest, BAPF: boreal/alpine Picea-Abies forest, TPTF: temperate Pinus tabulaeformis forest, TSPF: temperate/subtropical montane Populus-Betula deciduous forest, SPPF: subtropical montane Pinus yunnanensis and P. khasya forest, SPMF: subtropical Pinus massoniana forest, SMPF: subtropical montane Pinus armandii, P. taiwanensis and P. densada forest, SCLF: subtropical Cunninghamia lanceolata forest.
 is a p × 1 vector of unknown regression coefficients, v(·) > 0 is a known function, and is an n × 1 vector of random errors (Cade et al. 1999). The regression method solves an optimization problem of minimizing an asymmetric function of absolute errors (Cade et al. 1999; Zhang et al. 2005):
Over the eight forest types at the regional scale of China, if the slope of zero held universally, then (8 × 95%) = 7.6 of the estimated slopes should be zero according to sampling theory. Similarly, for the combined data set, (6 × 95%) = 5.7 of the six NPP–M (or N) slopes should be zero, if the slope of zero held universally at the national scale of China. Since ⎤
⎡
⎥ ⎢ ⎥ ⎢ p p ˆ ˆ ⎥ ⎢ ⎢ (1 − τ) yi − τ yi − min ⎢ βj xij + βj xij ⎥ ⎥ ⎥ ⎢ j=0 j=0 ˆ ˆ ⎦ ⎣ i∈ iyi ≥βj xij i∈ iyi <βj xij We used both ordinary least square (OLS) and standardized major axis (SMA) regression for selected boundary points, and quantile regression (QR, τ = 0.95) to estimate the slopes of regression lines. For OLS and SMA regression analysis, if the regression is not significant (P > 0.05), the dependent variable is not dependent on the independent variable (i.e. slope = 0). OLS regression of log10 -transformed data was conducted in PASW Statistics 18 (SPSS Inc., Chicago, IL, U.S.A.), and SMA regression of log10 -transformed data was conducted using SMATR Version 2.0 (Falster, Warton, & Wright 2006). For quantile regression analysis, statistically significant differences between observed and the proposed “0” exponent were tested based on 95% CIs of the estimates (Dai et al. 2009; Zhang et al. 2011). Quantile regression analysis was conducted using the ‘quantreg’ library (Koenker 2009) of R-statistical analysis software (R-2.14.0, 2011, distributed freely at http://www.R-project.org/).
forest counts are integers, in essence, this means that all slopes should be zero. In addition, a t-test of the slopes was run in PASW Statistics 18 (SPSS Inc., Chicago, IL, U.S.A.) to see if a slope of 0 can be excluded.
Results Quantile regression analysis indicated that four NPP–M exponents (i.e. BAPF, TPTF, TSPF and SCLF) and two NPP–N exponents (i.e. TPTF and SCLF) were significantly different from 0 based on the 95% CIs across the eight forest types (Table 2, Figs. 1 and 2). Ordinary least square (OLS) and standardized major axis (SMA) regression analysis obtained qualitatively similar results, both regression showed that four NPP–M (i.e. TPTF, TSPF, SMPF and SCLF) and two NPP–N exponents (i.e. TPTF and SMPF) were statistically
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Fig. 1. Scaling relationships between net primary productivity (NPP, kg/ha/yr) and mean tree biomass (M, kg/individual) across eight forest types as estimated by 95% quantile regression. (A) BTLF, (B) BAPF, (C) TPTF, (D) TSPF, (E) SPPF, (F) SPMF, (G) SMPF, (H) SCLF.
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Table 2. Scaling relationships of net primary productivity (NPP, kg/ha/yr) to mean tree biomass (M, kg/individual) and density (N, trees/ha) across eight forest types as estimated by 95% quantile regression. Forest types
NPP: M (N)
Slope (SE)
BTLF BAPF TPTF TSPF SPPF SPMF SMPF SCLF Complete BTLF BAPF TPTF TSPF SPPF SPMF SMPF SCLF Complete
NPP: M NPP: M NPP: M NPP: M NPP: M NPP: M NPP: M NPP: M NPP: M NPP: N NPP: N NPP: N NPP: N NPP: N NPP: N NPP: N NPP: N NPP: N
−0.029 (0.076) 0.054 (0.036) 0.033 (0.052) 0.106 (0.052) −0.020 (0.040) 0.156 (0.088) −0.051 (0.131) 0.589 (0.147) 0.016 (0.047) 0.035 (0.098) 0.016 (0.108) 0.072 (0.093) 0.010 (0.098) 0.060 (0.051) 0.195 (0.150) 0.076 (0.123) 0.571 (0.256) 0.225 (0.028)
95% CIs −0.054, 0.203 0.035, 0.112 0.004, 0.074 0.004, 0.136 −0.041, 0.054 −0.009, 0.497 −0.179, 0.319 0.270, 0.715 −0.075, 0.126 −0.187, 0.073 −0.082, 0.181 0.062, 0.186 −0.109, 0.266 −0.079, 0.209 −0.265, 0.460 −0.225, 0.211 0.015, 0.936 0.187, 0.262
Intercept (SE)
95% CIs
Number
4.215 (0.176) 3.929 (0.083) 3.915 (0.109) 4.058 (0.104) 4.158 (0.089) 4.041 (0.176) 4.309 (0.233) 3.288 (0.270) 4.260 (0.095) 4.045 (0.294) 4.019 (0.322) 3.784 (0.305) 4.242 (0.318) 3.961 (0.157) 3.767 (0.457) 3.970 (0.384) 2.547 (0.826) 3.578 (0.081)
3.846, 4.310 3.793, 3.937 3.844, 4.049 3.994, 4.268 4.008, 4.225 3.843, 4.618 3.878, 4.683 3.141, 3.870 4.031, 4.451 3.866, 4.993 3.607, 4.313 3.169, 3.824 3.563, 4.612 3.910, 4.372 3.047, 5.282 3.631, 5.360 1.312, 4.618 3.466, 3.673
48 168 154 127 46 66 55 98 762 48 168 154 127 46 66 55 98 762
Slopes in bold type are statistically indistinguishable from zero. SE represents the standard error for slopes and intercepts. 95% CIs are confidence intervals for slopes and intercepts. Number is sample size. BTLF, BAPF, TPTF, TSPF, SPPF, SPMF, SMPF, SCLF are defined as in Table 1.
distinguishable from 0 based on P value <0.05 (see Appendix A: Tables S1 and S2, Figs. S2 and S3). Although both QR and OLS showed that four NPP–M exponents were different from 0, the four forest types are not all the same for QR and OLS (Table 2, Fig. 1, see Appendix A: Table S1, Fig. S2). Similarly, both QR and OLS showed that two NPP–N exponents were different from 0, while the two forest types are not all the same (Table 2, Fig. 2, see Appendix A: Table S1, Fig. S3). When we combined all the data, QR showed that the scaling exponent of the NPP–M relationship was close to 0; however, the scaling exponent of the NPP–N relationship was not close to 0, and for OLS and SMA, the scaling exponents of neither relationships were close to 0 (Table 2, see Appendix A: Tables S1 and S2, Fig. S4). Among these non-zero exponents, many were remarkably close to 0 (Table 2, Fig. 3, see Appendix A: Tables S1 and S2). For example, the NPP–M exponents of QR regression in TPTF and BAPF were 0.033 and 0.054, respectively; the lower confidence intervals of NPP–M exponents in both TPTF and TSPF included 0.004 (Table 2, Fig. 3). The mean of the slopes in eight forest types was not significantly different from 0 (P = 0.197, 0.092, 0.248, 0.520, 0.173 and 0.491 for NPP–M (QR), NPP–N (QR), NPP–M (OLS), NPP–N (OLS), NPP–M (SMA) and NPP–N (SMA), respectively).
Discussion No universal exponent but with central tendency The ‘energetic equivalence rule’ has been criticized on both theoretical and empirical grounds (Deng et al. 2006;
King 2010; Zhang, Wang, Zheng, & Zhang 2010; Bai et al. 2010, 2011). In this study, QR, OLS and SMA regression all showed that four NPP–M exponents and two NPP–N exponents were different from 0 across the 8 forest types (Table 2, see Appendix A: Tables S1 and S2). In addition, when we combined all the data to determine a larger pattern that typifies Chinese forests, five out of the six NPP–M (or N) exponents deviated strongly from 0 (Table 2, see Appendix A: Tables S1 and S2, Fig. S4). Since the percentage of “0” exponent is less than 7.6/8 for the 8 forest types and 5.7/6 for all data pooled, the universality of the ‘energetic equivalence rule’ does not hold for forest communities at both the regional and the national scale in China. Using the same dataset (Luo 1996) as the present study, recent studies found little support for universal scaling of biomass–density relationship and metabolic theory across a range of forest types in China (Li et al. 2005, 2006; Zhang et al. 2011, 2012). Therefore, these studies indicate that no universal exponent applies to the predictions derived from metabolic scaling theory or the ‘energetic equivalence rule’. Recent studies showed that the exponents of standing leaf biomass–density relationship were different from 0 (Ogawa 2001; Zhang et al. 2012), and the maximum sustainable leaf biomass increased slightly with decreasing plant density along the self-thinning line in conifer plantations (Blake, Somers, & Ruark 1991). In particular, standing leaf mass (leaf biomass per unit area) was not constant during self-thinning across the same eight forest communities as the present study (Zhang et al. 2012); changes in light distribution in relation to canopy structure, particularly canopy depth and leaf clustering, may account for the increase in leaf biomass during self-thinning (Blake et al. 1991; Holdaway, Allen, Clinton,
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Fig. 2. Scaling relationships between net primary productivity (NPP, kg/ha/yr) and density (N, trees/ha) across eight forest types as estimated by 95% quantile regression. (A) BTLF, (B) BAPF, (C) TPTF, (D) TSPF, (E) SPPF, (F) SPMF, (G) SMPF, (H) SCLF.
Davis, & Coomes 2008; Pan et al. 2013). As forest growth and efficiency is significantly related to canopy leaf area (Waring 1983), changes in standing leaf biomass–density relationship may lead to no universal net primary productivity–biomass (or density) relationships.
Because the universal scaling of metabolic scaling theory (MST) has provoked much criticism (Russo et al. 2007; Bai et al. 2010), Enquist and colleagues have started to refer to their slopes as “basins of attraction” (Price, Enquist, & Savage 2007) rather than as an invariant or canonical value.
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Fig. 3. The NPP–M and NPP–N slopes for eight forest types, as estimated by 95% quantile regression (QR), ordinary least square (OLS) and standardized major axis (SMA) regression; bars show 95% CIs of QR regression. BTLF, BAPF, TPTF, TSPF, SPPF, SPMF, SMPF, SCLF are defined as in Table 1.
In our study, although some plots clearly deviated from the ‘energetic equivalence rule’ (e.g. SCLF), others were remarkably close (Table 2, Fig. 3, see Appendix A: Tables S1 and S2). For instance, the NPP–M exponents of QR regression in TPTF and BAPF were 0.033 and 0.054, respectively; the confidence intervals of NPP–M exponents in both TPTF and TSPF included 0.004 (Table 2, Fig. 3). In fact, the significant difference is almost certainly related to their larger sample size, as statistical power increases with sample size (Coomes & Allen 2009). This has led to a distinction being recognized between statistical significance and biological significance; large sample sizes can result in small, biologically unimportant effects being declared statistically significant (Quinn & Keough 2002). So in addition to the statistical tests for the value of the exponents, thought needs to be given to what degree of variation from a zero slope would be considered biologically important. The densities (or biomass) of the eight forest types range over approximately 1.5–2 orders of magnitude in our study; a true slope of 0.004 for example would mean that the NPP value at the lower end of this range would be approximately 98% (i.e. 1/100.004 × 1.5 or 1/100.004 × 2 ) of that at the higher end of the range of density (or biomass), a difference that could be considered biologically unimportant. A debate about what level of deviation from a slope of 0 would be considered biologically unimportant/important would assist in determining the usefulness of statistical tests. As an example, for slopes of 0 ± 0.02, the lower value of NPP would be 91–93% of the higher NPP value for the range of densities (or biomass) found in our study.
It appears that 0 is a “basin of attraction” for NPP–M (N) relationships in 7 out of 8 forest types (SCLF showed the greatest deviation), with slopes showing scatter over a small range around 0 (Fig. 3). Only 6 out of 54 exponents (i.e. NPP–M (QR), NPP–M (OLS) and NPP–N (OLS) slopes of SPPF, NPP–M (OLS) slope of SPMF, NPP–N (QR) slope of BAPF and TSPF) fell within the range −0.02 to +0.02 (Table 2, Fig. 3, see Appendix A: Tables S1 and S2), however, if this was accepted for example as the bounds of biologically unimportant deviation from 0. This exercise shows the difficulties in distinguishing between sampling variability of sample slopes around an underlying population slope of 0, and real deviations from a 0 slope that are small, but biologically important.
Factors related to deviation from the ‘energetic equivalence rule’ Deviation from the “energetic equivalence” possibly reflects multiple, unsound assumptions for “an average idealized forest” by metabolic scaling theory (MST), as well as unaccounted-for variations of site factors within forest communities (Deng et al. 2006; Russo et al. 2007; Dai et al. 2009). According to the predictions of metabolic scaling theory (MST) and the ‘energetic equivalence rule’, net primary production (NPP) remains constant during stand development (Enquist, West, & Brown 2009). However, it is well known that NPP typically changes with stand age; it reaches a peak (approximately at the time of canopy closure) and then
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declines as forests mature (Ryan, Binkley, & Fownes 1997). Age-related decline in forest productivity is often explained by the water/nutrient limitation hypothesis that large trees are unable to supply adequate resources to maintain productivity (Ryan et al. 1997; Coomes et al. 2012). These changes are contrary to the predictions of metabolic scaling theory (MST) and the ‘energetic equivalence rule’. Furthermore, a larger percentage of biomass is allocated to metabolic tissue (e.g. leaf) in trees seedlings; in contrast, most of the biomass is allocated to non-metabolic tissue (e.g. heartwood) in large trees (Mori et al. 2010), the transition from seedlings to large trees also indicates that forest age plays an important role in forest energy structure. The age class and stand development of forest communities are species-specific and depend on site quality (Luo 1996). For example, self-thinning even occurs in an overcrowded population of tree seedlings of hinoki (Chamaecyparis obtusa (Sieb. et Zucc.) Endl.) (Ogawa 2001). In our study, although mean forest age of 5 out of the 8 forest types are under 50 yrs, and two are under 30 yrs (i.e. SPMF and SCLF), mean and median forest age of the 8 forest types are close to or within the range of near-mature/mature forests (Table 1, see Appendix A: Fig. S1), according to the national forest resource inventory conducted by China’s State Forestry Administration (SFA) (Luo 1996). For instance, Pinus massoniana and C. lanceolata are widely planted in the subtropical area of China because of their fast growth (reach maturity at a relatively young age), high productivity, short rotation and high quality wood (Tao, Feng, & Ma 2011); therefore, subtropical Pinus massoniana forest (SPMF) and subtropical C. lanceolata forest (SCLF) are mainly composed of “fast-growing and high-yield plantations” (Luo 1996; Tao et al. 2011), consistently, some plots in SCLF and SPMF showed greatest NPP when all data was combined (see Appendix A: Fig. S4). More importantly, we used upper boundary to estimate NPP–M (N) lines, which can ultimately eliminate the bias of young forests. In addition, QR showed that NPP–M (N) slopes in SCLF were different from 0, while those in SPMF were indistinguishable from 0 (Table 2). Therefore, the strongest deviation from energetic equivalence of 23-year-old SCLF (but not 29-year-old SPMF) cannot all be attributed to young stand age. In addition to stand age, much of the variation in forest NPP and energy structure may be the result of site conditions not considered in the analyses, such as temperature, rainfall, standing biomass, disturbance or phylogeny within the forest type (Ni et al. 2001; Luo et al. 2002; Russo et al. 2007; Coomes et al. 2012). However, these effects were hard to detect since all these factors were mixed together. Recent literature indicates that the scaling exponents of biomass–density (M–N) relationship vary with environmental gradients, e.g., soil salinity (Zhang et al. 2010) and water availability (Deng et al. 2006; Dai et al. 2009; Bai et al. 2010). In contrast, most NPP–M (N) exponents were not correlated significantly with site factors (e.g. altitude and age in Table 1, P > 0.05). Our study used data only from benign
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environments, the range of climate and exponent variations were thus too narrow to detect a correlation. Though it is still unclear whether the deviations of NPP–M (N) slopes cooccur with site conditions and forest age, these factors should be taken into account when testing the ‘energetic equivalence rule’ across a large number of stands. In addition, resource division among forest types (i.e. different intercepts in Fig. S4) may result in the deviation from 0 at the national scale of China (White et al. 2007). Different forest types often consume different soil and/or light resources, and are different in their size; therefore, it is difficult to ensure that such distinct stands converge to a single boundary line when they are pooled.
Statistical methods and variability in estimating energy Statistical methods should be subject to strict scrutiny in estimating universal scaling relationships (Li et al. 2006). In our study, QR and OLS regression yielded qualitatively different results in some forest types (Table 2, see Appendix A: Tables S1 and S2). Using three regression (QR, OLS and SMA) methods, Li et al. (2006) also found qualitatively different biomass–density exponents in the same forest biome across tree-dominated communities. In addition, as a basis of the ‘energetic equivalence rule’, the self-thinning process (or biomass–density relationship) is usually deemed necessary to be quantified by fitting an upper boundary (Osawa & Allen 1993; Coomes et al. 2012), however, metabolic scaling theory (MST) always fits these relationships through the center of data (e.g. Enquist et al. 1998; Niklas et al. 2003), therefore, statistical methods underlying the basis of the ‘energetic equivalence rule’ are also questioned, which further undermined the validity and universality of the “energetic equivalence rule”. In our study, we used growth rate or NPP as an estimate of energy use, the best measure is presumably gross primary production (GPP) in plant communities. However, the ‘energetic equivalence rule’ may not hold universally for forests even if we used GPP, because GPP is closely related to total leaf area (Price et al. 2007), while tree leaf area index (LAI) does not remain constant during stand development in forest communities (Blake et al. 1991; Holdaway et al. 2008). Furthermore, the scaling exponents of GPP–M (N) may be different from those of NPP–M (N), since the ratio of GPP to NPP changes over ontogeny (becoming less favorable for older trees) (Mori et al. 2010). Future work is needed to test the ‘energetic equivalence rule’ using GPP as a variable.
Acknowledgements We thank Jacob Weiner, managing editor (Klaus Hövemeyer), Yan Chen and anonymous reviewers for their helpful comments and suggestions on this work. We also thank Dr.
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Tianxiang Luo (Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing, China) for providing helpful information to this work. This work was supported by Chinese Universities Scientific Fund (2014XJ031), National Natural Science Foundation of China (31330010), Zhejiang Provincial Natural Science Foundation of China (LZ13C030002) and Ministry of Science and Technology of the People’s Republic of China (2011AA100503).
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/ j.baae.2015.04.005.
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Testing predictions of the energetic equivalence rule in forest communities Wei-Ping Zhanga,∗ , E. Charles Morrisb , Xin Jiac , Sha Pand , Gen-Xuan Wange a Beijing Key Laboratory of Biodiversity and Organic Farming, College of Resources and Environmental Sciences, China Agricultural University, Beijing 100193, China b School of Science and Health, University of Western Sydney, Richmond 2753, NSW, Australia c Key Laboratory of Soil and Water Conservation and Desertification Combating, Ministry of Education, Beijing Forestry University, Beijing 100083, China d Department of Environmental Hygiene, School of Public Health, Guiyang Medical University, Guiyang 550004, China e Institute of Ecology, College of Life Sciences, Zhejiang University, Hangzhou 310058, China
Received 13 May 2014; received in revised form 24 March 2015; accepted 6 April 2015
Abstract As growth rate is a reasonable proxy measure of the rate of resource use per plant individual, the ‘energetic equivalence rule’ predicts that net primary productivity (the rate of biomass production per unit area, NPP) will be independent of plant biomass and maximum population density in plant communities. However, only a few studies have tested these relationships in plant communities. In this study, we investigated allometric scaling of net primary productivity (NPP) to tree biomass (M) and density (N) across a range of tree-dominated communities in China. The aim was to test the universality of the ‘energetic equivalence rule’ (i.e. whether the exponents of these relationships take a universal value of 0) in forest communities. We used both ordinary least square (OLS) and standardized major axis (SMA) regression for selected boundary points, and quantile regression (QR) to estimate the slopes of regression lines. QR, OLS and SMA regression all showed that four NPP–M and two NPP–N exponents were different from 0 across the 8 forest types. In addition, when we combined all the data to determine a larger pattern that typifies Chinese forests, five out of the six exponents of NPP–M and NPP–N relationships deviated strongly from 0. Therefore the universality of the ‘energetic equivalence rule’ does not hold for forest communities at both the regional and the national scale of China. However, the “zero” exponent seems to be a central tendency for NPP–M and NPP–N relationships in 7 out of 8 forest types. Deviation from the energetic equivalence possibly reflects multiple, unsound assumptions for “an average idealized forest” by metabolic scaling theory, as well as unaccounted-for variations of site factors (e.g. stand age and stand conditions) within forest communities. In addition, our study suggested that statistical methods should be subject to strict scrutiny in testing the ‘energetic equivalence rule’.
Zusammenfassung Da die Wachstumsrate näherungsweise ein gutes Maß für die Rate der Ressourcennutzung pro Pflanze ist, sagt die Regel von der energetischen Äquivalenz voraus, dass die Nettoprimärproduktion (die Rate der Biomasseproduktion je Flächeneinheit: NPP) von der Pflanzenbiomasse und der maximalen Populationsdichte von Pflanzengemeinschaften unabhängig sein ∗ Corresponding
author. Tel.: +86 10 62734684; fax: +86 10 62731016. E-mail address: [email protected] (W.-P. Zhang).
http://dx.doi.org/10.1016/j.baae.2015.04.005 1439-1791/© 2015 Gesellschaft für Ökologie. Published by Elsevier GmbH. All rights reserved.
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wird. Indessen haben nur wenige Studien diese Beziehungen in Pflanzengemeinschaften untersucht. Wir untersuchten die allometrische Beziehung zwischen der Nettoprimärproduktion (NPP) und der Baum-Biomasse (M) und -Dichte (N) über eine Reihe von baumdominierten Gemeinschaften in China. Unser Ziel war, die Allgemeingültigkeit der Regel von der energetischen Äquivalenz zu überprüfen, d.h., ob die Exponenten dieser Beziehungen einen universellen Wert von Null annehmen. Wir benutzten die Methode der kleinsten Quadrate (OLS), standardisierte Hauptachsen-Regression (SMA) für ausgewählte Grenzpunkte sowie Quantil-Regression (QR), um die Steigungen von Regressionsgeraden zu bestimmen. QR, OLS und SMA zeigten, dass über die acht untersuchten Waldtypen vier NPP-M- und zwei NPP-N-Exponenten von Null verschieden waren. Als wir alle Daten kombinierten, um das größere, für chinesische Wälder charakteristische Muster zu bestimmen, unterschieden sich fünf der sechs NPP-M- und NPP-N-Beziehungen stark von Null. Damit ist die Regel von der energetischen Äquivalenz für Waldgemeinschaften in China weder auf der regionalen noch auf der nationalen Ebene allgemein gültig. Indessen scheint der Null-Exponent eine zentrale Tendenz für die NPP-M und NPP-N-Beziehungen in sieben von acht Waldtypen zu sein. Die Abweichung von der energetischen Äquivalenz spiegelt möglicherweise mehrere unbegründete Annahmen der Theorie der metabolischen Skalierung hinsichtlich eines durchschnittlichen, idealisierten Waldes wider, sowie die unberücksichtigt gebliebenen Schwankungen von Standortfaktoren in Waldgemeinschaften (z.B.: Bestandsalter, Bestandsbedingungen). Darüberhinaus legt unsere Untersuchung nahe, dass bei der Prüfung der Regel von der energetischen Äquivalenz die statistischen Methoden einer äußerst eingehenden Überprüfung unterzogen werden sollten. © 2015 Gesellschaft für Ökologie. Published by Elsevier GmbH. All rights reserved.
Keywords: Scaling; Net primary productivity; Biomass; Density; Forest communities; Self-thinning; Upper boundary; Quantile regression; Standardized major axis regression; Metabolic scaling theory
Introduction Metabolic scaling theory (MST) argued that metabolic rate (or resource use rate, Q) per individual would scale approximately as the 3/4 power of body mass (M): (i.e. Q ∝ M3/4 ) (West, Brown, & Enquist 1997; Niklas & Enquist 2001). In addition, maximum population density (Nmax ) should scale with body mass M−3/4 (i.e. Nmax ∝ M−3/4 ) (Damuth 1981, 1987; Enquist, Brown, & West 1998; Niklas & Enquist 2001; Niklas, Midgley, & Enquist 2003). Since the maximum rate of resource use per unit area Rmax = Q × Nmax ∝ M 3/4 × 0 , the maximum rate of resource use per M −3/4 ∝ M 0 ∝ Nmax unit area will be independent of average plant biomass and density. The relationship is termed as the ‘energetic equivalence rule’ (Damuth 1981, 1987), and is considered as one of the fundamental laws of nature (Deng et al. 2008). The ‘energetic equivalence rule’ has been tested using different kinds of organisms and across a range of biological scales, but with mixed results (Enquist et al. 1998; White, Ernest, Kerkhoff, & Enquist 2007; King 2010). In addition, both the 3/4 scaling of metabolic theory (Q–M) and the −4/3 scaling of biomass–density relationship (M–Nmax ) are challenged on theoretical and empirical grounds (Russo, Wiser, & Coomes 2007; Dai et al. 2009; Mori et al. 2010; Bai et al. 2010, 2011). Therefore, controversy continues over the validity and universality of the ‘energetic equivalence rule’ (White et al. 2007; Deng et al. 2008; King 2010). It is mathematically difficult to examine the applicability of the ‘energetic equivalence rule’ directly in plant communities (Enquist et al. 1998; Deng et al. 2008). As growth rate is a reasonable proxy measure of the rate of resource use per plant individual (Q) (Niklas & Enquist 2001), net primary productivity (the rate of biomass production per unit area,
NPP) is predicted to scale proportionally to total energy use per unit area in plant communities. It follows from the ‘ener0 . getic equivalence rule’ that NPPmax ∝ Rmax ∝ M 0 ∝ Nmax More importantly, these relationships provide an alternative indicator for ‘energetic equivalence’. However, only a few studies have tested these relationships in plant communities (Niklas & Enquist 2001; Niklas et al. 2003). Recent literature indicates that both scaling of metabolic theory and biomass–density relationship are not a constant, but rather vary with environmental conditions or particular taxonomic groups across a range of forest communities in China (Li, Han, & Wu 2005; Li, Han, & Wu 2006; Zhang, Jia, Bai, & Wang 2011; Zhang, Jia, Morris, Bai, & Wang 2012). Based on the same dataset, we further investigated allometric scaling of net primary productivity (NPP) to tree biomass (M) and maximum density (Nmax ) across a range of tree-dominated communities in China. The aim of was to test the universality of the ‘energetic equivalence rule’ (i.e. whether the exponents of these relationships take a universal value of 0) in forest communities.
Materials and methods Data source We examined the allometric scaling of net primary productivity (NPP) to tree biomass (M) and density (N) by using the compilation of Luo (1996) for standing biomass and production. These data were compiled from the Chinese literature for continuous forest-inventory plots of the forestry departments between 1970 and 1994. The dataset includes
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biomass, production and associated climatic variables for 1266 selected forest plots over China (Luo 1996). In most plots, all biomass components of trees (i.e. stem, branch, leaf and root) were estimated by species-specific allometric equations based on tree height (H, m) and diameter at breast height (D, cm) (Luo, Li, & Zhu 2002). In addition, the tree biomass in a few plots was estimated by the “clearcutting method” (all trees in a plot were cut down) or “mean tree method”. For “mean tree method”: several “mean trees” in a plot were felled and standing biomass was then estimated by the product of these mean tree biomass and stand density (Luo 1996; Ni, Zhang, & Scurlock 2001). This method was mainly used for plantations and even-aged stands, in which the variations of tree size (height, diameter and biomass, etc,) were small. Therefore, the “mean trees” were usually accessed according to mean height, mean diameter and mean crown width empirically (Zhang et al. 2011). Net primary productivity (NPP, kg/ha/yr) for all tree components (i.e. stem, branch, leaf and root) were estimated from the sum of annual growth rate per tree (kg/yr). Due to the careful estimation of biomass and net primary productivity, the data of Luo (1996) has been used for analysis in many studies (e.g. Ni et al. 2001; Luo et al. 2002; Li et al. 2005, 2006; Zhang et al. 2011). More detailed information about the dataset can be found in Luo (1996), Luo et al. (2002) and Zhang et al. (2011). Luo (1996) stratified the 1266 plots into 17 major forest types. Among the 17 forest types, seven have only 9 to 22 plots, the sample size is thus too small to analyze these scaling relationships. In addition, self-thinning is a phenomenon found in dense monocultures, indicated by increasing stand biomass with decreasing stand density (Yoda, Kira, Ogawa, & Hozumi 1963; Zhang et al. 2011, 2012). There is a basic agreement that stands with different individuals (but with same species or life-form, and with similar conditions) and densities should converge to a single self-thinning line. If the stand conditions and species composition are not equivalent, there are no guarantees that all the stands will be on a single self-thinning line, even if they are fully stocked. Considering the possibility of different conditions of the sample stands, it is suggested to stratify the stands, or classify stands into groups depending on their conditions, and obtain a boundary line in each group of the stands (Zhang et al. 2011, 2012). Therefore, although another two forest types, i.e. temperate typical deciduous broadleaved forest and subtropical evergreen broadleaved forest, have more plots, they were not used in this study due to the variation in species composition. We selected sites which had at least 40 stands, represented to 3 or less species. Therefore, the remaining eight out of 17 forest types were further selected from the dataset as in Zhang et al. (2011). The forest types were: boreal/temperate Larix forest (BTLF), boreal/alpine Picea-Abies forest (BAPF), temperate Pinus tabulaeformis forest (TPTF), temperate/subtropical montane Populus-Betula deciduous forest (TSPF), subtropical montane Pinus yunnanensis and Pinus khasya forest (SPPF), subtropical Pinus massoniana
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forest (SPMF), subtropical montane Pinus armandii, Pinus taiwanensis and Pinus densada forest (SMPF), subtropical Cunninghamia lanceolata forest (SCLF) (Table 1, see Appendix A: Fig. S1). Mean tree biomass (kg/individual) was calculated as standing tree biomass divided by density (N, trees/ha). Net primary productivity (NPP, kg/ha/yr), mean tree biomass (M, kg/individual) and density (N, trees/ha) were then used to estimate net primary productivity–biomass (NPP–M) and net primary productivity–density (NPP–N) relationships, respectively. We also combined all the data to determine a larger pattern that typifies Chinese forests.
Fitting of regression lines It is suggested that the upper boundary of the log–log relationship between plant size (or growth) and density provides a stronger test of theory than fitting lines through the center of data (Coomes, Holdaway, Kobe, Lines, & Allen 2012). In addition, an upper boundary ultimately represents mature or near-mature forests, and mature forests have gone through an initial natural thinning due to species interactions (Zhang et al. 2011, 2013), with the biomass close to dynamic equilibrium (Luo 1996; Zhang et al. 2011). According to the traditional methods, the scaling of maximum NPP to biomass and density was estimated by selecting data points that lie close to the apparent upper boundary of bivariate plots (Osawa & Allen 1993). These selected data points are then used to estimate the slopes and intercepts by ordinary least square (OLS) or standardized major axis (SMA) regression (Osawa & Allen 1993; Enquist et al. 1998). However, some stands lay far above the cluster scatter and boundary lines, which indicated unusually large NPP for their biomass and densities. These stands were judged as outliers due to sampling problems, and they were excluded from the following analysis (Osawa & Allen 1993; Zhang et al. 2011). The traditional methods are sensitive to the data selected (Zhang, Bi, Gove, & Heath 2005), while this problem could be solved by taking quantile regression (QR), which uses all the data points, involves no subjective data selection and contains all information (Zhang et al. 2005). Therefore, upper boundary is also often estimated from quantile regression (QR) (Zhang et al. 2005; Coomes et al. 2012). Quantile regression can provide estimation for linear fit to any conditional quantile of a response distribution, including near the upper boundaries, and is free of stringent assumptions of the error distribution (Cade, Terrell, & Schroeder 1999; Zhang et al. 2005). The τth quantile (0 ≤ τ ≤ 1) of a random variable Y is inverse to the cumulative distribution function, F−1 (τ), which is defined as the smallest real value y such that the probability of obtaining any smaller values is greater than or equal to τ (Cade et al. 1999). The τth regression quantile for the linear model y = X + v(X) is defined as QY (τ|X) = X(τ) and (τ) =  + v(·)F −1 (τ), where y is an n × 1 vector of dependent responses, X is an n × p matrix of predictors,
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Table 1. Vegetation and climatic characteristics of eight forest types in Luo (1996). Variable
BTLF
BAPF
TPTF
Alt (m) Lon (◦ ) Lat (◦ ) T (◦ C) P (mm) Age (yr) Age (yr) Age of NMF Age of MF N (trees/ha) M (kg) MT (t/ha) NPP (t/ha/yr) NPP (t/ha/yr)
1409 (900) 114 (14) 46 (6) –1 (3) 601 (176) 93 (53) 30–195 81–100 101–140 2922 (4472) 200 (221) 152 (81) 9 (3) 3–16
3063 (765) 102 (9) 34 (6) 4 (3) 820 (257) 130 (46) 46–350 81–100 101–140 728 (567) 576 (805) 250 (145) 7 (2) 3–14
1457 (396) 113 (3) 37 (1) 6 (2) 611 (101) 48 (13) 15–95 51–60 61–80 1113 (1004) 94 (97) 68 (34) 5 (2) 2–12
TSPF 1567 (843) 115 (10) 39 (6) 4 (3) 661 (179) 50 (23) 25–222 16–20 21–30 1326 (1117) 145 (119) 120 (50) 11 (4) 3–24
SPPF
SPMF
SMPF
SCLF
2463 (626) 101 (2) 27 (1) 12 (3) 1049 (210) 61 (35) 20–150 51–60 61–80 995 (1444) 564 (684) 154 (71) 9 (3) 4–15
536 (307) 111 (4) 27 (3) 18 (3) 1441 (228) 29 (13) 15–101 51–60 61–80 1358 (766) 153 (128) 146 (72) 15 (4) 7–25
2060 (856) 105 (8) 29 (3) 10 (2) 1128 (460) 43 (29) 16–160 51–60 61–80 1622 (1259) 179 (248) 124 (66) 11 (3) 4–22
602 (377) 112 (5) 27 (3) 17 (3) 1526 (361) 23 (7) 16–55 21–25 26–35 2074 (729) 72 (58) 136 (93) 14 (6) 6–34
T is mean annual temperature. P is mean annual precipitation. Age and Age are forest age and its range, respectively. Age class was divided according to the national forest resource inventory conducted by China’s State Forestry Administration (SFA) (Luo 1996), NMF: near-mature forest, MF: mature forest. N is density. M is mean tree biomass. MT is standing biomass. NPP and NPP are net primary production and its range, respectively. Values with parentheses are means (standard deviation). BTLF: boreal/temperate Larix forest, BAPF: boreal/alpine Picea-Abies forest, TPTF: temperate Pinus tabulaeformis forest, TSPF: temperate/subtropical montane Populus-Betula deciduous forest, SPPF: subtropical montane Pinus yunnanensis and P. khasya forest, SPMF: subtropical Pinus massoniana forest, SMPF: subtropical montane Pinus armandii, P. taiwanensis and P. densada forest, SCLF: subtropical Cunninghamia lanceolata forest.
 is a p × 1 vector of unknown regression coefficients, v(·) > 0 is a known function, and is an n × 1 vector of random errors (Cade et al. 1999). The regression method solves an optimization problem of minimizing an asymmetric function of absolute errors (Cade et al. 1999; Zhang et al. 2005):
Over the eight forest types at the regional scale of China, if the slope of zero held universally, then (8 × 95%) = 7.6 of the estimated slopes should be zero according to sampling theory. Similarly, for the combined data set, (6 × 95%) = 5.7 of the six NPP–M (or N) slopes should be zero, if the slope of zero held universally at the national scale of China. Since ⎤
⎡
⎥ ⎢ ⎥ ⎢ p p ˆ ˆ ⎥ ⎢ ⎢ (1 − τ) yi − τ yi − min ⎢ βj xij + βj xij ⎥ ⎥ ⎥ ⎢ j=0 j=0 ˆ ˆ ⎦ ⎣ i∈ iyi ≥βj xij i∈ iyi <βj xij We used both ordinary least square (OLS) and standardized major axis (SMA) regression for selected boundary points, and quantile regression (QR, τ = 0.95) to estimate the slopes of regression lines. For OLS and SMA regression analysis, if the regression is not significant (P > 0.05), the dependent variable is not dependent on the independent variable (i.e. slope = 0). OLS regression of log10 -transformed data was conducted in PASW Statistics 18 (SPSS Inc., Chicago, IL, U.S.A.), and SMA regression of log10 -transformed data was conducted using SMATR Version 2.0 (Falster, Warton, & Wright 2006). For quantile regression analysis, statistically significant differences between observed and the proposed “0” exponent were tested based on 95% CIs of the estimates (Dai et al. 2009; Zhang et al. 2011). Quantile regression analysis was conducted using the ‘quantreg’ library (Koenker 2009) of R-statistical analysis software (R-2.14.0, 2011, distributed freely at http://www.R-project.org/).
forest counts are integers, in essence, this means that all slopes should be zero. In addition, a t-test of the slopes was run in PASW Statistics 18 (SPSS Inc., Chicago, IL, U.S.A.) to see if a slope of 0 can be excluded.
Results Quantile regression analysis indicated that four NPP–M exponents (i.e. BAPF, TPTF, TSPF and SCLF) and two NPP–N exponents (i.e. TPTF and SCLF) were significantly different from 0 based on the 95% CIs across the eight forest types (Table 2, Figs. 1 and 2). Ordinary least square (OLS) and standardized major axis (SMA) regression analysis obtained qualitatively similar results, both regression showed that four NPP–M (i.e. TPTF, TSPF, SMPF and SCLF) and two NPP–N exponents (i.e. TPTF and SMPF) were statistically
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Fig. 1. Scaling relationships between net primary productivity (NPP, kg/ha/yr) and mean tree biomass (M, kg/individual) across eight forest types as estimated by 95% quantile regression. (A) BTLF, (B) BAPF, (C) TPTF, (D) TSPF, (E) SPPF, (F) SPMF, (G) SMPF, (H) SCLF.
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Table 2. Scaling relationships of net primary productivity (NPP, kg/ha/yr) to mean tree biomass (M, kg/individual) and density (N, trees/ha) across eight forest types as estimated by 95% quantile regression. Forest types
NPP: M (N)
Slope (SE)
BTLF BAPF TPTF TSPF SPPF SPMF SMPF SCLF Complete BTLF BAPF TPTF TSPF SPPF SPMF SMPF SCLF Complete
NPP: M NPP: M NPP: M NPP: M NPP: M NPP: M NPP: M NPP: M NPP: M NPP: N NPP: N NPP: N NPP: N NPP: N NPP: N NPP: N NPP: N NPP: N
−0.029 (0.076) 0.054 (0.036) 0.033 (0.052) 0.106 (0.052) −0.020 (0.040) 0.156 (0.088) −0.051 (0.131) 0.589 (0.147) 0.016 (0.047) 0.035 (0.098) 0.016 (0.108) 0.072 (0.093) 0.010 (0.098) 0.060 (0.051) 0.195 (0.150) 0.076 (0.123) 0.571 (0.256) 0.225 (0.028)
95% CIs −0.054, 0.203 0.035, 0.112 0.004, 0.074 0.004, 0.136 −0.041, 0.054 −0.009, 0.497 −0.179, 0.319 0.270, 0.715 −0.075, 0.126 −0.187, 0.073 −0.082, 0.181 0.062, 0.186 −0.109, 0.266 −0.079, 0.209 −0.265, 0.460 −0.225, 0.211 0.015, 0.936 0.187, 0.262
Intercept (SE)
95% CIs
Number
4.215 (0.176) 3.929 (0.083) 3.915 (0.109) 4.058 (0.104) 4.158 (0.089) 4.041 (0.176) 4.309 (0.233) 3.288 (0.270) 4.260 (0.095) 4.045 (0.294) 4.019 (0.322) 3.784 (0.305) 4.242 (0.318) 3.961 (0.157) 3.767 (0.457) 3.970 (0.384) 2.547 (0.826) 3.578 (0.081)
3.846, 4.310 3.793, 3.937 3.844, 4.049 3.994, 4.268 4.008, 4.225 3.843, 4.618 3.878, 4.683 3.141, 3.870 4.031, 4.451 3.866, 4.993 3.607, 4.313 3.169, 3.824 3.563, 4.612 3.910, 4.372 3.047, 5.282 3.631, 5.360 1.312, 4.618 3.466, 3.673
48 168 154 127 46 66 55 98 762 48 168 154 127 46 66 55 98 762
Slopes in bold type are statistically indistinguishable from zero. SE represents the standard error for slopes and intercepts. 95% CIs are confidence intervals for slopes and intercepts. Number is sample size. BTLF, BAPF, TPTF, TSPF, SPPF, SPMF, SMPF, SCLF are defined as in Table 1.
distinguishable from 0 based on P value <0.05 (see Appendix A: Tables S1 and S2, Figs. S2 and S3). Although both QR and OLS showed that four NPP–M exponents were different from 0, the four forest types are not all the same for QR and OLS (Table 2, Fig. 1, see Appendix A: Table S1, Fig. S2). Similarly, both QR and OLS showed that two NPP–N exponents were different from 0, while the two forest types are not all the same (Table 2, Fig. 2, see Appendix A: Table S1, Fig. S3). When we combined all the data, QR showed that the scaling exponent of the NPP–M relationship was close to 0; however, the scaling exponent of the NPP–N relationship was not close to 0, and for OLS and SMA, the scaling exponents of neither relationships were close to 0 (Table 2, see Appendix A: Tables S1 and S2, Fig. S4). Among these non-zero exponents, many were remarkably close to 0 (Table 2, Fig. 3, see Appendix A: Tables S1 and S2). For example, the NPP–M exponents of QR regression in TPTF and BAPF were 0.033 and 0.054, respectively; the lower confidence intervals of NPP–M exponents in both TPTF and TSPF included 0.004 (Table 2, Fig. 3). The mean of the slopes in eight forest types was not significantly different from 0 (P = 0.197, 0.092, 0.248, 0.520, 0.173 and 0.491 for NPP–M (QR), NPP–N (QR), NPP–M (OLS), NPP–N (OLS), NPP–M (SMA) and NPP–N (SMA), respectively).
Discussion No universal exponent but with central tendency The ‘energetic equivalence rule’ has been criticized on both theoretical and empirical grounds (Deng et al. 2006;
King 2010; Zhang, Wang, Zheng, & Zhang 2010; Bai et al. 2010, 2011). In this study, QR, OLS and SMA regression all showed that four NPP–M exponents and two NPP–N exponents were different from 0 across the 8 forest types (Table 2, see Appendix A: Tables S1 and S2). In addition, when we combined all the data to determine a larger pattern that typifies Chinese forests, five out of the six NPP–M (or N) exponents deviated strongly from 0 (Table 2, see Appendix A: Tables S1 and S2, Fig. S4). Since the percentage of “0” exponent is less than 7.6/8 for the 8 forest types and 5.7/6 for all data pooled, the universality of the ‘energetic equivalence rule’ does not hold for forest communities at both the regional and the national scale in China. Using the same dataset (Luo 1996) as the present study, recent studies found little support for universal scaling of biomass–density relationship and metabolic theory across a range of forest types in China (Li et al. 2005, 2006; Zhang et al. 2011, 2012). Therefore, these studies indicate that no universal exponent applies to the predictions derived from metabolic scaling theory or the ‘energetic equivalence rule’. Recent studies showed that the exponents of standing leaf biomass–density relationship were different from 0 (Ogawa 2001; Zhang et al. 2012), and the maximum sustainable leaf biomass increased slightly with decreasing plant density along the self-thinning line in conifer plantations (Blake, Somers, & Ruark 1991). In particular, standing leaf mass (leaf biomass per unit area) was not constant during self-thinning across the same eight forest communities as the present study (Zhang et al. 2012); changes in light distribution in relation to canopy structure, particularly canopy depth and leaf clustering, may account for the increase in leaf biomass during self-thinning (Blake et al. 1991; Holdaway, Allen, Clinton,
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Fig. 2. Scaling relationships between net primary productivity (NPP, kg/ha/yr) and density (N, trees/ha) across eight forest types as estimated by 95% quantile regression. (A) BTLF, (B) BAPF, (C) TPTF, (D) TSPF, (E) SPPF, (F) SPMF, (G) SMPF, (H) SCLF.
Davis, & Coomes 2008; Pan et al. 2013). As forest growth and efficiency is significantly related to canopy leaf area (Waring 1983), changes in standing leaf biomass–density relationship may lead to no universal net primary productivity–biomass (or density) relationships.
Because the universal scaling of metabolic scaling theory (MST) has provoked much criticism (Russo et al. 2007; Bai et al. 2010), Enquist and colleagues have started to refer to their slopes as “basins of attraction” (Price, Enquist, & Savage 2007) rather than as an invariant or canonical value.
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Fig. 3. The NPP–M and NPP–N slopes for eight forest types, as estimated by 95% quantile regression (QR), ordinary least square (OLS) and standardized major axis (SMA) regression; bars show 95% CIs of QR regression. BTLF, BAPF, TPTF, TSPF, SPPF, SPMF, SMPF, SCLF are defined as in Table 1.
In our study, although some plots clearly deviated from the ‘energetic equivalence rule’ (e.g. SCLF), others were remarkably close (Table 2, Fig. 3, see Appendix A: Tables S1 and S2). For instance, the NPP–M exponents of QR regression in TPTF and BAPF were 0.033 and 0.054, respectively; the confidence intervals of NPP–M exponents in both TPTF and TSPF included 0.004 (Table 2, Fig. 3). In fact, the significant difference is almost certainly related to their larger sample size, as statistical power increases with sample size (Coomes & Allen 2009). This has led to a distinction being recognized between statistical significance and biological significance; large sample sizes can result in small, biologically unimportant effects being declared statistically significant (Quinn & Keough 2002). So in addition to the statistical tests for the value of the exponents, thought needs to be given to what degree of variation from a zero slope would be considered biologically important. The densities (or biomass) of the eight forest types range over approximately 1.5–2 orders of magnitude in our study; a true slope of 0.004 for example would mean that the NPP value at the lower end of this range would be approximately 98% (i.e. 1/100.004 × 1.5 or 1/100.004 × 2 ) of that at the higher end of the range of density (or biomass), a difference that could be considered biologically unimportant. A debate about what level of deviation from a slope of 0 would be considered biologically unimportant/important would assist in determining the usefulness of statistical tests. As an example, for slopes of 0 ± 0.02, the lower value of NPP would be 91–93% of the higher NPP value for the range of densities (or biomass) found in our study.
It appears that 0 is a “basin of attraction” for NPP–M (N) relationships in 7 out of 8 forest types (SCLF showed the greatest deviation), with slopes showing scatter over a small range around 0 (Fig. 3). Only 6 out of 54 exponents (i.e. NPP–M (QR), NPP–M (OLS) and NPP–N (OLS) slopes of SPPF, NPP–M (OLS) slope of SPMF, NPP–N (QR) slope of BAPF and TSPF) fell within the range −0.02 to +0.02 (Table 2, Fig. 3, see Appendix A: Tables S1 and S2), however, if this was accepted for example as the bounds of biologically unimportant deviation from 0. This exercise shows the difficulties in distinguishing between sampling variability of sample slopes around an underlying population slope of 0, and real deviations from a 0 slope that are small, but biologically important.
Factors related to deviation from the ‘energetic equivalence rule’ Deviation from the “energetic equivalence” possibly reflects multiple, unsound assumptions for “an average idealized forest” by metabolic scaling theory (MST), as well as unaccounted-for variations of site factors within forest communities (Deng et al. 2006; Russo et al. 2007; Dai et al. 2009). According to the predictions of metabolic scaling theory (MST) and the ‘energetic equivalence rule’, net primary production (NPP) remains constant during stand development (Enquist, West, & Brown 2009). However, it is well known that NPP typically changes with stand age; it reaches a peak (approximately at the time of canopy closure) and then
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declines as forests mature (Ryan, Binkley, & Fownes 1997). Age-related decline in forest productivity is often explained by the water/nutrient limitation hypothesis that large trees are unable to supply adequate resources to maintain productivity (Ryan et al. 1997; Coomes et al. 2012). These changes are contrary to the predictions of metabolic scaling theory (MST) and the ‘energetic equivalence rule’. Furthermore, a larger percentage of biomass is allocated to metabolic tissue (e.g. leaf) in trees seedlings; in contrast, most of the biomass is allocated to non-metabolic tissue (e.g. heartwood) in large trees (Mori et al. 2010), the transition from seedlings to large trees also indicates that forest age plays an important role in forest energy structure. The age class and stand development of forest communities are species-specific and depend on site quality (Luo 1996). For example, self-thinning even occurs in an overcrowded population of tree seedlings of hinoki (Chamaecyparis obtusa (Sieb. et Zucc.) Endl.) (Ogawa 2001). In our study, although mean forest age of 5 out of the 8 forest types are under 50 yrs, and two are under 30 yrs (i.e. SPMF and SCLF), mean and median forest age of the 8 forest types are close to or within the range of near-mature/mature forests (Table 1, see Appendix A: Fig. S1), according to the national forest resource inventory conducted by China’s State Forestry Administration (SFA) (Luo 1996). For instance, Pinus massoniana and C. lanceolata are widely planted in the subtropical area of China because of their fast growth (reach maturity at a relatively young age), high productivity, short rotation and high quality wood (Tao, Feng, & Ma 2011); therefore, subtropical Pinus massoniana forest (SPMF) and subtropical C. lanceolata forest (SCLF) are mainly composed of “fast-growing and high-yield plantations” (Luo 1996; Tao et al. 2011), consistently, some plots in SCLF and SPMF showed greatest NPP when all data was combined (see Appendix A: Fig. S4). More importantly, we used upper boundary to estimate NPP–M (N) lines, which can ultimately eliminate the bias of young forests. In addition, QR showed that NPP–M (N) slopes in SCLF were different from 0, while those in SPMF were indistinguishable from 0 (Table 2). Therefore, the strongest deviation from energetic equivalence of 23-year-old SCLF (but not 29-year-old SPMF) cannot all be attributed to young stand age. In addition to stand age, much of the variation in forest NPP and energy structure may be the result of site conditions not considered in the analyses, such as temperature, rainfall, standing biomass, disturbance or phylogeny within the forest type (Ni et al. 2001; Luo et al. 2002; Russo et al. 2007; Coomes et al. 2012). However, these effects were hard to detect since all these factors were mixed together. Recent literature indicates that the scaling exponents of biomass–density (M–N) relationship vary with environmental gradients, e.g., soil salinity (Zhang et al. 2010) and water availability (Deng et al. 2006; Dai et al. 2009; Bai et al. 2010). In contrast, most NPP–M (N) exponents were not correlated significantly with site factors (e.g. altitude and age in Table 1, P > 0.05). Our study used data only from benign
9
environments, the range of climate and exponent variations were thus too narrow to detect a correlation. Though it is still unclear whether the deviations of NPP–M (N) slopes cooccur with site conditions and forest age, these factors should be taken into account when testing the ‘energetic equivalence rule’ across a large number of stands. In addition, resource division among forest types (i.e. different intercepts in Fig. S4) may result in the deviation from 0 at the national scale of China (White et al. 2007). Different forest types often consume different soil and/or light resources, and are different in their size; therefore, it is difficult to ensure that such distinct stands converge to a single boundary line when they are pooled.
Statistical methods and variability in estimating energy Statistical methods should be subject to strict scrutiny in estimating universal scaling relationships (Li et al. 2006). In our study, QR and OLS regression yielded qualitatively different results in some forest types (Table 2, see Appendix A: Tables S1 and S2). Using three regression (QR, OLS and SMA) methods, Li et al. (2006) also found qualitatively different biomass–density exponents in the same forest biome across tree-dominated communities. In addition, as a basis of the ‘energetic equivalence rule’, the self-thinning process (or biomass–density relationship) is usually deemed necessary to be quantified by fitting an upper boundary (Osawa & Allen 1993; Coomes et al. 2012), however, metabolic scaling theory (MST) always fits these relationships through the center of data (e.g. Enquist et al. 1998; Niklas et al. 2003), therefore, statistical methods underlying the basis of the ‘energetic equivalence rule’ are also questioned, which further undermined the validity and universality of the “energetic equivalence rule”. In our study, we used growth rate or NPP as an estimate of energy use, the best measure is presumably gross primary production (GPP) in plant communities. However, the ‘energetic equivalence rule’ may not hold universally for forests even if we used GPP, because GPP is closely related to total leaf area (Price et al. 2007), while tree leaf area index (LAI) does not remain constant during stand development in forest communities (Blake et al. 1991; Holdaway et al. 2008). Furthermore, the scaling exponents of GPP–M (N) may be different from those of NPP–M (N), since the ratio of GPP to NPP changes over ontogeny (becoming less favorable for older trees) (Mori et al. 2010). Future work is needed to test the ‘energetic equivalence rule’ using GPP as a variable.
Acknowledgements We thank Jacob Weiner, managing editor (Klaus Hövemeyer), Yan Chen and anonymous reviewers for their helpful comments and suggestions on this work. We also thank Dr.
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Tianxiang Luo (Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing, China) for providing helpful information to this work. This work was supported by Chinese Universities Scientific Fund (2014XJ031), National Natural Science Foundation of China (31330010), Zhejiang Provincial Natural Science Foundation of China (LZ13C030002) and Ministry of Science and Technology of the People’s Republic of China (2011AA100503).
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/ j.baae.2015.04.005.
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Please cite this article in press as: Zhang, W. -P., et al. Testing predictions of the energetic equivalence rule in forest communities. Basic and Applied Ecology (2015), http://dx.doi.org/10.1016/j.baae.2015.04.005