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Scaling relations based on the geometric and metabolic theories in woody plant species: A review
Perspectives in Plant Ecology, Evolution and Systematics 40 (2019) 125480
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Scaling relations based on the geometric and metabolic theories in woody plant species: A review
T
Kazuharu Ogawa Laboratory of Forest Ecology, Graduate School of Bioagricultural Sciences, Nagoya University, Nagoya 464-8601, Japan
ARTICLE INFO
ABSTRACT
Keywords: von Bertalanffy’s equation Geometric model Metabolic theory Self-thinning Translocation Woody plant species
Extensive studies by Enquist and colleagues have indicated that many characteristics of organisms vary with body size, as described by allometric equations of power form. There are two representative models in relation to a scaling exponent for self-thinning from the viewpoints of resource allocation and geometry. This study was performed to discuss the reasonability of the geometric basis by providing several experimental examples based on a geometric analysis of forest and seedling stands where self-thinning does or does not occur. The metabolic scaling is reevaluated based on observations for forest trees where self-thinning does or does not occur, as well as seedlings. This metabolic scaling leads to a generalization of von Bertalanffy’s model in view of not only plant growth, but also fruit growth. Although a scaling exponent of 3/4 for respiratory scaling is supported by the West/Brown/Enquist (WBE), proofs have shown that the scaling exponent for metabolic rates should be 2/3 according to dimension analysis. In addition, the value of respiratory scaling exponent was considered with a diagrammatic representation in which the forest trees become giant. This study shows that the scaling exponent of 3/4 for the metabolic scaling relationships is not common for forest trees.
1. Introduction
Y = Y0 M 4
Extensive studies by Enquist and colleagues (e.g., West et al., 1997; Enquist et al., 1998; Brown and West, 2000) have indicated that many characteristics of organisms vary with body size, as described by allometric equations of the form:
They then argued that plants grow until they are limited by resources, so that the rate of resource use by plants approximates the rate of resource supply, and so if Q is the rate of resource supply per unit area and ρ is the maximum allowable density of plants, then:
Y = Y0 M b
3
(1)
where Y is a dependent variable, Y0 is a normalization constant, M is an independent variable (typically body mass), and b is the scaling exponent. The West/Brown/Enquist (WBE) theory, i.e., a metabolic scaling theory, provides a general model to explain why many anatomical and physiological scaling exponents of both plants and animals scale as quarter-powers of mass, e.g., b = 3/4 for metabolic rate, gross photosynthetic rate, resource use, and total leaf mass, –3/4 for population density, and 1/4 for lifespan. According to Begon et al. (2006), there are two representative models, including the WBE theory, for self-thinning in relation to the scaling exponent b in Eq. (1) as follows: (1) Resource allocation basis The WBE theory suggests that the rate of resource use per individual, Y, should be related to mean plant mass, M, according to the following equation (cf. Brown and West, 2000; Begon et al., 2006):
(2)
(3)
Q= Y or, from Eq. (2): 3
Q = Y0 M 4
(4)
At equilibrium, when the rate of resource use approximates rates of resource supply, Q should be constant. Hence, Eq. (4) leads to the following self-thinning function in which the expected value of the selfthinning scaling exponent is –4/3:
M = K1
4 3
(5)
where K1 is a constant. (2) Geometric basis Geometric analysis (Yoda et al., 1963) and the dimensional rule (Miyanishi et al., 1979) have both assigned a value of 3/2 to the selfthinning scaling exponent when the average ground area occupied by a plant is denoted by s:
E-mail address: [email protected]. https://doi.org/10.1016/j.ppees.2019.125480 Received 6 February 2019; Received in revised form 1 July 2019; Accepted 2 August 2019 Available online 26 August 2019 1433-8319/ © 2019 Elsevier GmbH. All rights reserved.
Perspectives in Plant Ecology, Evolution and Systematics 40 (2019) 125480
K. Ogawa
s=
2006).
1 (6)
2.2. Regression analysis
If plants at any growth stage are assumed to be always geometrically similar in shape:
s
2
Bivariate relationships, such as the Y–M (Eqs. (1), (10), (14), and (15)) relationships, were analyzed using standardized major axis (SMA) regression or model II regression, with the R package SMATR (version 3.5.2; R Development Core Team, 2018). SMA regression is more suitable than model I regression or ordinary least squares (OLS) regression for examining the scaling relationship between two variables that are both measured with errors (Warton et al., 2006). In the analysis of power (i.e., scaling) exponents, the statistical significance of differences was based on the 95% confidence interval (CI) of the exponents. On the other hand, the following nonlinear equation was fitted to the data on mean seedling height (Y) and mean shoot mass (M):
(7)
M3
Eqs. (6) and (7) lead to:
M = K2
3 2
(8)
where K2 is a constant, and the self-thinning scaling exponent has been reported in various studies as having a value of approximately –3/2 for a wide range of species (e.g., White, 1980; Westoby, 1984; Zeide, 1987; Ducey, 2012). Having reached its maximum density, a stand is considered to follow the –3/2 power law of self-thinning (Yoda et al., 1963; White and Harper, 1970), otherwise known as the self-thinning rule (Westoby, 1984). The WBE theory (3/4 scaling exponent) has been confirmed emirically by more recent studies (Brown et al., 2004; Enquist et al., 2007; Price et al., 2007; Duursma et al., 2010). Since Ferguson and Archibald (2002) confirmed the WBE theory rather than the allometric scaling (2/3 exponent), Zeide (2010) argued cautiously againt a constant scaling exponent. Condés et al. (2017) and Kim et al. (2017) also argued that the allometric scaling in general needs to be differentiated by species, which is much in the line of Pretzsch et al. (2013), or environment. Initial density also seems to play a role in biomass-density and allometric scaling relations (Li et al., 2013). There are indeed recent studies supporting the 3/2 self-thinning rule (Ducey, 2012). Reich et al. (2006) argued that the respiratory scaling exponent of Eq. (1) is close to unity regarding small trees or specific shading strategies (Cheng et al., 2010; Pretzsch et al., 2013; Pretzsch and Dieler, 2012). This rule that supports the argumentation in Reich et al. (2006) deviates from 3/4 scaling exponent of the WBE theory. It is thus widely accepted that the scaling exponent is not universally valid (Coomes and Allen, 2007, 2009; Muller-Landau et al., 2006), and there is controversy regarding the degree of generality in the self-thinning law (cf. Brown and West, 2000; Begon et al., 2006). Therefore, this study aims to demonstrate the validities of metabolic and geometric scaling theories by providing the published results of a research group at Nagoya University. On the basis of the published results, the appropriateness of the geometric basis is discussed for the seedling growth of hinoki cypress [Chamaecyparis obtusa (Sieb. et Zucc.) Endl.] stands where self-thinning does or does not occur (Ogawa, 2009b). In addition, we reevaluated the metabolic scaling of respiration and gross photosynthesis based on the observed results of forest trees where self-thinning does (Ninomiya and Hozumi, 1983a, 1983b; Hagihara and Hozumi, 1986; Yokota and Hagihara, 1998) or does not occur (Ninomiya and Hozumi, 1981; Yokota et al., 1994), as well as seedlings (Ogawa et al., 1985a, b; Ogawa, 1989). This metabolic scaling leads to a generalization of von Bertalanffy’s (1949) model from the viewpoint of not only plant growth (Hozumi, 1985), but also fruit growth (Ogawa, 2009a).
1 1 1 = + Y Y1 M b1 Y2 M b2
(9)
The fitting by Eq. (9) was performed using the software KaleidaGraph (v. 4.1.2; Synergy Software, Reading, PA), which is based on the Levenberg–Marquardt algorithm (Press et al., 1992), and the coefficient of determination (R2) was used to test for goodness of fit. 3. Results 3.1. Allometric scaling For the seasonal growth of a dense stand of C. obtusa seedling undergoing self-thinning, Ogawa (2009b) proposed that the allometric scaling relationship between mean seedling height Y and mean shoot mass M is expressed as a mixed-power function given by Eq. (9), which captures the transition between two simple power functions (Ogawa and Kira, 1977; Shinozaki, 1979). In Eq. (9), the values of coefficients Y1 and Y2 were estimated to be 1711 cmg b1 and 35.29 cmg b2 , respectively, and the values of scaling exponents b1 and b2 were estimated to be 3.954 and 0.3249, respectively (Fig. 1). The curves in Eq. (9) gradually approaches the two allometries of Y = Y1 M b1 and Y = Y2 M b2 for small and large values of M. Because the 1
estimated value of b2 is close to 1/3, Y is proportional to M 3 , i.e., growth in the height direction is isotropic with increasing shoot mass (Ogawa, 2009b), from the viewpoints of the geometric (Yoda et al., 1963) or dimensional (Miyanishi et al., 1979) rule. At the growth stage where the estimated value of b2 in Eq. (9) is close to 1/3, the selfthinning scaling exponent (Eq. (8)) is determined to be –3/2 (Ogawa, 2009b), indicating Yoda’s geometric self-thinning law rather than the 4/3 resource-based thinning law (Brown and West, 2000) as can be seen in Fig. 2 (Ogawa, 2009b). In the current-year C. obtusa seedling stand (1989), with full open spaces where self-thinning did not occur, the allometric scaling relationship of mean seedling height Y to mean shoot mass M were fitted by a simple power function of Eq. (1), where Y0 and b were estimated to be 25.17 cmg b and 0.3972 with a 95%CI of 0.3747 to 0.4211 (R2 = 0.988, P = 2.22×10−16), respectively (Fig. 3). The value of the allometric scaling exponent b was close to 1/3 (Ogawa, 2009b), indicating the reasonability of the geometric (Yoda et al., 1963) or dimensional (Miyanishi et al., 1979) rule.
2. Materials and methods 2.1. Study materials This study introduces the published results on the scaling relations for five woody species of Pinus densi-thunbergii Uyeki, Chamaecyparis obtusa (Sieb. et Zucc.) Endl., Cryptomeria japonica (L.f.) D. Don, Larix leptolepis Gordon and Durio zibethinus Murray performed by a research group at Nagoya University (Table 1), which was not cited by the group of Enquist and colleagues. For C. obtusa seedlings (Ogawa et al., 1985b, 2009b) and D. zibethinus fruits (Ogawa, 2009a), linear regression analysis was performed using standardized major axis (SMA) regression (Warton et al.,
3.2. Metabolic scaling 3.2.1. Respiration Mori et al. (2010) also empirically reported a mixed-power equation for expressing the respiratory scaling relation from seedlings to giant trees using Eq. (9), in which the values of the scaling exponents b1 and b2 for aboveground parts were estimated to be 1.082 and 0.780, respectively. 2
Perspectives in Plant Ecology, Evolution and Systematics 40 (2019) 125480
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Table 1 Information about the present scaling study. 95% confidence intervals of scaling exponent are shown in parentheses. Locality
Species
Nagoya, Japan Inabu, Japan Inabu, Japan Minokamo, Japan Minokamo, Japan
Pinus densi-thunbergii trees Chamaecyparis obtusa trees Chamaecyparis obtusa trees Chamaecyparis obtusa seedlings Chamaecyparis obtusa seedlings
Nagoya, Japan Nagoya, Japan Nagoya, Japan Inabu, Japan Inabu, Japan Inabu, Japan Inabu, Japan Serdang, Malaysia
Chamaecyparis obtusa trees Chamaecyparis obtusa trees Chamaecyparis obtusa trees Chamaecyparis obtusa trees Larix leptolepis trees Cryptomeria japonica trees a single Cryptomeria japonica tree Durio zibethinus fruits
Equation
Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq.
(1) (10) (1) (1) (1) (1) (1) (1) (1) (9) (1) (10) (10) (10) (1) (1) (1) (11) (11) (12) (12) (12)
General features of stands Age [yrs]
Density [ha
7–11 24$ 25$ 2 0 1$ 8 17 18 25$ 14–18 14–17 12–20 –
6400 7378 7252 620000 13820000 8760000 10400 9091 10000 7274 2654 2160 2158 –
Scaling relationships Dependent variable
Scaling exponent
Respiration Respiration Gross photosynthesis Respiration Death GPP GPP CLE Height Height Respiration Gross photosynthesis Gross photosynthesis Respiration Litterfall Litterfall Litterfall Litterfall NPP Respiration Gross photosynthesis Translocation
1.03 0.634 1.84 (1.75, 1.93) 1.034 (1.019, 1.049) 1.680 (1.434, 1.967) 1.131 (1.090, 1.173)# 1.130 (1.104, 1.184) −0.1616 (−0.1351, −0.1930) 0.3972 (0.3747, 0.4211) b1=3.954, b2=0.3249 0.986 0.529# 0.602 0.667 – 0.788 1.01 0.86–1.04 1.20# 2.16 0.594 0.715 (0.461, 1.111) 0.697 (0.393, 1.234) 0.693 (0.590, 0.814)
Mean height [m]
Mean DBH [cm]
3.25 8.13 8.36 0.519 0.176 0.430 3.35 5.10 5.10 8.36 6.88 8.37 6.84 8.8&
3.75 7.67 7.84 0.636* – 0.198* 3.93¥ 6.23¥ 6.56¥ 7.84 7.48 11.43 11.86 25.0&
−1
]
n
References
4 5 136 80 80 80 80 80 17 12 5 5 5 5 5 10 5 1 1 5 5 5
Ninomiya and Hozumi, 1981 Ninomiya and Hozumi, 1983a, 1983b Hagihara and Hozumi, 1986 Ogawa et al., 1985b
Ogawa, 1989 Ogawa 2009b Yokota et al., 1994 Yokota and Hagihara, 1996 Yokota and Hagihara, Miyaura and Hozumi, Miyaura and Hozumi, Miyaura and Hozumi, Miyaura and Hozumi,
1998 1988 1988 1993 1993
Ogawa, 2009a
$
Self-thinning stand. Diameter at 1 m above ground. * Diameter at crown base. & values for a sample tree. # Leaf dry mass is adopted as an independent variable. ¥
2-year-old (Ogawa et al., 1985a; Ogawa, 1989) and young trees of P. densithunbergii Uyeki (7–11 years old) (Ninomiya and Hozumi, 1981) and C. obtusa (8 years old) (Yokota et al., 1994), as well as several saplings (Reich et al., 2006). However, Ogawa et al. (1985b) reported that the scaling exponent of the annual respiration rate to mean total mass of 2-year-old C. obtusa was significantly greater than unity (1.034 with 95%CI of 1.019–1.049) by measuring respiration with an open system chamber and an infra-red gas analyzer (URAS-1; Hartmann and Braun, Frankfurt, Germany), indicating that respiratory activity increases as the seedling becomes larger in size (Fig. 4). A respiratory scaling exponent of 3/4 at the later growth stage was recently supported by the WBE or metabolic scaling theories (West et al., 1997; Enquist et al., 1998; Brown and West, 2000). In contrast, Ninomiya and Hozumi (1983a, b) proposed a general scaling relation of aboveground respiration rate Y to aboveground mass M in a self-thinning C. obtusa forest stand as follows:
Fig. 1. Relationship between mean seedling height Y and mean shoot mass M for an overcrowded population of 1-year-old Chamaecyparis obtusa seedlings (Ogawa, 2009b). The regression curve is given by Eq. (9) (R2 = 0.988).
In Mori et al. (2010), the respiration rate was proportional to the plant mass during the early growth stage, and was proportional to 3/4 of the plant mass during the later growth stage. The former phenomenon during the early growth stage has also been observed in current to
Y = Y0 (M
Mmin )b
(10)
where Y0 and b are constants, and Mmin is a critical value for aboveground mass, below which trees may not be able to survive in the 3
Perspectives in Plant Ecology, Evolution and Systematics 40 (2019) 125480
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Fig. 4. Relationship between annual respiration Y and mean seedling mass M for 80 seedlings of 2-year-old Chamaecyparis obtusa seedlings (Ogawa et al., 1985b). The linear line shows the regression on log–log coordinates: Y = 3.950M1.034 (R2 = 0.996, P = 2.22×10–16).
Fig. 2. Time trajectory of changes in mean shoot mass Y and stand density M for an overcrowded population of 1-year-old Chamaecyparis obtusa seedlings (Ogawa, 2009b). The curves show the regression: Y = KM (R2 = 0.926).
1.515
(1 ( ) )
0.0741 0.0105 M 1504.1
Fig. 3. Relationship between mean seedling height Y and mean shoot mass M for current-year Chamaecyparis obtusa seedlings under full open space growth conditions (Ogawa, 2009b). For details of the regression line, see the text.
stand. The biological meaning of Eq. (10) is that as M approaches Mmin, the respiration rate drops abruptly, while the relationship satisfies the simple scaling relation of Eq. (1) where M is sufficiently larger than Mmin (Fig. 5). Ninomiya and Hozumi (1983b) concluded that the respiration rate of individual trees is nearly proportional to their surface area, where M is sufficiently larger than Mmin, because the scaling exponent b of Eq. (10) is close to 2/3. Deriviation of proportionality between the respiration rate and the surface area of a plant from the b exponent of Eq. (10) is caused by the dimensional analysis (Miyanishi et al., 1979) that the mass and surface area are respectively based on three- and two-dimensional model. Yokota and Hagihara (1998) also reported that the scaling exponent b of Eq. (10) ranged from 0.667 to 0.788 over three years, rather indicating that it is somewhere between 2/3 and 3/4.
Fig. 5. Conceptual diagram of the general scaling relation of Y to M in Eq. (10). Mmin is a critical value for tree mass, below which the trees may not be able to survive in the forest stand.
tended to increase with increasing seedling size, because the scaling exponent b is significantly greater than unity. In addition, the scaling of photosynthetic metabolism with whole seedling dry mass, instead of leaf dry mass, can also be approximated by Eq. (1), where Y0 and b were calculated to be 4.215 g1–b yr–1 and 1.130 with 95%CI (1.104 to 1.184), respectively (R2 = 0.976, P = 2.22×10–16), indicating that the scaling exponent b was significantly different from unity (Fig. 6). A similar result was reported for 136 C. obtusa trees, excluding suppressed individuals, by Hagihara and Hozumi (1986).
3.2.2. Gross photosynthesis By estimating the annual rate of gross primary production (GPP, i.e., gross photosynthesis) of 80 C. obtusa seedlings by the summation method (Ogawa, 1977; Waring et al., 1998), Ogawa et al. (1985b) suggested that the scaling relationship between GPP and leaf dry mass is expressed as Eq. (1), in which Y0 and b were calculated to be 8.628 g1–b yr–1 and 1.131 with a 95%CI of 1.090 to 1.173, respectively. They concluded that the photosynthetic activity per unit leaf dry mass 4
Perspectives in Plant Ecology, Evolution and Systematics 40 (2019) 125480
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Fig. 8. Relationship between annual attached dead leaves Y and mean seedling mass M for 80 seedlings of 2-year-old Chamaecyparis obtusa seedlings (Ogawa et al., 1985b). The linear line shows the regression on log–log coordinates: Y = 0.0156M1.680 (R2 = 0.506, P = 1.45×10–13).
Fig. 6. Relationship between annual gross primary production (GPP) Y and mean seedling mass M for 80 seedlings of 2-year-old Chamaecyparis obtusa seedlings (Ogawa et al., 1985b). For details of the regression line, see the text.
3.3. von Bertalanffy’s equation
For a self-thinning plantation of C. obtusa (Yokota and Hagihara, 1996), the scaling of gross photosynthesis with the aboveground mass was well described by a general power function, with the critical variable Mmin in Eq. (11) (cf. Fig. 5) proposed by Ninomiya and Hozumi (1983a, b). The scaling exponent b for annual gross photosynthesis was calculated to be 0.602 by Yokota and Hagihara (1996), suggesting that the photosynthetic scaling exponent b is close to 2/3 rather than 3/4.
Hozumi (1985) generalized von Bertalanffy’s model (von Bertalanffy, 1949) from the viewpoint of plant growth as follows:
(11) where v1, v2, and v3 represent the velocity of gross photosynthesis (or GPP), respiration, and litterfall, respectively. Miyaura and Hozumi (1988) demonstrated the scaling relationship between litterfall rate v3 and the aboveground mass M in Larix leptolepis Gordon and C. obtusa trees, where the scaling exponent b3 was 0.86–1.04 for L. leptolepis and 1.01 for C. obtusa. Ogawa et al. (1985b) also reported the scaling relation of the death rate of leaves v3, which was estimated by the monthly clipping of new dead leaves attached to 80 sample seedlings over one year, to the whole seedling mass M in a C. obtusa seedling population (Fig. 8), where the scaling exponent b3 was 1.680 with 95%CI (1.434–1.967). Because the rates of gross photosynthesis v1 and respiration v2 scale as M b1 and M b2 where M is sufficiently larger than Mmin in Eq. (10) (cf. Fig. 5), these three lines of experimental evidence for gross photosynthesis, respiration, and litterfall support the generalization of von Bertalanffy’s model in Eq. (11) proposed by Hozumi (1985). Miyaura and Hozumi (1993) confirmed experimentally that annual aboveground growth v in Eq. (11) is given by the sum of the rates of net production and death (monthly new dead leaves and branches) by monitoring the growth of a single C. japonica D. Don over six years. Ogawa (2009a) reported that the fruit growth rate v can be expressed as scaling rules considering translocation (dTr dt ) into the fruit from other plant organs by measuring the growth and CO2 exchange rates, such as gross photosynthesis (dP dt ) and respiration (dR dt ) of the tropical fruit, D. zibethinus Murray, as follows:
3.2.3. Carbon loss efficiency The present study defines the carbon loss efficiency (CLE) as the ratio of the annual respiration loss (R) to annual gross primary production (GPP). The CLE is equal to the difference between unity and the carbon use efficiency CUE, which corresponds to the primary production efficiency (Kira, 1977). Ogawa et al. (1985b) reported that the scaling relation of CLE to the mean whole seedling mass (M) of 2-year-old C. obtusa over a whole year is negatively correlated (Fig. 7, scaling exponent (b) = –0.1616, with a 95%CI of –0.1930 to –0.1351). This indicates that the larger the seedling mass is, the lower the CLE becomes.
(12) where Y4 and b4, Y1 and b1, and Y2 and b2 were determined to be 0.313 g1 b4 day 1 and 0.693 with 95%CI of 0.590 to 0.814 (R2 = 0.992, P = 0.000291), 0.0387 g1 b1 day 1 and 0.697 with 95%CI of 0.393–1.234 (R2 = 0.811, P = 0.0143), and 0.0217 g1 b2 day 1 and
Fig. 7. Relationship between annual carbon loss efficiency (CLE) Y and mean seedling mass M for 80 seedlings of 2-year-old Chamaecyparis obtusa seedlings (Ogawa et al., 1985b). The linear line shows the regression on log–log coordinates: Y = 0.0232M 0.162 (R2 = 0.369, P = 2.28×10–9). 5
Perspectives in Plant Ecology, Evolution and Systematics 40 (2019) 125480
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Fig. 10. Diagrammatic representation on log–log coordinates for the respiratory scaling exponent (b in Eq. 1) values of 2/3 and 3/4 after the canopy fully closes where the forest trees become giant.
rate is low in birds and mammals (cf. Fig. 2, Brown and West, 2000). This difference is mainly due to the temperature effect, because Q10 values change seasonally (Ninomiya and Hozumi, 1983a, b; Ogawa, 1989; Ogawa et al., 1985a) in woody plant species. 3) The degree of data scattering for respiration also varies among species as indicated by the measurement results of leaf dark respiration in woody plant species as summarized by Larcher (2003). 4) Considering these reasons for respiration data scattering, the biological significance of the fitted equations, such as Reich et al. (2006) and Mori et al. (2010), is unclear although we can easily comprehend the comparison of scaling relations of fruit CO2 gas exchange between Cinnamomum camphora (Ogawa and Takano, 1997) growing in temperate regions and D. zibethinus (Ogawa et al., 1996) growing in tropical regions. 5) Mori et al. (2013) and Mori (2014) reported that a time interval of 100 s is sufficient for determining the time trend of the CO2 concentration within the respiration chamber. However, Ogawa (1989) measured the respiration rate of current-year C. obtusa seedlings over 26 h by gas chromatography (GC-9A; Shimadzu, Kyoto, Japan) and with gastight syringes (A-Type; Dynatech Precision Sampling Corp., Baton Rouge, LA) according to the enclosed-chamber method for monitoring respiration of the wolf spider (Pardosa astrigera (L. Koch)) developed by Tanaka and Saito (1984). After measuring the respiration of C. obtusa seedlings, Ogawa (1989) showed an experimental example of the time trend of specific respiration rate, where the specific respiration rate tended to decrease continuously immediately after being enclosed in the respiration chamber
Fig. 9. Relationship between net translocation rate (dTr/dt), net respiration rate (dP/dt), and nighttime respiration rate (dR/dt) and fruit dry mass M in tropical fruits of Durio zibethinus (Ogawa, 2009a). For details of the regression line, see the text.
0.715 with 95%CI of 0.461–1.111 (R2 = 0.893, P = 0.00448), respectively (Fig. 9). 4. Discussion 4.1. Methodological aspects for respiration In comparison with Ninomiya and Hozumi (1981, 1983a), Yokota et al. (1994), and Yokota and Hagihara (1996, 1998), the methodological aspects for respiration measurement by Mori et al. (2010) were not presented in detail, and therefore the present study considered the respiratory scaling exponent b in Eq. (1), with a diagrammatical representation after the forest canopy had fully closed and where the forest trees became giant. Here, we consider the forest trees at the growth stage near giant trees, and list several problems: 1) The regression line with the greater value of b as x approaches infinity yields a higher respiration rate than the regression line with the lower value of b. Therefore, when respiration is measured for a plant with a mass approaching that of, for example, an exceptionally large tree, the respiration rate r2 (b = 3/4) is greater than r1 (b = 2/3) (Fig. 10). This difference between r1 and r2 may indicate that the measured respiration rate of an exceptionally large tree is overestimated by the assumption that b = 3/4, due to a high CO2 concentration sampled within the chamber, possibly caused by increased wound respiration from the excision of plant organs (Sprugel and Benecke, 1991), if CO2 is diffused sufficiently throughout the chamber. 2) The degree of data scattering for respiration is high (around an order of 102) at a given tree mass in woody plant species (Fig. 2, Mori et al., 2010), while the degree of data scattering for the metabolic
Fig. 11. Time trend of specific respiration rate of current-year Chamaecyparis obtusa seedlings after enclosing the respiration chamber by gas chromatography (Ogawa, 1989). 6
Perspectives in Plant Ecology, Evolution and Systematics 40 (2019) 125480
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(Fig. 11). This could be the issue of CO2 diffusion out of the chamber. This issue may be related to all enclosed-chamber measurements of respiration, not specifically related to trees. Therefore, the enclosed-chamber method may not be useful for measuring the respiration of woody plant species.
range of values of the respiratory scaling exponent b in Eq. (10) by Yokota and Hagihara (1998). However, the scaling exponent b of Eq. (1) was significantly greater than unity (Ogawa et al., 1985b) in a C. obtusa seedling population where self-thinning did not occur, and the leaf area index (LAI) ranged from 1.54 to 5.16 m2 m–2 (Ogawa et al., 1986). According to Tadaki (1977), the LAI ranges between 5 and 7 ha ha–1 in C. obtusa plantations with closed canopies. Therefore, it is noteworthy that the final value (5.16 m2 m–2) of LAI in the seedling population was comparable to the values of mature forests. Considering the seedling growth conditions from LAI values, the competition for light among individuals is thought to be high, and so larger seedlings show higher rates of photosynthesis, and vice versa. In conclusion, the allometric scaling relation of mean seedling height and mean shoot mass (Figs. 1 and 3) supports the geometric rule the same as the 3/2 self-thinning law (Fig. 2). However, it should be noted that the allometric scaling varied among densities, because initial density affects the allometric scaling exponent between mean height and mean shoot mass (Li et al., 2013). West et al. (1997, 2009) presented a general explanation of allometric scaling with exponents to be drived from 1/4 based on the fractal nature of transportation network systems that supply the metabolism in organisms. On the contrary, the metabolic scaling relation such as respiration cannot be explained only from the geometrical basis, because the respiratory metabolic scaling showed the exponent value of nearly 3/4 in some cases. Considering the argumentation of Reich et al. (2006); Enquist et al. (2007) made some modification to their original WBE model and restated that scaling exponents are close to 1.0 for seedlings owing to the violation of WBE assumptions in seedlings and shift to three-quarters in larger plants. However, the results of annual respiration (Fig. 4) and gross photosynthesis (Fig. 6) for seedlings shows the scaling exponents higher than 1.0. This empirical evidence also indicates that there is no universal metabolic scaling in plants (Coomes and Allen, 2007, 2009; Muller-Landau et al., 2006) and the scaling exponent of 3/4 for the metabolic scaling relationships is not common for forest trees (Cheng et al., 2010).
4.2. von Bertalanffy’s equation von Bertalanffy (1949) first studied the case with b1 = 2/3 and b2 = 1 without the term v3 in Eq. (11), whereas the equation considered by West et al. (2001) is a special case with b1 = 3/4 and b2 = 1. The present study empirically supports b1 = b2 = 2/3, indicating the appropriateness of the geometric (Yoda et al., 1963) and dimensional (Miyanishi et al., 1979) rules. However, it should be noted that the case of the seedling stage, where b1 and b2 is greater than unity, especially for photosynthesis, is different from adult trees. In the generalized von Bertalanffy growth model of Eq. (11) proposed by Hozumi (1985), an additional term of death (or litterfall) rate v3 has a value of the scaling exponent b3 close to unity at the growth stage of adult trees (Miyaura and Hozumi, 1988). However, the value of the scaling exponent b3 is significantly greater than unity at the early growth stage of seedlings (Ogawa et al., 1985b), although the observed values showed deviations from the regression (Fig. 8, R2 = 0.506, P = 0.000). In another generalized von Bertalanffy model Eq. (12) of a fruit growth proposed by Ogawa (2009a), the values of three scaling exponents (b1 and b4) were close to 2/3 rather than 3/4, while the value of b2 for respiration scaling was closer to 3/4 than to 2/3. This also indicates the appropriateness of the geometric (Yoda et al., 1963) and dimensional (Miyanishi et al., 1979) rules for translocation and photosynthesis scaling. 4.3. Allometric and metabolic scaling The results presented here indicated the applicability of a mixedpower scaling equation to the allometric scaling relationship between seedling height and mass, which is the same as the respiratory scaling relations of seedlings to giant trees proposed by Mori et al. (2010). However, it should be noted that the values of the scaling exponent are different between the two scaling relations during the later stages of growth. That is, the present study had a scaling exponent value of ca. 1/3, while that reported by Mori et al. (2010) was 3/4. Mori et al. (2010) based the value of the scaling exponent on the WBE or metabolic theories (West et al., 1997; Enquist et al., 1998; Brown and West, 2000). In contrast, the present study used the geometric or dimensional analysis. The scaling exponent value of 0.3249 reported by Ogawa (2009b) is close to 1/3. This biological phenomenon can be explained based on geometrical considerations. That is, the seedling height is a one-dimensional model, and the seedling mass is a three-dimensional model. The allometric scaling of seedling height and mass can be explained based on geometric considerations, although West et al. (1997) presented a general explanation of allometric scaling with exponents to be derived from 1/4 based on the fractal nature of the transportation network systems that supply the metabolism in organisms. According to the experimental observations of Ninomiya and Hozumi (1983b), the respiration rate is proportional to the surface area of self-thinning forest trees because the scaling exponent of Eq. (10) was estimated to be nearly 2/3. The same tendency was observed by Yokota and Hagihara (1996) for the scaling of gross photosynthesis with the aboveground mass in self-thinning forest trees. Therefore, our observations support not only the allometric scaling equation (Eq. (9)), but also the respiratory and photosynthetic scaling equation (Eq. (10)) from the viewpoints of the geometric (Yoda et al., 1963) or dimensional (Miyanishi et al., 1979) rule, in addition to the WBE theory (West et al., 1997; Enquist et al., 1998; Brown and West, 2000), which includes the
Declaration of Competing Interest None declared. Acknowledgements I thank the staff of the Nagoya Regional Forest Office and the Midorigaoka Nursery attached to the Gifu District Forest Office for providing the seedlings, and my colleagues for their assistance during field work. Thanks are also due to Daniel Falster for his helpful advice on the use of the freely available software program SMATR (Warton et al., 2006). References Begon, M., Townsend, C.R., Happer, J.L., 2006. Ecology, 4th ed. Blackwell, Oxford. von Bertalanffy, L., 1949. Problems of organic growth. Nature 163, 156–158. Brown, J.H., West, G.B. (Eds.), 2000. Scaling in Biology. Oxford University Press, New York. Brown, J.H., Gillooly, J.F., Allen, A.P., Savage, V.M., West, G.B., 2004. Toward a metabolic theory of ecology. Ecology 85, 1771–1789. Cheng, D.L., Li, T., Zhong, Q.L., Wang, G.X., 2010. Scaling relationship between tree respiration rates and biomass. Biol. Lett. 6, 715–717. Condés, S., Vallet, P., Bielak, K., Bravo-Oviedo, A., Coll, L., Ducey, M.J., Pach, M., Pretzsch, H., Sterba, H., Vayreda, J., Rio, M., 2017. Climate influences on the maximum size-density relationship in Scots pine (Pinus sylvestris L.) and European beech (Fagus sylvatica L.) stands. For. Ecol. Manage. 385, 295–307. Coomes, D.A., Allen, R.B., 2007. Effects of size, competition and altitude on tree growth. J. Ecol. 95, 1084–1097. Coomes, D.A., Allen, R.B., 2009. Testing the metabolic scaling theory of tree growth. J. Ecol. 97, 1369–1373. Ducey, M.J., 2012. Evergreenness and wood density predict height-diameter scaling in trees of the northeastern United States. For. Ecol. Manage. 279, 21–26.
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K. Ogawa Duursma, R.A., Mäkelä, A., Reid, D.E.B., Jokela, E.J., Porté, A.J., Roberts, S.D., 2010. Selfshading affects allometric scaling in trees. Funct. Ecol. 24, 723–730. Enquist, B.J., Brown, J.H., West, G.B., 1998. Allometric scaling of plant energetics and population density. Nature 395, 163–165. Enquist, B.J., Allen, A.P., Brown, J.H., Gillooly, J.F., 2007. Biological scaling: does the exception prove the rule? Nature 445, E9–E10. Ferguson, S.H., Archibald, D.J., 2002. The 3/4 power law in forest management: how to grow dead trees. For. Ecol. Manage. 169, 283–292. Hagihara, A., Hozumi, K., 1986. An estimation of the photosynthetic production of individual trees in a Chamaecyparis obtusa plantation. Tree Physiol. 1, 9–20. Hozumi, K., 1985. Phase diagramatica approach to the analysis of growth curve using the u–w diagram. – basic aspects. Bot. Mag. Tokyo 98, 239–250. Kim, M., Lee, W.-K., Choi, G.-M., Song, C., Lim, C.-H., Moon, J., Piao, D., Kraxner, F., Shividenko, A., Forsell, N., 2017. Modeling stand-level mortality based on maximum stem number and seasonal temperature. For. Ecol. Manage. 386, 37–50. Kira, T., 1977. Production rates. In: Shidei, T., Kira, T. (Eds.), Primary pRoductivity of Japanese Forests. University of Tokyo Press, Tokyo, pp. 101–114. Larcher, W., 2003. Physiological Plant Ecology, 4th ed. Springer, Berlin. Li, L., Weiner, J., Zhou, D., Huang, Y., Sheng, L., 2013. Initial density affects biomass–density and allometric relationships in self-thinning populations of Fagopyrum esculentum. J. Ecol. 101, 475–483. Miyanishi, K., Hoy, A.R., Cavers, P.B., 1979. A general law of self-thinning in plant population. J. Theor. Biol. 78, 439–442. Miyaura, T., Hozumi, K., 1988. Measurement of litterfall in a Japanese larch (Larix leptolepis Gordon) plantation by the cloth-trap method. J. Jpn. For. Soc. 70, 11–19. Miyaura, T., Hozumi, K., 1993. A growth model of a single sugi (Cryptomeria japonica) tree based on the dry matter budget of its aboveground parts. Tree Physiol. 13, 263–274. Mori, S., 2014. Respiratory scaling from seedlings to giant trees. Photosynth. Study 24, 27–33. Mori, S., Koyama, K., Yagi, M., Fukumori, K., 2013. Ecological significance of whole-plant metabolic scaling. Jpn. J. Ecol. 63, 125–132. Mori, S., Yamaji, K., Ishida, A., et al., 2010. Mixed-power scaling of whole-plant respiration from seedlings to giant trees. Proc. Natl. Acad. Sci. U. S. A. 107, 1447–1451. Muller-Landau, H.C., Condit, R.S., Chave, J., et al., 2006. Testing metabolic ecology theory for allometric scaling of tree size, growth and mortality in tropical forests. Ecol. Lett. 9, 575–588. Ninomiya, I., Hozumi, K., 1981. Respiration of forest trees. I. Measurement of respiration in Pinus densi-thunbergii Uyeki by an enclosed standing tree method. J. Jpn. For. Soc. 63, 8–18. Ninomiya, I., Hozumi, K., 1983a. Respiration of forest trees. II. Measurement of nighttime respiration in a Chamecyparis obtusa plantation. J. Jpn. For. Soc. 65, 193–200. Ninomiya, I., Hozumi, K., 1983b. Respiration of forest trees. III. Estimation of community respiration. J. Jpn. For. Soc. 65, 275–281. Ogawa, H., 1977. Principles and methods of estimating primary production in forests. In: Shidei, T., Kira, T. (Eds.), Primary Productivity of Japanese Forests. University of Tokyo Press, Tokyo, pp. 29–37. Ogawa, H., Kira, T., 1977. Methods of estimating forest biomass. In: Shidei, T., Kira, T. (Eds.), Primary Productivity of Japanese Forests. University of Tokyo Press, Tokyo pp 15-25/35-36. Ogawa, K., 1989. Growth and Matter Economy of Hinoki Cypress (Chamaecyparis Obtusa) Seedling Populations Grown under Nursery Conditions. Ph.D. thesis. Nagoya University. Ogawa, K., 2009a. Consideration of translocation into a growth model of a plant organ. Ecol. Model 220, 1492–1494. Ogawa, K., 2009b. Theoretical analysis of the interrelationship between self-thinning, biomass density, and plant form in dense populations of hinoki (Chamaecyparis obtusa) seedlings. Eur. J. For. Res. 128, 447–453. Ogawa, K., Takano, Y., 1997. Seasonal courses of CO2 exchange and carbon balance in fruits of Cinnamomum camphora. Tree Physiol. 17, 415–420.
Ogawa, K., Hagihara, A., Hozumi, K., 1985a. Growth analysis of a seedling community of Chamaecyparis obtusa. I. Respiratiuon consumption. J. Jpn. For. Soc. 67, 218–227. Ogawa, K., Hagihara, A., Hozumi, K., 1985b. Growth analysis of a seedling community of Chamaecyparis obtusa. VI. Estimation of individual gross primary production by the summation method. Transactions of the 30th Meeting of Chubu Branch of Japanese Forestry Society 179–181. Ogawa, K., Hagihara, A., Hozumi, K., 1986. Growth analysis of a seedling community of Chamaecyparis obtusa. II. Primary production. J. Jpn. For. Soc. 68, 135–142. Ogawa, K., Furukawa, A., Abdullah, A.M., Awang, M., 1996. Analysis of translocatory balance in durian (Durio zibethinus) fruit. Tree Physiol. 16, 315–318. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1992. Numerical Recipes in C, 2nd ed. Cambridge University Press, Cambridge. Pretzsch, H., Dauber, E., Biber, P., 2013. Species-specific and ontogeny-related stem allometry of European Forest Trees: evidence from extensive stem analyses. For. Sci. 59, 290–302. Pretzsch, H., Dieler, J., 2012. Evidence of variant intra- and interspecific scaling of tree crown structure and relevance for allometric theory. Oecologia 169, 637–649. Price, C.A., Enquist, B.J., Savage, V.M., 2007. A general model for allometric covariation in botanical form and function. Proc. Natl. Acad. Sci. U. S. A. 104, 13204–13209. R Core Team, 2018. R: A Language and Environment for Statistical Computing. URL. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org/. Reich, P.B., Tjoelker, M.G., Machado, J.L., Oleksyn, J., 2006. Universal scaling of respiratory metabolism, size, and nitrogen in plants. Nature 439, 457–461. Shinozaki, K., 1979. Summation of simple power functions. Chem. Biol. 17, 331–333. Sprugel, D.G., Benecke, U., 1991. Measuring woody-tissue respiration and photosynthesis. In: Lassoie, J.P., Hinckley, T.M. (Eds.), Techniques and Approaches in Forest Tree Ecophysiology. CRC Press, Boca Raton, pp. 329–355. Tadaki, Y., 1977. Leaf biomass. In: Shidei, T., Kira, T. (Eds.), Primary pRoductivity of Japanese Forests. University of Tokyo Press, Tokyo pp 39-44/57-62. Tanaka, K., Saito, T., 1984. A method for the determination of respiratory quotient by gas chromatography. Appl. Entomol. Zool. 19, 456–460. Waring, R.H., Landseberg, J.J., Williams, M., 1998. Net primary production of forests: a constant fraction of gross primary production? Tree Physiol. 18, 129–134. Warton, D.I., Wright, I.J., Falster, D.S., Westoby, M., 2006. Bivariate line fitting methods for allometry. Biol. Rev. 81, 259–291. West, G.B., Brown, J.H., Enquist, B.J., 1997. A general model for the origin of allometric scaling laws in biology. Science 276, 122–126. West, G.B., Brown, J.H., Enquist, B.J., 2001. A general model for ontogenetic growth. Nature 413, 628–631. West, G.B., Enquist, B.J., Brown, J.H., 2009. A general quantitative theory of forest structure and dynamics. Proc. Natl. Acad. Sci. U. S. A. 106, 7040–7045. Westoby, M., 1984. The self-thinning rule. Adv. Ecol. Res. 14, 167–225. White, J., 1980. Demographic factors in populations of plants. In: Solbrig, O.T. (Ed.), Demography and Evolution in Plant Populations. Blackwell, Oxford, pp. 21–48. White, J., Harper, J.L., 1970. Correlated changes in plant size and number in plant populations. J. Ecol. 58, 467–485. Yoda, K., Kira, T., Ogawa, H., Hozumi, K., 1963. Self-thinning in overcrowded pure stands under cultivated and natural conditions. (Intraspecific competition among higher plants XI). J. Biol. Osaka City Univ. 14, 107–129. Yokota, T., Hagihara, A., 1996. Dependence of the aboveground CO2 exchange rate on tree size in field-grown hinoki cypress (Chamaecyparis obtusa). J. Plant Res. 109, 177–184. Yokota, T., Hagihara, A., 1998. Changes in the relationship between tree size and aboveground respiration in field-grown hinoki cypress (Chamaecyparis obtusa) trees over three years. Tree Physiol. 18, 37–43. Yokota, T., Ogawa, K., Hagihara, A., 1994. Dependence of the aboveground respiration of hinoki cypress (Chamaecyparis obtusa) on tree size. Tree Physiol. 14, 467–479. Zeide, B., 1987. Analysis of the 3/2 power law of self-thinning. For. Sci. 32, 517–537. Zeide, B., 2010. Comparison of self-thinning models: an exercise in reasoning. Trees 24, 1117–1126.
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Scaling relations based on the geometric and metabolic theories in woody plant species: A review
T
Kazuharu Ogawa Laboratory of Forest Ecology, Graduate School of Bioagricultural Sciences, Nagoya University, Nagoya 464-8601, Japan
ARTICLE INFO
ABSTRACT
Keywords: von Bertalanffy’s equation Geometric model Metabolic theory Self-thinning Translocation Woody plant species
Extensive studies by Enquist and colleagues have indicated that many characteristics of organisms vary with body size, as described by allometric equations of power form. There are two representative models in relation to a scaling exponent for self-thinning from the viewpoints of resource allocation and geometry. This study was performed to discuss the reasonability of the geometric basis by providing several experimental examples based on a geometric analysis of forest and seedling stands where self-thinning does or does not occur. The metabolic scaling is reevaluated based on observations for forest trees where self-thinning does or does not occur, as well as seedlings. This metabolic scaling leads to a generalization of von Bertalanffy’s model in view of not only plant growth, but also fruit growth. Although a scaling exponent of 3/4 for respiratory scaling is supported by the West/Brown/Enquist (WBE), proofs have shown that the scaling exponent for metabolic rates should be 2/3 according to dimension analysis. In addition, the value of respiratory scaling exponent was considered with a diagrammatic representation in which the forest trees become giant. This study shows that the scaling exponent of 3/4 for the metabolic scaling relationships is not common for forest trees.
1. Introduction
Y = Y0 M 4
Extensive studies by Enquist and colleagues (e.g., West et al., 1997; Enquist et al., 1998; Brown and West, 2000) have indicated that many characteristics of organisms vary with body size, as described by allometric equations of the form:
They then argued that plants grow until they are limited by resources, so that the rate of resource use by plants approximates the rate of resource supply, and so if Q is the rate of resource supply per unit area and ρ is the maximum allowable density of plants, then:
Y = Y0 M b
3
(1)
where Y is a dependent variable, Y0 is a normalization constant, M is an independent variable (typically body mass), and b is the scaling exponent. The West/Brown/Enquist (WBE) theory, i.e., a metabolic scaling theory, provides a general model to explain why many anatomical and physiological scaling exponents of both plants and animals scale as quarter-powers of mass, e.g., b = 3/4 for metabolic rate, gross photosynthetic rate, resource use, and total leaf mass, –3/4 for population density, and 1/4 for lifespan. According to Begon et al. (2006), there are two representative models, including the WBE theory, for self-thinning in relation to the scaling exponent b in Eq. (1) as follows: (1) Resource allocation basis The WBE theory suggests that the rate of resource use per individual, Y, should be related to mean plant mass, M, according to the following equation (cf. Brown and West, 2000; Begon et al., 2006):
(2)
(3)
Q= Y or, from Eq. (2): 3
Q = Y0 M 4
(4)
At equilibrium, when the rate of resource use approximates rates of resource supply, Q should be constant. Hence, Eq. (4) leads to the following self-thinning function in which the expected value of the selfthinning scaling exponent is –4/3:
M = K1
4 3
(5)
where K1 is a constant. (2) Geometric basis Geometric analysis (Yoda et al., 1963) and the dimensional rule (Miyanishi et al., 1979) have both assigned a value of 3/2 to the selfthinning scaling exponent when the average ground area occupied by a plant is denoted by s:
E-mail address: [email protected]. https://doi.org/10.1016/j.ppees.2019.125480 Received 6 February 2019; Received in revised form 1 July 2019; Accepted 2 August 2019 Available online 26 August 2019 1433-8319/ © 2019 Elsevier GmbH. All rights reserved.
Perspectives in Plant Ecology, Evolution and Systematics 40 (2019) 125480
K. Ogawa
s=
2006).
1 (6)
2.2. Regression analysis
If plants at any growth stage are assumed to be always geometrically similar in shape:
s
2
Bivariate relationships, such as the Y–M (Eqs. (1), (10), (14), and (15)) relationships, were analyzed using standardized major axis (SMA) regression or model II regression, with the R package SMATR (version 3.5.2; R Development Core Team, 2018). SMA regression is more suitable than model I regression or ordinary least squares (OLS) regression for examining the scaling relationship between two variables that are both measured with errors (Warton et al., 2006). In the analysis of power (i.e., scaling) exponents, the statistical significance of differences was based on the 95% confidence interval (CI) of the exponents. On the other hand, the following nonlinear equation was fitted to the data on mean seedling height (Y) and mean shoot mass (M):
(7)
M3
Eqs. (6) and (7) lead to:
M = K2
3 2
(8)
where K2 is a constant, and the self-thinning scaling exponent has been reported in various studies as having a value of approximately –3/2 for a wide range of species (e.g., White, 1980; Westoby, 1984; Zeide, 1987; Ducey, 2012). Having reached its maximum density, a stand is considered to follow the –3/2 power law of self-thinning (Yoda et al., 1963; White and Harper, 1970), otherwise known as the self-thinning rule (Westoby, 1984). The WBE theory (3/4 scaling exponent) has been confirmed emirically by more recent studies (Brown et al., 2004; Enquist et al., 2007; Price et al., 2007; Duursma et al., 2010). Since Ferguson and Archibald (2002) confirmed the WBE theory rather than the allometric scaling (2/3 exponent), Zeide (2010) argued cautiously againt a constant scaling exponent. Condés et al. (2017) and Kim et al. (2017) also argued that the allometric scaling in general needs to be differentiated by species, which is much in the line of Pretzsch et al. (2013), or environment. Initial density also seems to play a role in biomass-density and allometric scaling relations (Li et al., 2013). There are indeed recent studies supporting the 3/2 self-thinning rule (Ducey, 2012). Reich et al. (2006) argued that the respiratory scaling exponent of Eq. (1) is close to unity regarding small trees or specific shading strategies (Cheng et al., 2010; Pretzsch et al., 2013; Pretzsch and Dieler, 2012). This rule that supports the argumentation in Reich et al. (2006) deviates from 3/4 scaling exponent of the WBE theory. It is thus widely accepted that the scaling exponent is not universally valid (Coomes and Allen, 2007, 2009; Muller-Landau et al., 2006), and there is controversy regarding the degree of generality in the self-thinning law (cf. Brown and West, 2000; Begon et al., 2006). Therefore, this study aims to demonstrate the validities of metabolic and geometric scaling theories by providing the published results of a research group at Nagoya University. On the basis of the published results, the appropriateness of the geometric basis is discussed for the seedling growth of hinoki cypress [Chamaecyparis obtusa (Sieb. et Zucc.) Endl.] stands where self-thinning does or does not occur (Ogawa, 2009b). In addition, we reevaluated the metabolic scaling of respiration and gross photosynthesis based on the observed results of forest trees where self-thinning does (Ninomiya and Hozumi, 1983a, 1983b; Hagihara and Hozumi, 1986; Yokota and Hagihara, 1998) or does not occur (Ninomiya and Hozumi, 1981; Yokota et al., 1994), as well as seedlings (Ogawa et al., 1985a, b; Ogawa, 1989). This metabolic scaling leads to a generalization of von Bertalanffy’s (1949) model from the viewpoint of not only plant growth (Hozumi, 1985), but also fruit growth (Ogawa, 2009a).
1 1 1 = + Y Y1 M b1 Y2 M b2
(9)
The fitting by Eq. (9) was performed using the software KaleidaGraph (v. 4.1.2; Synergy Software, Reading, PA), which is based on the Levenberg–Marquardt algorithm (Press et al., 1992), and the coefficient of determination (R2) was used to test for goodness of fit. 3. Results 3.1. Allometric scaling For the seasonal growth of a dense stand of C. obtusa seedling undergoing self-thinning, Ogawa (2009b) proposed that the allometric scaling relationship between mean seedling height Y and mean shoot mass M is expressed as a mixed-power function given by Eq. (9), which captures the transition between two simple power functions (Ogawa and Kira, 1977; Shinozaki, 1979). In Eq. (9), the values of coefficients Y1 and Y2 were estimated to be 1711 cmg b1 and 35.29 cmg b2 , respectively, and the values of scaling exponents b1 and b2 were estimated to be 3.954 and 0.3249, respectively (Fig. 1). The curves in Eq. (9) gradually approaches the two allometries of Y = Y1 M b1 and Y = Y2 M b2 for small and large values of M. Because the 1
estimated value of b2 is close to 1/3, Y is proportional to M 3 , i.e., growth in the height direction is isotropic with increasing shoot mass (Ogawa, 2009b), from the viewpoints of the geometric (Yoda et al., 1963) or dimensional (Miyanishi et al., 1979) rule. At the growth stage where the estimated value of b2 in Eq. (9) is close to 1/3, the selfthinning scaling exponent (Eq. (8)) is determined to be –3/2 (Ogawa, 2009b), indicating Yoda’s geometric self-thinning law rather than the 4/3 resource-based thinning law (Brown and West, 2000) as can be seen in Fig. 2 (Ogawa, 2009b). In the current-year C. obtusa seedling stand (1989), with full open spaces where self-thinning did not occur, the allometric scaling relationship of mean seedling height Y to mean shoot mass M were fitted by a simple power function of Eq. (1), where Y0 and b were estimated to be 25.17 cmg b and 0.3972 with a 95%CI of 0.3747 to 0.4211 (R2 = 0.988, P = 2.22×10−16), respectively (Fig. 3). The value of the allometric scaling exponent b was close to 1/3 (Ogawa, 2009b), indicating the reasonability of the geometric (Yoda et al., 1963) or dimensional (Miyanishi et al., 1979) rule.
2. Materials and methods 2.1. Study materials This study introduces the published results on the scaling relations for five woody species of Pinus densi-thunbergii Uyeki, Chamaecyparis obtusa (Sieb. et Zucc.) Endl., Cryptomeria japonica (L.f.) D. Don, Larix leptolepis Gordon and Durio zibethinus Murray performed by a research group at Nagoya University (Table 1), which was not cited by the group of Enquist and colleagues. For C. obtusa seedlings (Ogawa et al., 1985b, 2009b) and D. zibethinus fruits (Ogawa, 2009a), linear regression analysis was performed using standardized major axis (SMA) regression (Warton et al.,
3.2. Metabolic scaling 3.2.1. Respiration Mori et al. (2010) also empirically reported a mixed-power equation for expressing the respiratory scaling relation from seedlings to giant trees using Eq. (9), in which the values of the scaling exponents b1 and b2 for aboveground parts were estimated to be 1.082 and 0.780, respectively. 2
Perspectives in Plant Ecology, Evolution and Systematics 40 (2019) 125480
K. Ogawa
Table 1 Information about the present scaling study. 95% confidence intervals of scaling exponent are shown in parentheses. Locality
Species
Nagoya, Japan Inabu, Japan Inabu, Japan Minokamo, Japan Minokamo, Japan
Pinus densi-thunbergii trees Chamaecyparis obtusa trees Chamaecyparis obtusa trees Chamaecyparis obtusa seedlings Chamaecyparis obtusa seedlings
Nagoya, Japan Nagoya, Japan Nagoya, Japan Inabu, Japan Inabu, Japan Inabu, Japan Inabu, Japan Serdang, Malaysia
Chamaecyparis obtusa trees Chamaecyparis obtusa trees Chamaecyparis obtusa trees Chamaecyparis obtusa trees Larix leptolepis trees Cryptomeria japonica trees a single Cryptomeria japonica tree Durio zibethinus fruits
Equation
Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq.
(1) (10) (1) (1) (1) (1) (1) (1) (1) (9) (1) (10) (10) (10) (1) (1) (1) (11) (11) (12) (12) (12)
General features of stands Age [yrs]
Density [ha
7–11 24$ 25$ 2 0 1$ 8 17 18 25$ 14–18 14–17 12–20 –
6400 7378 7252 620000 13820000 8760000 10400 9091 10000 7274 2654 2160 2158 –
Scaling relationships Dependent variable
Scaling exponent
Respiration Respiration Gross photosynthesis Respiration Death GPP GPP CLE Height Height Respiration Gross photosynthesis Gross photosynthesis Respiration Litterfall Litterfall Litterfall Litterfall NPP Respiration Gross photosynthesis Translocation
1.03 0.634 1.84 (1.75, 1.93) 1.034 (1.019, 1.049) 1.680 (1.434, 1.967) 1.131 (1.090, 1.173)# 1.130 (1.104, 1.184) −0.1616 (−0.1351, −0.1930) 0.3972 (0.3747, 0.4211) b1=3.954, b2=0.3249 0.986 0.529# 0.602 0.667 – 0.788 1.01 0.86–1.04 1.20# 2.16 0.594 0.715 (0.461, 1.111) 0.697 (0.393, 1.234) 0.693 (0.590, 0.814)
Mean height [m]
Mean DBH [cm]
3.25 8.13 8.36 0.519 0.176 0.430 3.35 5.10 5.10 8.36 6.88 8.37 6.84 8.8&
3.75 7.67 7.84 0.636* – 0.198* 3.93¥ 6.23¥ 6.56¥ 7.84 7.48 11.43 11.86 25.0&
−1
]
n
References
4 5 136 80 80 80 80 80 17 12 5 5 5 5 5 10 5 1 1 5 5 5
Ninomiya and Hozumi, 1981 Ninomiya and Hozumi, 1983a, 1983b Hagihara and Hozumi, 1986 Ogawa et al., 1985b
Ogawa, 1989 Ogawa 2009b Yokota et al., 1994 Yokota and Hagihara, 1996 Yokota and Hagihara, Miyaura and Hozumi, Miyaura and Hozumi, Miyaura and Hozumi, Miyaura and Hozumi,
1998 1988 1988 1993 1993
Ogawa, 2009a
$
Self-thinning stand. Diameter at 1 m above ground. * Diameter at crown base. & values for a sample tree. # Leaf dry mass is adopted as an independent variable. ¥
2-year-old (Ogawa et al., 1985a; Ogawa, 1989) and young trees of P. densithunbergii Uyeki (7–11 years old) (Ninomiya and Hozumi, 1981) and C. obtusa (8 years old) (Yokota et al., 1994), as well as several saplings (Reich et al., 2006). However, Ogawa et al. (1985b) reported that the scaling exponent of the annual respiration rate to mean total mass of 2-year-old C. obtusa was significantly greater than unity (1.034 with 95%CI of 1.019–1.049) by measuring respiration with an open system chamber and an infra-red gas analyzer (URAS-1; Hartmann and Braun, Frankfurt, Germany), indicating that respiratory activity increases as the seedling becomes larger in size (Fig. 4). A respiratory scaling exponent of 3/4 at the later growth stage was recently supported by the WBE or metabolic scaling theories (West et al., 1997; Enquist et al., 1998; Brown and West, 2000). In contrast, Ninomiya and Hozumi (1983a, b) proposed a general scaling relation of aboveground respiration rate Y to aboveground mass M in a self-thinning C. obtusa forest stand as follows:
Fig. 1. Relationship between mean seedling height Y and mean shoot mass M for an overcrowded population of 1-year-old Chamaecyparis obtusa seedlings (Ogawa, 2009b). The regression curve is given by Eq. (9) (R2 = 0.988).
In Mori et al. (2010), the respiration rate was proportional to the plant mass during the early growth stage, and was proportional to 3/4 of the plant mass during the later growth stage. The former phenomenon during the early growth stage has also been observed in current to
Y = Y0 (M
Mmin )b
(10)
where Y0 and b are constants, and Mmin is a critical value for aboveground mass, below which trees may not be able to survive in the 3
Perspectives in Plant Ecology, Evolution and Systematics 40 (2019) 125480
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Fig. 4. Relationship between annual respiration Y and mean seedling mass M for 80 seedlings of 2-year-old Chamaecyparis obtusa seedlings (Ogawa et al., 1985b). The linear line shows the regression on log–log coordinates: Y = 3.950M1.034 (R2 = 0.996, P = 2.22×10–16).
Fig. 2. Time trajectory of changes in mean shoot mass Y and stand density M for an overcrowded population of 1-year-old Chamaecyparis obtusa seedlings (Ogawa, 2009b). The curves show the regression: Y = KM (R2 = 0.926).
1.515
(1 ( ) )
0.0741 0.0105 M 1504.1
Fig. 3. Relationship between mean seedling height Y and mean shoot mass M for current-year Chamaecyparis obtusa seedlings under full open space growth conditions (Ogawa, 2009b). For details of the regression line, see the text.
stand. The biological meaning of Eq. (10) is that as M approaches Mmin, the respiration rate drops abruptly, while the relationship satisfies the simple scaling relation of Eq. (1) where M is sufficiently larger than Mmin (Fig. 5). Ninomiya and Hozumi (1983b) concluded that the respiration rate of individual trees is nearly proportional to their surface area, where M is sufficiently larger than Mmin, because the scaling exponent b of Eq. (10) is close to 2/3. Deriviation of proportionality between the respiration rate and the surface area of a plant from the b exponent of Eq. (10) is caused by the dimensional analysis (Miyanishi et al., 1979) that the mass and surface area are respectively based on three- and two-dimensional model. Yokota and Hagihara (1998) also reported that the scaling exponent b of Eq. (10) ranged from 0.667 to 0.788 over three years, rather indicating that it is somewhere between 2/3 and 3/4.
Fig. 5. Conceptual diagram of the general scaling relation of Y to M in Eq. (10). Mmin is a critical value for tree mass, below which the trees may not be able to survive in the forest stand.
tended to increase with increasing seedling size, because the scaling exponent b is significantly greater than unity. In addition, the scaling of photosynthetic metabolism with whole seedling dry mass, instead of leaf dry mass, can also be approximated by Eq. (1), where Y0 and b were calculated to be 4.215 g1–b yr–1 and 1.130 with 95%CI (1.104 to 1.184), respectively (R2 = 0.976, P = 2.22×10–16), indicating that the scaling exponent b was significantly different from unity (Fig. 6). A similar result was reported for 136 C. obtusa trees, excluding suppressed individuals, by Hagihara and Hozumi (1986).
3.2.2. Gross photosynthesis By estimating the annual rate of gross primary production (GPP, i.e., gross photosynthesis) of 80 C. obtusa seedlings by the summation method (Ogawa, 1977; Waring et al., 1998), Ogawa et al. (1985b) suggested that the scaling relationship between GPP and leaf dry mass is expressed as Eq. (1), in which Y0 and b were calculated to be 8.628 g1–b yr–1 and 1.131 with a 95%CI of 1.090 to 1.173, respectively. They concluded that the photosynthetic activity per unit leaf dry mass 4
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Fig. 8. Relationship between annual attached dead leaves Y and mean seedling mass M for 80 seedlings of 2-year-old Chamaecyparis obtusa seedlings (Ogawa et al., 1985b). The linear line shows the regression on log–log coordinates: Y = 0.0156M1.680 (R2 = 0.506, P = 1.45×10–13).
Fig. 6. Relationship between annual gross primary production (GPP) Y and mean seedling mass M for 80 seedlings of 2-year-old Chamaecyparis obtusa seedlings (Ogawa et al., 1985b). For details of the regression line, see the text.
3.3. von Bertalanffy’s equation
For a self-thinning plantation of C. obtusa (Yokota and Hagihara, 1996), the scaling of gross photosynthesis with the aboveground mass was well described by a general power function, with the critical variable Mmin in Eq. (11) (cf. Fig. 5) proposed by Ninomiya and Hozumi (1983a, b). The scaling exponent b for annual gross photosynthesis was calculated to be 0.602 by Yokota and Hagihara (1996), suggesting that the photosynthetic scaling exponent b is close to 2/3 rather than 3/4.
Hozumi (1985) generalized von Bertalanffy’s model (von Bertalanffy, 1949) from the viewpoint of plant growth as follows:
(11) where v1, v2, and v3 represent the velocity of gross photosynthesis (or GPP), respiration, and litterfall, respectively. Miyaura and Hozumi (1988) demonstrated the scaling relationship between litterfall rate v3 and the aboveground mass M in Larix leptolepis Gordon and C. obtusa trees, where the scaling exponent b3 was 0.86–1.04 for L. leptolepis and 1.01 for C. obtusa. Ogawa et al. (1985b) also reported the scaling relation of the death rate of leaves v3, which was estimated by the monthly clipping of new dead leaves attached to 80 sample seedlings over one year, to the whole seedling mass M in a C. obtusa seedling population (Fig. 8), where the scaling exponent b3 was 1.680 with 95%CI (1.434–1.967). Because the rates of gross photosynthesis v1 and respiration v2 scale as M b1 and M b2 where M is sufficiently larger than Mmin in Eq. (10) (cf. Fig. 5), these three lines of experimental evidence for gross photosynthesis, respiration, and litterfall support the generalization of von Bertalanffy’s model in Eq. (11) proposed by Hozumi (1985). Miyaura and Hozumi (1993) confirmed experimentally that annual aboveground growth v in Eq. (11) is given by the sum of the rates of net production and death (monthly new dead leaves and branches) by monitoring the growth of a single C. japonica D. Don over six years. Ogawa (2009a) reported that the fruit growth rate v can be expressed as scaling rules considering translocation (dTr dt ) into the fruit from other plant organs by measuring the growth and CO2 exchange rates, such as gross photosynthesis (dP dt ) and respiration (dR dt ) of the tropical fruit, D. zibethinus Murray, as follows:
3.2.3. Carbon loss efficiency The present study defines the carbon loss efficiency (CLE) as the ratio of the annual respiration loss (R) to annual gross primary production (GPP). The CLE is equal to the difference between unity and the carbon use efficiency CUE, which corresponds to the primary production efficiency (Kira, 1977). Ogawa et al. (1985b) reported that the scaling relation of CLE to the mean whole seedling mass (M) of 2-year-old C. obtusa over a whole year is negatively correlated (Fig. 7, scaling exponent (b) = –0.1616, with a 95%CI of –0.1930 to –0.1351). This indicates that the larger the seedling mass is, the lower the CLE becomes.
(12) where Y4 and b4, Y1 and b1, and Y2 and b2 were determined to be 0.313 g1 b4 day 1 and 0.693 with 95%CI of 0.590 to 0.814 (R2 = 0.992, P = 0.000291), 0.0387 g1 b1 day 1 and 0.697 with 95%CI of 0.393–1.234 (R2 = 0.811, P = 0.0143), and 0.0217 g1 b2 day 1 and
Fig. 7. Relationship between annual carbon loss efficiency (CLE) Y and mean seedling mass M for 80 seedlings of 2-year-old Chamaecyparis obtusa seedlings (Ogawa et al., 1985b). The linear line shows the regression on log–log coordinates: Y = 0.0232M 0.162 (R2 = 0.369, P = 2.28×10–9). 5
Perspectives in Plant Ecology, Evolution and Systematics 40 (2019) 125480
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Fig. 10. Diagrammatic representation on log–log coordinates for the respiratory scaling exponent (b in Eq. 1) values of 2/3 and 3/4 after the canopy fully closes where the forest trees become giant.
rate is low in birds and mammals (cf. Fig. 2, Brown and West, 2000). This difference is mainly due to the temperature effect, because Q10 values change seasonally (Ninomiya and Hozumi, 1983a, b; Ogawa, 1989; Ogawa et al., 1985a) in woody plant species. 3) The degree of data scattering for respiration also varies among species as indicated by the measurement results of leaf dark respiration in woody plant species as summarized by Larcher (2003). 4) Considering these reasons for respiration data scattering, the biological significance of the fitted equations, such as Reich et al. (2006) and Mori et al. (2010), is unclear although we can easily comprehend the comparison of scaling relations of fruit CO2 gas exchange between Cinnamomum camphora (Ogawa and Takano, 1997) growing in temperate regions and D. zibethinus (Ogawa et al., 1996) growing in tropical regions. 5) Mori et al. (2013) and Mori (2014) reported that a time interval of 100 s is sufficient for determining the time trend of the CO2 concentration within the respiration chamber. However, Ogawa (1989) measured the respiration rate of current-year C. obtusa seedlings over 26 h by gas chromatography (GC-9A; Shimadzu, Kyoto, Japan) and with gastight syringes (A-Type; Dynatech Precision Sampling Corp., Baton Rouge, LA) according to the enclosed-chamber method for monitoring respiration of the wolf spider (Pardosa astrigera (L. Koch)) developed by Tanaka and Saito (1984). After measuring the respiration of C. obtusa seedlings, Ogawa (1989) showed an experimental example of the time trend of specific respiration rate, where the specific respiration rate tended to decrease continuously immediately after being enclosed in the respiration chamber
Fig. 9. Relationship between net translocation rate (dTr/dt), net respiration rate (dP/dt), and nighttime respiration rate (dR/dt) and fruit dry mass M in tropical fruits of Durio zibethinus (Ogawa, 2009a). For details of the regression line, see the text.
0.715 with 95%CI of 0.461–1.111 (R2 = 0.893, P = 0.00448), respectively (Fig. 9). 4. Discussion 4.1. Methodological aspects for respiration In comparison with Ninomiya and Hozumi (1981, 1983a), Yokota et al. (1994), and Yokota and Hagihara (1996, 1998), the methodological aspects for respiration measurement by Mori et al. (2010) were not presented in detail, and therefore the present study considered the respiratory scaling exponent b in Eq. (1), with a diagrammatical representation after the forest canopy had fully closed and where the forest trees became giant. Here, we consider the forest trees at the growth stage near giant trees, and list several problems: 1) The regression line with the greater value of b as x approaches infinity yields a higher respiration rate than the regression line with the lower value of b. Therefore, when respiration is measured for a plant with a mass approaching that of, for example, an exceptionally large tree, the respiration rate r2 (b = 3/4) is greater than r1 (b = 2/3) (Fig. 10). This difference between r1 and r2 may indicate that the measured respiration rate of an exceptionally large tree is overestimated by the assumption that b = 3/4, due to a high CO2 concentration sampled within the chamber, possibly caused by increased wound respiration from the excision of plant organs (Sprugel and Benecke, 1991), if CO2 is diffused sufficiently throughout the chamber. 2) The degree of data scattering for respiration is high (around an order of 102) at a given tree mass in woody plant species (Fig. 2, Mori et al., 2010), while the degree of data scattering for the metabolic
Fig. 11. Time trend of specific respiration rate of current-year Chamaecyparis obtusa seedlings after enclosing the respiration chamber by gas chromatography (Ogawa, 1989). 6
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(Fig. 11). This could be the issue of CO2 diffusion out of the chamber. This issue may be related to all enclosed-chamber measurements of respiration, not specifically related to trees. Therefore, the enclosed-chamber method may not be useful for measuring the respiration of woody plant species.
range of values of the respiratory scaling exponent b in Eq. (10) by Yokota and Hagihara (1998). However, the scaling exponent b of Eq. (1) was significantly greater than unity (Ogawa et al., 1985b) in a C. obtusa seedling population where self-thinning did not occur, and the leaf area index (LAI) ranged from 1.54 to 5.16 m2 m–2 (Ogawa et al., 1986). According to Tadaki (1977), the LAI ranges between 5 and 7 ha ha–1 in C. obtusa plantations with closed canopies. Therefore, it is noteworthy that the final value (5.16 m2 m–2) of LAI in the seedling population was comparable to the values of mature forests. Considering the seedling growth conditions from LAI values, the competition for light among individuals is thought to be high, and so larger seedlings show higher rates of photosynthesis, and vice versa. In conclusion, the allometric scaling relation of mean seedling height and mean shoot mass (Figs. 1 and 3) supports the geometric rule the same as the 3/2 self-thinning law (Fig. 2). However, it should be noted that the allometric scaling varied among densities, because initial density affects the allometric scaling exponent between mean height and mean shoot mass (Li et al., 2013). West et al. (1997, 2009) presented a general explanation of allometric scaling with exponents to be drived from 1/4 based on the fractal nature of transportation network systems that supply the metabolism in organisms. On the contrary, the metabolic scaling relation such as respiration cannot be explained only from the geometrical basis, because the respiratory metabolic scaling showed the exponent value of nearly 3/4 in some cases. Considering the argumentation of Reich et al. (2006); Enquist et al. (2007) made some modification to their original WBE model and restated that scaling exponents are close to 1.0 for seedlings owing to the violation of WBE assumptions in seedlings and shift to three-quarters in larger plants. However, the results of annual respiration (Fig. 4) and gross photosynthesis (Fig. 6) for seedlings shows the scaling exponents higher than 1.0. This empirical evidence also indicates that there is no universal metabolic scaling in plants (Coomes and Allen, 2007, 2009; Muller-Landau et al., 2006) and the scaling exponent of 3/4 for the metabolic scaling relationships is not common for forest trees (Cheng et al., 2010).
4.2. von Bertalanffy’s equation von Bertalanffy (1949) first studied the case with b1 = 2/3 and b2 = 1 without the term v3 in Eq. (11), whereas the equation considered by West et al. (2001) is a special case with b1 = 3/4 and b2 = 1. The present study empirically supports b1 = b2 = 2/3, indicating the appropriateness of the geometric (Yoda et al., 1963) and dimensional (Miyanishi et al., 1979) rules. However, it should be noted that the case of the seedling stage, where b1 and b2 is greater than unity, especially for photosynthesis, is different from adult trees. In the generalized von Bertalanffy growth model of Eq. (11) proposed by Hozumi (1985), an additional term of death (or litterfall) rate v3 has a value of the scaling exponent b3 close to unity at the growth stage of adult trees (Miyaura and Hozumi, 1988). However, the value of the scaling exponent b3 is significantly greater than unity at the early growth stage of seedlings (Ogawa et al., 1985b), although the observed values showed deviations from the regression (Fig. 8, R2 = 0.506, P = 0.000). In another generalized von Bertalanffy model Eq. (12) of a fruit growth proposed by Ogawa (2009a), the values of three scaling exponents (b1 and b4) were close to 2/3 rather than 3/4, while the value of b2 for respiration scaling was closer to 3/4 than to 2/3. This also indicates the appropriateness of the geometric (Yoda et al., 1963) and dimensional (Miyanishi et al., 1979) rules for translocation and photosynthesis scaling. 4.3. Allometric and metabolic scaling The results presented here indicated the applicability of a mixedpower scaling equation to the allometric scaling relationship between seedling height and mass, which is the same as the respiratory scaling relations of seedlings to giant trees proposed by Mori et al. (2010). However, it should be noted that the values of the scaling exponent are different between the two scaling relations during the later stages of growth. That is, the present study had a scaling exponent value of ca. 1/3, while that reported by Mori et al. (2010) was 3/4. Mori et al. (2010) based the value of the scaling exponent on the WBE or metabolic theories (West et al., 1997; Enquist et al., 1998; Brown and West, 2000). In contrast, the present study used the geometric or dimensional analysis. The scaling exponent value of 0.3249 reported by Ogawa (2009b) is close to 1/3. This biological phenomenon can be explained based on geometrical considerations. That is, the seedling height is a one-dimensional model, and the seedling mass is a three-dimensional model. The allometric scaling of seedling height and mass can be explained based on geometric considerations, although West et al. (1997) presented a general explanation of allometric scaling with exponents to be derived from 1/4 based on the fractal nature of the transportation network systems that supply the metabolism in organisms. According to the experimental observations of Ninomiya and Hozumi (1983b), the respiration rate is proportional to the surface area of self-thinning forest trees because the scaling exponent of Eq. (10) was estimated to be nearly 2/3. The same tendency was observed by Yokota and Hagihara (1996) for the scaling of gross photosynthesis with the aboveground mass in self-thinning forest trees. Therefore, our observations support not only the allometric scaling equation (Eq. (9)), but also the respiratory and photosynthetic scaling equation (Eq. (10)) from the viewpoints of the geometric (Yoda et al., 1963) or dimensional (Miyanishi et al., 1979) rule, in addition to the WBE theory (West et al., 1997; Enquist et al., 1998; Brown and West, 2000), which includes the
Declaration of Competing Interest None declared. Acknowledgements I thank the staff of the Nagoya Regional Forest Office and the Midorigaoka Nursery attached to the Gifu District Forest Office for providing the seedlings, and my colleagues for their assistance during field work. Thanks are also due to Daniel Falster for his helpful advice on the use of the freely available software program SMATR (Warton et al., 2006). References Begon, M., Townsend, C.R., Happer, J.L., 2006. Ecology, 4th ed. Blackwell, Oxford. von Bertalanffy, L., 1949. Problems of organic growth. Nature 163, 156–158. Brown, J.H., West, G.B. (Eds.), 2000. Scaling in Biology. Oxford University Press, New York. Brown, J.H., Gillooly, J.F., Allen, A.P., Savage, V.M., West, G.B., 2004. Toward a metabolic theory of ecology. Ecology 85, 1771–1789. Cheng, D.L., Li, T., Zhong, Q.L., Wang, G.X., 2010. Scaling relationship between tree respiration rates and biomass. Biol. Lett. 6, 715–717. Condés, S., Vallet, P., Bielak, K., Bravo-Oviedo, A., Coll, L., Ducey, M.J., Pach, M., Pretzsch, H., Sterba, H., Vayreda, J., Rio, M., 2017. Climate influences on the maximum size-density relationship in Scots pine (Pinus sylvestris L.) and European beech (Fagus sylvatica L.) stands. For. Ecol. Manage. 385, 295–307. Coomes, D.A., Allen, R.B., 2007. Effects of size, competition and altitude on tree growth. J. Ecol. 95, 1084–1097. Coomes, D.A., Allen, R.B., 2009. Testing the metabolic scaling theory of tree growth. J. Ecol. 97, 1369–1373. Ducey, M.J., 2012. Evergreenness and wood density predict height-diameter scaling in trees of the northeastern United States. For. Ecol. Manage. 279, 21–26.
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Perspectives in Plant Ecology, Evolution and Systematics 40 (2019) 125480
K. Ogawa Duursma, R.A., Mäkelä, A., Reid, D.E.B., Jokela, E.J., Porté, A.J., Roberts, S.D., 2010. Selfshading affects allometric scaling in trees. Funct. Ecol. 24, 723–730. Enquist, B.J., Brown, J.H., West, G.B., 1998. Allometric scaling of plant energetics and population density. Nature 395, 163–165. Enquist, B.J., Allen, A.P., Brown, J.H., Gillooly, J.F., 2007. Biological scaling: does the exception prove the rule? Nature 445, E9–E10. Ferguson, S.H., Archibald, D.J., 2002. The 3/4 power law in forest management: how to grow dead trees. For. Ecol. Manage. 169, 283–292. Hagihara, A., Hozumi, K., 1986. An estimation of the photosynthetic production of individual trees in a Chamaecyparis obtusa plantation. Tree Physiol. 1, 9–20. Hozumi, K., 1985. Phase diagramatica approach to the analysis of growth curve using the u–w diagram. – basic aspects. Bot. Mag. Tokyo 98, 239–250. Kim, M., Lee, W.-K., Choi, G.-M., Song, C., Lim, C.-H., Moon, J., Piao, D., Kraxner, F., Shividenko, A., Forsell, N., 2017. Modeling stand-level mortality based on maximum stem number and seasonal temperature. For. Ecol. Manage. 386, 37–50. Kira, T., 1977. Production rates. In: Shidei, T., Kira, T. (Eds.), Primary pRoductivity of Japanese Forests. University of Tokyo Press, Tokyo, pp. 101–114. Larcher, W., 2003. Physiological Plant Ecology, 4th ed. Springer, Berlin. Li, L., Weiner, J., Zhou, D., Huang, Y., Sheng, L., 2013. Initial density affects biomass–density and allometric relationships in self-thinning populations of Fagopyrum esculentum. J. Ecol. 101, 475–483. Miyanishi, K., Hoy, A.R., Cavers, P.B., 1979. A general law of self-thinning in plant population. J. Theor. Biol. 78, 439–442. Miyaura, T., Hozumi, K., 1988. Measurement of litterfall in a Japanese larch (Larix leptolepis Gordon) plantation by the cloth-trap method. J. Jpn. For. Soc. 70, 11–19. Miyaura, T., Hozumi, K., 1993. A growth model of a single sugi (Cryptomeria japonica) tree based on the dry matter budget of its aboveground parts. Tree Physiol. 13, 263–274. Mori, S., 2014. Respiratory scaling from seedlings to giant trees. Photosynth. Study 24, 27–33. Mori, S., Koyama, K., Yagi, M., Fukumori, K., 2013. Ecological significance of whole-plant metabolic scaling. Jpn. J. Ecol. 63, 125–132. Mori, S., Yamaji, K., Ishida, A., et al., 2010. Mixed-power scaling of whole-plant respiration from seedlings to giant trees. Proc. Natl. Acad. Sci. U. S. A. 107, 1447–1451. Muller-Landau, H.C., Condit, R.S., Chave, J., et al., 2006. Testing metabolic ecology theory for allometric scaling of tree size, growth and mortality in tropical forests. Ecol. Lett. 9, 575–588. Ninomiya, I., Hozumi, K., 1981. Respiration of forest trees. I. Measurement of respiration in Pinus densi-thunbergii Uyeki by an enclosed standing tree method. J. Jpn. For. Soc. 63, 8–18. Ninomiya, I., Hozumi, K., 1983a. Respiration of forest trees. II. Measurement of nighttime respiration in a Chamecyparis obtusa plantation. J. Jpn. For. Soc. 65, 193–200. Ninomiya, I., Hozumi, K., 1983b. Respiration of forest trees. III. Estimation of community respiration. J. Jpn. For. Soc. 65, 275–281. Ogawa, H., 1977. Principles and methods of estimating primary production in forests. In: Shidei, T., Kira, T. (Eds.), Primary Productivity of Japanese Forests. University of Tokyo Press, Tokyo, pp. 29–37. Ogawa, H., Kira, T., 1977. Methods of estimating forest biomass. In: Shidei, T., Kira, T. (Eds.), Primary Productivity of Japanese Forests. University of Tokyo Press, Tokyo pp 15-25/35-36. Ogawa, K., 1989. Growth and Matter Economy of Hinoki Cypress (Chamaecyparis Obtusa) Seedling Populations Grown under Nursery Conditions. Ph.D. thesis. Nagoya University. Ogawa, K., 2009a. Consideration of translocation into a growth model of a plant organ. Ecol. Model 220, 1492–1494. Ogawa, K., 2009b. Theoretical analysis of the interrelationship between self-thinning, biomass density, and plant form in dense populations of hinoki (Chamaecyparis obtusa) seedlings. Eur. J. For. Res. 128, 447–453. Ogawa, K., Takano, Y., 1997. Seasonal courses of CO2 exchange and carbon balance in fruits of Cinnamomum camphora. Tree Physiol. 17, 415–420.
Ogawa, K., Hagihara, A., Hozumi, K., 1985a. Growth analysis of a seedling community of Chamaecyparis obtusa. I. Respiratiuon consumption. J. Jpn. For. Soc. 67, 218–227. Ogawa, K., Hagihara, A., Hozumi, K., 1985b. Growth analysis of a seedling community of Chamaecyparis obtusa. VI. Estimation of individual gross primary production by the summation method. Transactions of the 30th Meeting of Chubu Branch of Japanese Forestry Society 179–181. Ogawa, K., Hagihara, A., Hozumi, K., 1986. Growth analysis of a seedling community of Chamaecyparis obtusa. II. Primary production. J. Jpn. For. Soc. 68, 135–142. Ogawa, K., Furukawa, A., Abdullah, A.M., Awang, M., 1996. Analysis of translocatory balance in durian (Durio zibethinus) fruit. Tree Physiol. 16, 315–318. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1992. Numerical Recipes in C, 2nd ed. Cambridge University Press, Cambridge. Pretzsch, H., Dauber, E., Biber, P., 2013. Species-specific and ontogeny-related stem allometry of European Forest Trees: evidence from extensive stem analyses. For. Sci. 59, 290–302. Pretzsch, H., Dieler, J., 2012. Evidence of variant intra- and interspecific scaling of tree crown structure and relevance for allometric theory. Oecologia 169, 637–649. Price, C.A., Enquist, B.J., Savage, V.M., 2007. A general model for allometric covariation in botanical form and function. Proc. Natl. Acad. Sci. U. S. A. 104, 13204–13209. R Core Team, 2018. R: A Language and Environment for Statistical Computing. URL. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org/. Reich, P.B., Tjoelker, M.G., Machado, J.L., Oleksyn, J., 2006. Universal scaling of respiratory metabolism, size, and nitrogen in plants. Nature 439, 457–461. Shinozaki, K., 1979. Summation of simple power functions. Chem. Biol. 17, 331–333. Sprugel, D.G., Benecke, U., 1991. Measuring woody-tissue respiration and photosynthesis. In: Lassoie, J.P., Hinckley, T.M. (Eds.), Techniques and Approaches in Forest Tree Ecophysiology. CRC Press, Boca Raton, pp. 329–355. Tadaki, Y., 1977. Leaf biomass. In: Shidei, T., Kira, T. (Eds.), Primary pRoductivity of Japanese Forests. University of Tokyo Press, Tokyo pp 39-44/57-62. Tanaka, K., Saito, T., 1984. A method for the determination of respiratory quotient by gas chromatography. Appl. Entomol. Zool. 19, 456–460. Waring, R.H., Landseberg, J.J., Williams, M., 1998. Net primary production of forests: a constant fraction of gross primary production? Tree Physiol. 18, 129–134. Warton, D.I., Wright, I.J., Falster, D.S., Westoby, M., 2006. Bivariate line fitting methods for allometry. Biol. Rev. 81, 259–291. West, G.B., Brown, J.H., Enquist, B.J., 1997. A general model for the origin of allometric scaling laws in biology. Science 276, 122–126. West, G.B., Brown, J.H., Enquist, B.J., 2001. A general model for ontogenetic growth. Nature 413, 628–631. West, G.B., Enquist, B.J., Brown, J.H., 2009. A general quantitative theory of forest structure and dynamics. Proc. Natl. Acad. Sci. U. S. A. 106, 7040–7045. Westoby, M., 1984. The self-thinning rule. Adv. Ecol. Res. 14, 167–225. White, J., 1980. Demographic factors in populations of plants. In: Solbrig, O.T. (Ed.), Demography and Evolution in Plant Populations. Blackwell, Oxford, pp. 21–48. White, J., Harper, J.L., 1970. Correlated changes in plant size and number in plant populations. J. Ecol. 58, 467–485. Yoda, K., Kira, T., Ogawa, H., Hozumi, K., 1963. Self-thinning in overcrowded pure stands under cultivated and natural conditions. (Intraspecific competition among higher plants XI). J. Biol. Osaka City Univ. 14, 107–129. Yokota, T., Hagihara, A., 1996. Dependence of the aboveground CO2 exchange rate on tree size in field-grown hinoki cypress (Chamaecyparis obtusa). J. Plant Res. 109, 177–184. Yokota, T., Hagihara, A., 1998. Changes in the relationship between tree size and aboveground respiration in field-grown hinoki cypress (Chamaecyparis obtusa) trees over three years. Tree Physiol. 18, 37–43. Yokota, T., Ogawa, K., Hagihara, A., 1994. Dependence of the aboveground respiration of hinoki cypress (Chamaecyparis obtusa) on tree size. Tree Physiol. 14, 467–479. Zeide, B., 1987. Analysis of the 3/2 power law of self-thinning. For. Sci. 32, 517–537. Zeide, B., 2010. Comparison of self-thinning models: an exercise in reasoning. Trees 24, 1117–1126.
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