REVIEW OF STUDIES ON MAXIMUM SIZE-DENSITY RULES

doi:10.3773/j.issn.1005-264x.2008.02.030

Abstract

Abstract:

We summarized the research on theory and method related to maximum size-density rules. There are two main theories, Yoda's 3/2 power law based on Euclidian geometry and West, Brown and Enquist's fractal scaling rules (WBE model). However, both are based on static analysis rather than dynamic competition between plants, which is the direction attempted by recent researchers. In spite of this, some researches have not eliminated the reliance on these untenable assumptions, such as the instead of whole population by the average plant size. Further work is needed. In addition, there is ongoing debate regarding assumptions, mathematical deduction and original data points used for estimating parameters. Each model is formulated based on special situations and assumptions, it is not all-purpose; therefore, these models can be coupled, depending on practical situations. Moreover, various methods are used for estimating parameters because of different perceptions and criteria. Therefore uniform criteria need to be established.

Key words: maximum size-density line, 3/2 power law, WBE's model, stand density

FU Li-Hua, ZHANG Jian-Guo, DUAN Ai-Guo, SUN Hong-Gang, HE Cai-Yun. REVIEW OF STUDIES ON MAXIMUM SIZE-DENSITY RULES[J]. Chin J Plant Ecol, 2008, 32(2): 501-511.

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URL: https://www.plant-ecology.com/EN/10.3773/j.issn.1005-264x.2008.02.030

https://www.plant-ecology.com/EN/Y2008/V32/I2/501

Figures/Tables 3

Fig.1 Diagram of WBE's model (West et al., 1997)

Table 1 Relationships between stand density and mass of plant's part

	植株重量 Pant body mass	地上生物量 Shoot biomass	地下生物量 Root biomass	叶子生物量 Leaf biomass	生长速率 Whole-plant growth rate
林分密度 Stand density	M_T∝N^-4^/³ $M ˙$ ∝N^-1^/³	M_sh∝N^-4^/³ $M ˙ sh$ ∝N^-1^/³	M_R∝N^-4^/³ $M ˙ R$ ∝N^-1^/³	M_L ∝N^-1 $M ˙ L$ ∝N0	G_T∝N-1 $G ˙$ ∝N⁰

Table 1 Relationships between stand density and mass of plant's part

	植株重量 Pant body mass	地上生物量 Shoot biomass	地下生物量 Root biomass	叶子生物量 Leaf biomass	生长速率 Whole-plant growth rate
林分密度 Stand density	M_T∝N^-4^/³ $M ˙$ ∝N^-1^/³	M_sh∝N^-4^/³ $M ˙ sh$ ∝N^-1^/³	M_R∝N^-4^/³ $M ˙ R$ ∝N^-1^/³	M_L ∝N^-1 $M ˙ L$ ∝N0	G_T∝N-1 $G ˙$ ∝N⁰

Table 2 Values of exponents in different models

指数值 Values	Reinke方程 Reinke's equation	Yoda的3/2法则 3/2 power law	WBE模型 WBE's model	Pretzsch经验值 Pretzsch's experimental values
W~d^a		3	8/3	8/3
N~ $d ? b$	-1.605	-2	-2	-1.605
$w ?$ ~N^c		-3/2	-4/3	-3/2

Table 2 Values of exponents in different models

指数值 Values	Reinke方程 Reinke's equation	Yoda的3/2法则 3/2 power law	WBE模型 WBE's model	Pretzsch经验值 Pretzsch's experimental values
W~d^a		3	8/3	8/3
N~ $d ? b$	-1.605	-2	-2	-1.605
$w ?$ ~N^c		-3/2	-4/3	-3/2

References 77

[1]	Adler FR (1996). A model of self-thinning through local competition. Proceedings of the National Academy of Science of United States of America, 93, 9980-9984.
[2]	Arenas F, Fernandes C (2000). Size structure and dynamics in a population of Sargassum muticum (Phaeophyceae). Journal of Psychology, 36, 1012-1020.
[3]	Bégin E, Bégin J, Bélanger L, Rivest LP, Tremblay S (2001). Balsam fir self-thinning relationship and its constancy among different ecological regions. Canadian Journal of Forest Research, 31, 950-959.
[4]	Belgrano A, Allen AP, Enquist BJ, Gillooly JF (2002). Allometric scaling of maximum population density: a common rule for marine phytoplankton and terrestrial plants. Ecology Letters, 5, 611-613.
[5]	Bi HQ (2001). The self-thinning surface. Forest Science, 47, 361-370.
[6]	Bi HQ (2004). Stochastic frontier analysis of a classic self-thinning experiment. Austral Ecology, 29, 408-417.
[7]	Bi HQ, Turvey ND (1997). A method of selecting data points for fitting the maximum biomass-density line for stand undergoing self-thinning. Australian Journal of Ecology, 22, 356-359.
[8]	Bi HQ, Wan G, Turvey ND (2000). Estimating the self-thinning boundary line as a density-dependent stochastic biomass frontier. Ecology, 81, 1477-1483.
[9]	Blackburn TM, Lawton JH, Perry JN (1992). A method of estimating the slope of upper bounds of plots of body size and abundance in natural animal assemblages. Oikos, 65, 107-112.
[10]	Cousens R, Hutchings MJ (1983). The relationship between density and mean frond weight in monospecific seaweed stands. Nature, 301, 240-241.
[11]	del Río M, Montero G, Bravo F (2001). Analysis of diameter-density relationships and self-thinning in non-thinned even-aged Scots pine stands. Forest Ecology and Management, 142, 79-87.
[12]	Deng JM, Wang GX, Morris EC, Wei XP, Li DX, Chen BM, Zhao CM, Liu J, Wang Y (2006). Plant mass-density relationship along a moisture gradient in north-west China. Journal of Ecology, 94, 1-6. URL PMID
[13]	Enquist B (2002). Universal scaling in tree and vascular plant allometry: toward a general quantitative theory linking plant form and function from cells to ecosystems. Tree Physiology, 22, 1045-1064.
[14]	Enquist BJ, Brown JH, West GB (1998). Allometric scaling of plant energetics and population density. Nature, 395, 163-165.
[15]	Enquist BJ, Niklas KJ (2001). Invariant scaling relations across tree-dominated communities. Nature, 410, 655-660. DOI URL PMID
[16]	Enquist BJ, West GB, Charnov EL, Brown JH (1999). Allometric scaling of production and life-history variation in vascular plants. Nature, 401, 907-911.
[17]	Fang JY (方精云), Kan M, Yamakura T (1991). Relationships between population growth and population density in monocultures of Larix leptolepis. Acta Botanica Sinica (植物学报), 33, 949-957. (in Chinese with English abstract)
[18]	Fang JY (方精云) (1992). Self-thinning rule in plant population. Rural Eco-Environment (农村生态环境), (2), 7-12. (in Chinese with English abstract)
[19]	Franco M, Kelly CK (1998). The interspecific mass-density relationship and plant geometry. Proceedings of the National Academy of Science of United States of America, 95, 7830-7835.
[20]	Gorham E (1979). Shoot height, weight and standing crop in relation to density of monospecific plant stands. Nature, 279, 148-150.
[21]	Harper JL (1977). Population Biology of Plants. Acadamical Press, London, New York.
[22]	Hutchings MJ (1983). Ecology's law in search of a theory. New Scientist, 98, 765-767.
[23]	Inoue A, Nishizono T (2004). Allometric model of the Reineke equation for Japanese cypress (Chamaecyparis obtuse) and red pine (Pinus densiflora) stands. Journal of Forest Research, 9, 319-324.
[24]	Kikuzawa K (1999). Theoretical relationships between mean plant size, size distribution and self thinning under one-sided competition. Annals of Botany, 83, 11-18.
[25]	Kopp RJ, Mullahy J (1993). Least squares estimation of econometric frontier models: consistent estimation and inference. Scandinavian Journal of Economics, 95, 125-132.
[26]	Kozlowski J, Konarzewski M (2004). Is West, Brown and Enquist's model of allometric scaling mathematically correct and biologically relevant? Functional Ecology, 18, 283-289.
[27]	Kozlowski J, Konarzewski M (2005). West, Brown and Enquist's model of allometric scaling again: the same questions remain. Functional Ecology, 19, 739-743.
[28]	Li BL, Wu Hi, Zou G (2000). Self-thinning rule: a causal interpretation from ecological field theory. Ecological Modelling, 132, 167-173.
[29]	Li FR (李凤日) (1994). A Simulation System of Stand Dynamics for Larix olgensis Plantation (落叶松人工林林分动态模拟系统的研究). PhD dissertation, Beijing Forestry University, Beijing. 36. (in Chinese with English abstract)
[30]	Li FR (李凤日) (1995). A review on stand density-about the 3/2 Power Law. Forest Research (林业科学研究), 8(1), 25-32. (in Chinese with English abstract)
[31]	Li HT, Han XG, Wu JG (2005). Lack of evidence for 3/4 scaling of metabolism in terrestrial plants. Journal of Integrative Plant Biology, 47, 1173-1183.
[32]	Li HT, Han XG, Wu JG (2006). Variant scaling relationship for mass-density across tree-dominated communities. Journal of Integrative Plant Biology, 48, 268-277.
[33]	Lonsdale W (1990). The self-thinning rule: dead or alive? Ecology, 71, 1373-1388.
[34]	Mohler CL, Marks PL, Sprugel DG (1978). Stand structure and allometry of tree during self-thing of pure stand. Journal of Ecology, 68, 598-614.
[35]	Morris EC (2002). Self-thinning lines differ with fertility level. Ecological Research 17, 7-28.
[36]	Newton PE (2006). Asymptotic size-density relationships within self-thinning black spruce and jack pine stand-types: parameter estimation and model reformulations. Forest Ecology and Management, 226, 49-59.
[37]	Niklas K (1994). Plant Allometry and the Scaling of Form and Process. The University of Chicago Press, Chicago.
[38]	Niklas KJ, Midgley JJ, Enquist BJ (2003). A general model for mass-growth-density relations across tree-dominated communities. Evolutionary Ecology Research, 5, 459-468.
[39]	Noberg RA (1988). Theory of growth geometry of plants and self-thinning of plant populations: geometric similarity, elastic similarity, and different growth modes of plant parts. The American Naturalist, 131, 220-256.
[40]	Ogawa K (2005a). Relationship between mean shoot and root masses and density in an overcrowded population of hinoki (Chamaecyparis obtusa (Sieb. et Zucc.) Endl.) seedlings. Forest Ecology and Management, 213, 391-398.
[41]	Ogawa K (2005b). Time-trajectory of mean phytomass and density during a course of self-thinning in a sugi (Cryptomeria japonica D. Don) plantation. Forest Ecology and Management, 214, 104-110.
[42]	Ogawa K, Hagihara A (2003). Self-thinning and size variation in a sugi (Cryptomeria japonica D. Don) plantation. Forest Ecology and Management, 174, 413-421.
[43]	Osawa A (1995). Inverse relationship of crown fractal dimension to self-thinning exponent of tree population: a hypothesis. Canadian Journal of Forest Research, 25, 1608-1617.
[44]	Osawa A, Sugita S (1989). The self-thinning rule: another interpretation of Weller's results. Ecology, 70, 279-283.
[45]	Pretzsch H (2002). A unified law of spatial allometry for woody and herbaceous plants. Plant Biology, 4, 159-166.
[46]	Pretzsch H (2006). Species-specific allometric scaling under self-thinning: evidence from long-term plots in forest stands. Oecologia, 146, 572-583.
[47]	Pretzsch H, Biber P (2005). A re-evaluation of Reineke's rule and stand density index. Forest Science, 51, 304-320.
[48]	Reich PB, Tjoelker MG, Machado JL, Oleksyn J (2006). Universal scaling of respiratory metabolism, size and nitrogen in plants. Nature, 439, 457-461. DOI URL PMID
[49]	Reineke LH (1933). Perfecting a stand-density index for even-aged forests. Journal of Agricultural Research, 46, 627-638.
[50]	Reynolds JH, Forde D (2005). Improving competition representation in theoretical models of self-thinning: a critical review. Journal of Ecology, 93, 362-372.
[51]	Roderick ML, Barnes B (2004). Self-thinning of plant population from a dynamic viewpoint. Functional Ecology, 18, 197-203.
[52]	Sackville Hamilton NR, Matthew C, Lemaire G (1995). In defence of the -3/2 boundary rule: a re-evaluation of self-thinning concepts and status. Annals of Botany, 76, 569-577.
[53]	Scharf FS, Juanes F, Sutherland M (1998). Inferring ecological relationships from the edges of scatter diagrams: comparison of regression techniques. Ecology, 79, 448-460.
[54]	Solomon DS, Zhang LJ (2002). Maximum size-density relationships for mixed softwoods in the northeastern USA. Forest Ecology and Management, 155, 163-170.
[55]	Stoll P, Weiner J, Muller-Landau H, Müller E, Hara T (2002). Size symmetry of competition alters biomass-density relationships. Proceedings of the Royal Society Biology Science Series B, 269, 2191-2195.
[56]	Torres JL (2001). On the conceptual basis of the self-thinning rule. Oikos, 95, 544-548.
[57]	Wang RZ (王仁忠), Fang L (方林), Lu WX (卢文祥), Jiang SC (姜世成), Wang P (王平) (1998). Some comments on the -3/2 power law of self-thinning. Journal of Northeast Normal University (Natural Science Edition) (东北师大学报(自然科学版)). (4), 58-62. (in Chinese with English abstract)
[58]	Weller DE (1987a). A reevaluation of the -3/2 power rule of plant self-thinning. Ecological Monographs, 57, 23-43.
[59]	Weller DE (1987b). Self-thinning exponent correlated with allometric measures of plant geometry. Ecology, 68, 813-821.
[60]	Weller DE (1990). Will the real self-thinning rule please stand up? A reply to Osawa and Sugita. Ecology, 71, 1204-1207.
[61]	West GB, Brown JH, Enquist BJ (1997). A general model for the origin of allometric scaling laws in biology. Science, 276, 122-126. DOI URL PMID
[62]	West GB, Brown JH, Enquist BJ (1999). A general model for the structure and allometry of plant vascular systems. Nature, 400, 664-667.
[63]	Westoby M (1984). The self-thinning rule. Advances in Ecology Research, 14, 167-225.
[64]	White J (1980). Demographic Factors in Populations of Plants. University of California Press, Berkeley.
[65]	White J (1981). The allometric interpretation of self-thinning rule. Journal of Theoretical Biology, 89, 475-500.
[66]	White J, Harper JL (1970). Correlated changes in plant size and number in plant populations. Journal of Ecology, 58, 467-485.
[67]	Whitfield J (2001). All creatures great and small. Nature, 413, 342-344. DOI URL PMID
[68]	Wu CZ (吴承祯), Hong W (洪伟) (2000). A study on the self-thinning law of Chinese fir plantation. Scientia Silvae Sinicae (林业科学), 36(4), 97-101. (in Chinese with English abstract)
[69]	Xue L (薛立), Hagihara A (2001). Summary and evaluation of the researches on the self-thinning pure stands. Acta Ecologica Sinica (生态学报), 21, 835-838. (in Chinese with English abstract)
[70]	Xue L, Ogawa K, Hagihara A, Liang S, Bai J (1999). Self-thinning exponents based on the allometric model in Chinese pine (Pinus tabulaeformis Carr.) and Prince Rupprecht's larch (Larix principis-rupprechtii Mayr) stands. Forest Ecology and Management, 117, 87-93.
[71]	Yang Y, Titus SJ (2002). Maximum size-density relationship for constraining individual tree mortality functions. Forest Ecology and Management, 168, 259-273.
[72]	Yoda K, Kira T, Ogawa H, Hozumi K (1963). Self-thinning in overcrowded pure stand under cultivated and natural conditions. Journal of Biology of Osaka City University, 14, 107-129.
[73]	Zeide B (1985). Tolerance and self-tolerance of trees. Forest Ecology and Management, 13, 149-166.
[74]	Zeide B (1987). Analysis of the 3/2 power law of self-thinning. Forest Science, 33, 517-537.
[75]	Zeide B (2001). Natural thinning and environmental change: an ecological process model. Forest Ecology and Management, 154, 165-177.
[76]	Zeide B (2005). How to measure stand density. Trees, 19, 1-14.
[77]	Zhang LJ, Bi HQ, Jeffrey HG, Linda SH (2005). A comparison of alternative methods for estimating the self-thinning boundary line. Canadian Journal of Forest Research, 35, 1507-1514.