Abstract
The findings regarding the self-thinning rule in forest stands indicate that the rule may have a sort of ‘asymptotic’ status. It means the admittance of two theses: (1) the slopes of self-thinning trajectories may deviate from −3/2 value; (2) in the space ‘size–density,’ a limiting line exists that serves as a ‘goal’ for population dynamics which this dynamics tends to. An application of the simple geometrical model to the Douglas-fir and Scots pine data suggests that the slope of the self-thinning curve will not remain constant during the course of growth and self-thinning of a single forest stand. Most probably, at the initial stages of stand growth, the slope will be less than −3/2, and at old ages of the stand, the slope will be higher than −3/2. The slope −3/2 is thus an obligatory state in the course of self-thinning of a forest stand. At the very time of −3/2 slope, two particular features fit together. One is that the total bole surface area remains constant. Another feature of the −3/2 slope time is that geometric similarity holds in the growth of the forest stand, which is not in a contradiction with the ‘−3/2’ rule as it had been formulated by its authors. That is, the slope −3/2 (1) is a very specific and obligatory state in the process of forest stand growth and (2) is not an asymptote-like ‘goal’ but rather a transitional point (or may be a span) in the time of growth. These two assertions may be called a transitional status of the ‘−3/2’ rule.
Similar content being viewed by others
References
Curtis RO, Marshall DD (2009) Levels-of-growing-stock Cooperative Study in Douglas-fir: Report No. 19–the Iron Creek Study, 1966–2006. United States Department of Agriculture, Forest Service, Pacific Northwest Research Station
Curtis RO, Marshall DD (2009) Levels-of-growing-stock Cooperative Study in Douglas-fir: Report No. 18-Rocky Brook, 1963–2006. US Department of Agriculture, Forest Service, Pacific Northwest Research Station
Frothingham EH (1914) White pine under forest management. Bull US Dep Agric 13:1–70
Gavrikov VL (2014) A simple theory to link bole surface area, stem density and average tree dimensions in a forest stand. Eur J For Res 133:1087–1094. doi:10.1007/s10342-014-0824-4
Hamilton NRS, Matthew C, Lemair G (1995) In defense of the −3/2 boundary rule: a re-evaluation of self-thinning concepts and status. Ann Bot 76:569–577
Hilmi GF (1955) Biogeophysical theory and prognosis of forest self-thinning. Izd. AN SSSR, Moscow (in Russian)
Inoue A (2004) Relationships of stem surface area to other stem dimensions for Japanese cedar (Cryptomeria japonica D. Don) and Japanese cypress (Chamaecyparis obtusa Endl.) trees. J For Res 9:45–50
King JE, Marshall DD, Bell JF (2002) Levels-of-growing-stock Cooperative Study in Douglas-fir: Report No. 17-the Skykomish Study, 1961–93; the Clemons Study, 1963–94. Pacific Northwest Research Station, USDA Forest Service
Kofman GB (1986) Growth and form of trees. Nauka, Novosibirsk (in Russian)
Larjavaara M (2010) Maintenance cost, toppling risk and size of trees in a self-thinning stand. J Theor Biol 265(1):63–67
Lonsdale WM (1990) The self-thinning rule: dead or alive? Ecology 71:1373–1388
Marshall DD, Curtis RO (2001) Levels-of-growing-stock cooperative study in Douglas-fir: report no. 15-Hoskins: 1963–1998. United States Department of Agriculture, Forest Service
Newton PF, Smith VG (1990) Reformulated self-thinning exponents as applied to black spruce. Can J For Res 20(7):887–893
Perala DA, Leary RA, Cieszewski CJ (1999) Self-thinning and stockability of the circumboreal aspens (Populus tremuloides Michx. and P. tremula L.). Res. Pap. NC-335., U.S. Department of Agriculture, Forest Service, North Central Research Station, St. Paul, MN
Pretzsch H (2006) Species-specific allometric scaling under self-thinning: evidence from long-term plots in forest stands. Oecologia 146(4):572–583
Pretzsch H, Biber P (2005) A re-evaluation of Reineke’s rule and stand density index. For Sci 51(4):304–320
Reineke LH (1933) Perfecting a stand-density index for even-aged forests. J Agric Res 46(7):627–638
Sterba H (1987) Estimating potential density from thinning experiments and inventory data. For Sci 33(4):1022–1034
Usoltsev VA (2003) Forest phytomass of Northern Eurasia: limited productivity and geography. Ural Branch of Russian Academy of Sciences, Yekaterinburg (in Russian)
Usoltsev VA (2010) Eurasian forest biomass and primary production data. Ural Branch of Russian Academy of Sciences, Yekaterinburg (in Russian)
Vanclay JK, Sands PJ (2009) Calibrating the self-thinning frontier. For Ecol Man 259(1):81–85
Vospernik S, Sterba H (2014) Do competition-density rule and self-thinning rule agree? Ann For Sci. doi:10.1007/s13595-014-0433-x
West GB, Brown JH, Enquist BJ (1997) A general model for the origin of allometric scaling laws in biology. Science 276(5309):122–126
Whittaker RH, Woodwell GM (1967) Surface area relations of woody plants and forest communities. Am J Bot 54(8):931–939
Yoda K, Kira T, Ogawa H, Hozumi K (1963) Intraspecific competition among higher plants. XI. Self-thinning in over-crowded pure stands under cultivated and natural conditions. J Biol Osaka City Univ 14:107–129
Zeide B (1987) Analysis of the 3/2 power law of self-thinning. For Sci 33(2):517–537
Acknowledgments
The author is grateful to Prof. Dr. R.G. Khlebopros and Dr. O.P. Sekretenko who read the draft manuscript and made useful suggestions. The comments and suggestions made by two anonymous reviewers helped to improve the manuscript and are also greatly appreciated. The study was in part supported by the Russian Foundation for Basic Research, research Grant 14-05-00831 ‘The landscape features and the integral assessment of carbon storing function in protected forest areas of Siberian southern taiga.’ The author had no financial relationships with the sponsoring organization.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Arne Nothdurft.
Rights and permissions
About this article
Cite this article
Gavrikov, V.L. An application of bole surface growth model: a transitional status of ‘−3/2’ rule. Eur J Forest Res 134, 715–724 (2015). https://doi.org/10.1007/s10342-015-0885-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10342-015-0885-z