A Unified Law of Spatial Allometry for Woody and Herbaceous Plants

Abstract

The objective of the present paper is to provide both proof and theoretical deduction of an overlapping, valid law of allometry for woody and herbaceous plants used in agriculture and forestry. In his attempt to find an adequate expression for stand density, independent of site quality and age, Reineke (1933[18]) developed the following equation for even-aged and fully stocked forest stands in the northwest of the USA: ln(N) = a - 1.605 · ln(dg), based on the relationship between the average diameter dg and the number N of trees per unit area. With no knowledge of these results, Kira et al. (1953[9]) and Yoda et al. (1957[31] and 1963[32]) found the boundary line ln(m) = b - 3/2 · ln(N) in their study of herbaceous plants. This self-thinning rule - also called the - 3/2-power rule - describes the relationship between the average weight m of a plant and the density N in even-aged herbaceous plant populations growing under natural development conditions. It is possible to make a transition from Yoda's rule to Reineke's stand density rule if mass m in the former rule is substituted by the diameter dg. From biomass analyses for the tree species spruce (Picea abies [L.] Karst.) and beech (Fagus sylvatica L.), allometric relationships between biomass m and diameter d are derived. Using the latter in the equation ln(m) = b - 3/2 · ln(N) leads to allometric coefficients for spruce (Picea abies [L.] Karst.) and beech (Fagus sylvatica L.), that come very close to the Reineke coefficient. Thus Reineke's rule (1933[18]) proves to be a special case of Yoda's rule. Both rules are based on the simple allometric law governing the volume of a sphere v and its surface of projection s: v = c₁ · s^3/2. If the surface of projection s, is substituted by the reciprocal value of the number of stems s = 1/N and the isometric relationship between volume v and biomass m is considered v = c₂ · m^1.0 we come to Yoda's rule m = c₃ · N^-3/2 or, in logarithmic terms, ln(m) = ln c₃ - 3/2 · ln(N).

Key words

Self-thinning - allometry - - 3/2-power rule - stand density rule - woody and herbaceous plants

References

• 01 Bergel,  D.. (1985) Douglasien-Ertragstafel für Nordwestdeutschland. Niedersächsische Forstliche Versuchsanstalt, Abt. Waldwachstum 72 p.
• 02 Bertalanffy,  v. L.. (1951) Theoretische Biologie II. Band, Stoffwechsel, Wachstum. London; A. Francke AG, Verlag 418 p.
• 03 Ellenberg,  H.,, Mayer,  R.,, and Schauermann,  J.. (1996) Ökosystemforschung - Ergebnisse des Sollingprojektes. Stuttgart; Ulmer Verlag
• 04 Franz,  F.. (1963);  Durchmesserdifferenzprozent und Vornutzungsprozent: Ein neues Verfahren zur Herleitung der Gesamtwuchsleistung von Fichtenbeständen aus einmalig erhobenen Bestandeskennwerten.  Allgemeine Forst- und Jagdzeitung. 134 (7, 8) 181-197a, 201 - 214
• 05 Franz,  F.. (1965);  Ermittlung von Schätzwerten der natürlichen Grundfläche mit Hilfe ertragskundlicher Bestimmungsgrößen des verbleibenden Bestandes.  Forstw. Cbl.. 84 357-386
• 06 Franz,  F.. (1967);  Ertragsniveau-Schätzverfahren für die Fichte anhand einmalig erhobener Bestandesgrößen.  Forstw. Cbl.. 86 (2) 98-125
• 07 Franz,  F.. (1968) Das EDV-Programm STAOET - zur Herleitung mehrgliedriger Standort-Leistungstafeln. München, unveröff.; Manuskriptdruck
• 08 Harper,  J. L.. (1977) Population Biology of Plants. London, New-York, San-Francisco; Academic Press 892 p.
• 09 Kira,  T.,, Ogawa,  H.,, and Sakazaki,  N.. (1953) Intraspecific competition among higher plants, I. Competition-yield-density interrelationship in regularly dispersed populations, Vol. 4, No. 1, Series D. Osaka City University; Journal of the Institute of Polytechnics pp. 1-16
• 10 Kramer,  H., and Helms,  J. A.. (1985);  Zur Verwendung und Aussagefähigkeit von Bestandesdichteindizes bei Douglasie.  Forstw. Cbl.. 104 36-49
• 11 Matyssek,  R., and Elstner,  E. F.. (1997) Wachstum oder Parasitenabwehr? Wettbewerb um Nutzpflanzen aus Land- und Forstwirtschaft. Beantragung eines im Forschungsraum München geplanten Sonderforschungsbereiches (SFB 1642 - 98) 586 p.
• 12 Meschederu,  M.. (1997) Bilanzierung von Biomassen, Kohlenstoff- und Stickstoffgehalten mit dem Wuchsmodell SILVA 2.1. Grünberg; Bericht von der Jahrestagung der Sektion Ertragskunde im Deutschen Verband Forstlicher Forschungsanstalten 1997 pp. 35-44
• 13 Niklas,  K. J.. (1994) Plant Allometry. The scaling of form and process. Chicago; The University of Chicago Press 395 p.
• 14 Pellinen,  P.. (1986) Biomasseuntersuchungen im Kalkbuchenwald. Göttingen; Dissertation, Forstwiss. Fak. der Universität Göttingen 145 p.
• 15 Pretzsch,  H.. (1999);  Waldwachstum im Wandel, Konsequenzen für Forstwissenschaft und Forstwirtschaft.  Forstwiss. Cbl.. 118 228-250
• 16 Pretzsch,  H.. (2000);  Die Regeln von Reineke, Yoda und das Gesetz der räumlichen Allometrie.  Allgemeine Forst- und Jagdzeitung. 171 (11) 205-210
• 17 Pretzsch,  H.. (2001) Modellierung des Waldwachstums. Berlin, Wien; Blackwell Wissenschafts-Verlag 341 p.
• 18 Reineke,  L. H.. (1933);  Perfecting A Stand-Density Index For Even-Aged Forests.  Journal of Agricultural Research. 46 (7) 627-638
• 19 del Río,  M.,, Montero,  G.,, and 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
• 20 Sackville Hamilton,  N. R.,, Matthew,  C.,, and 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
• 21 Silvertown,  J. W.. (1992) Introduction to plant population ecology. Essex; Longman Scientific and Technical
• 22 Spencer,  H.. (1864) The principles of Biology, Vol. 1. London; Williams and Norgate
• 23 Sterba,  H.. (1975);  Assmanns Theorie der Grundflächenhaltung und die “Competition-Density-Rule” der Japaner Kira, Ando und Tadaki.  Centralblatt für das gesamte Forstwesen. 92 (1) 46-62
• 24 Sterba,  H.. (1981);  Natürlicher Bestockungsgrad und Reinekes SDI.  Centralblatt für das gesamte Forstwesen. 98 (2) 101-116
• 25 Sterba,  H.. (1987);  Estimating potential density from thinning experiments and inventory data.  Forest Science. 33 (4) 1022-1034
• 26 Thompson,  D. W.. (1917) On growth and form. Cambridge; Cambridge University Press
• 27 Weller,  D. E.. (1987);  A Reevaluation Of The - 3/2 Power Rule Of Plant Self-Thinning.  Ecological Monographs. 57 (1) 23-43
• 28 Weller,  D. E.. (1990);  Will The Real Self-Thinning Rule Please Stand Up? A Reply To Osawa And Sugita.  Ecology. 71 (3) 1204-1207
• 29 White,  J.. (1981);  The allometric interpretation of the self-thinning rule.  Journal of theoretical Biology. 89 475-500
• 30 Whittington,  R.. (1984);  Laying down the - 3/2 power law.  Nature. 311 217
• 31 Yoda,  K.,, Kira,  T.,, and Hozimu,  K.. (1957) Intraspecific competition among higher plants, IX. Further analysis of the competitive interaction between adjacent individuals, Vol. 8. Osaka City University; Journal of the Institute of Polytechnics pp. 161-178
• 32 Yoda,  K.,, Kira,  T.,, Ogawa,  H.,, and Hozumi,  K.. (1963);  Self-thinning in overcrowded pure stands under cultivated and natural conditions (Intraspecific competition among higher plants XI).  Journal of the Institute of Polytechnics, Osaka City University, Series D. 14 107-129
• 33 Zeide,  B.. (1987);  Analysis of the 3/2 power law of self-thinning.  Forest Science. 33 (2) 517-537

Section Editor: U. Lüttge