| Rozenberg Gennady Samuilovich | D.Sc., Institute of ecology of the Volga basin RAS, 10, Komzin st., Njgliatti, Samarsky region, Russia, 445003, genarozenberg@yandex.ru |
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Keywords: fractal multifractal community the structure of the ecosystem taxonomic diversity Samarskaya Luka |
Summary: The order maintaining fundamental physical and biological processes is the basis of the diversity of life and the complexity of ecosystems. Power laws describe empirical scaling relationships that are interpreted as quantitative characteristics of biodiversity. For these purposes, self-similarity concepts and the elements of fractal geometry are used. Self-similarity suggests that copying and scaling a certain “reference” image allows nature to easily create a complex multi-scale structure. Real objects have a quite clearly limited range of scales, in which they manifest their fractal nature. Power laws allow for extrapolation and prediction in a wide range of scales. Some of them appear to be universal and are found in almost all taxa of organisms and environmental types. We present a brief biography of an outstanding mathematician, Professor Benoit Mandelbrot (1924-2010), the originator of a new direction in geometry – fractal geometry. The theory of fractals developed by him had a significant impact on various areas of human activity and knowledge: computer graphics, finance, ecology, etc. The theory of multi-fractals characterized by an infinite hierarchy of dimensions provides the most general description of the internal structure of self-similar objects. The differences of the mono-fractal and multi-fractal approaches are considered. In applying multi-fractal formalism to a community structure, it is considered as a set consisting of individual fractal subsets which can be interpreted as the sets of individuals belonging to species with similar representation. For such subsets it is possible to calculate the fractal dimension characterizing the species diversity. The results of the application of the Willis rule to the analysis of taxonomic diversity of the flora of Samarskaya Luka (the Volga river basin) are discussed. It is stated that the patterns identified for the multi-fractal spectrum of the species structure of the community are also maintained for the spectrum of a generic structure to a certain extent. © Petrozavodsk State University |
Published on: 29 December 2018 | |
References
A maverick’s apprenticeship. The Wolf Prize for Physics, Ed. by D. Thouless. Singapore: World Scientific, 2004.
Arthur R. H. Wilson E. O. 2nd ed. Princ. Univ. Press, 2001. 203 p..
Azovskiy A. I. Chertoprud M. V. Analysis of the Spatial Organization of Communities and the Fractal Structure of the Littoral Benthos, Dokl. Akademii nauk (DAN). 1997. T. 356. No. 5. P. 713–715.
Azovskiy A. I. Chertoprud M. V. Scale-Oriented Approach to the Analysis of the Spatial Structure of Communities, Zhurn. obsch. biol. 1998. T. 59. No. 2. P. 117–136.
Azovsky A. I. Concept of scale in marine ecology: linking the words or the worlds?, Web Ecol. 2000. No. 1. R. 28–34. URL: http://www.oikos.ekol.lu.se/we/we.html (data obrascheniya: 25.10.2007).
Azovsky A. I., Chertoprood M. V., Kucheruk N. V., Rybnikov P. V., Sapozhnikov F. V. Fractal properties of spatial distribution of intertidal benthic communities, Marine Biol. 2000. Vol. 136. No 3. P. 581–590.
Bak P., Tang C., Weisenfeld K. Self-organized criticality, Phys. Rev. 1988. A 38. P. 364–374.
Beklemishev V. N. On the general principles of the organization of life, Byul. MOIP. Otd. biol. 1964. T. 69. Vyp. 2. P. 22–38.
Bigon M. Taunsend K. Ecology: Individuals, populations, communities: V 2 t. M.: Mir, 1989. T. 1. 667 p.; T. 2. 477 p.
Bossuyt B., Hermy M. Species turnover at small scales in dune slack plant communities, Basic and Applied Ecol. 2004. Vol. 5. P. 321–329.
Brown J. H., Gupta V. K., Li B. L., Milne B. T., Restrepo C., West G. B. The fractal nature of nature: power laws, ecological complexity and biodiversity, Philos. Trans. R. Soc. Lond. B. Biol. Sci. 2002. Vol. 357. No 1421. P. 619–626.
Buck R. C., Hull D. L. The logical structure of the Linnean hierarchy, Syst. Zool. 1966. Vol. 15. P. 97–111.
Bulgakov N. G. Levich A. P. Maksimov V. N. Regional Environmental Control Based on Biotic and Abiotic Monitoring Data, Ekologicheskiy monitoring. Metody biologicheskogo i fiziko-himicheskogo monitoringa. Ch. V. N. Novgorod: Izd-vo NNGU, 2003. C. 93–259.
Burlando B. The fractal dimension of taxonomic systems, J. Theor. Biol. 1990. Vol. 146. P. 99–114.
Burlando B. The fractal geometry of evolution., J. Theor. Biol. 1993. Vol. 163. P. 161–172.
Chertoprud M. V. Azovskiy A. I. Location of the macrobenthos of the White Sea littoral in various scales of the space, Zhurn. obsch. biol. 2000. T. 61. No. 1. P. 47–63.
Chislenko L. L. On the structure of taxa and taxonomic diversity, Zhurn. obsch. biol. 1977. T. 38. No. 3. P. 348–358.
Dicke M., Burrough P. A. Using fractal dimensions for characterizing tortuosity of animal trails, Physiol. Entomol. 1988. Vol. 13. P. 393–398.
Dzhiller P. Community structure and ecological niche. M.: Mir, 1988. 184 p.
Feder E. Fractals. M.: Mir, 1991. 254 p. (Feder J. Fractals. N. Y.: Plenum Pub. Corp., 1988. 283 r.).
Fedorov V. D. Relative abundance of sympatric species and the model of an exponentially broken rod, Chelovek i biosfera. M.: Izd-vo MGU, 1978. Vyp. 2. P. 17–41.
Filippov A. E. Discrete Speciation and the Willis Law, Zhurn. obsch. biol. 1984. T. 45. No. 3. P. 410–418.
Gelashvili D. B. Dmitriev A. I. Iudin D. I. Rozenberg G. S. Solncev L. A. Multifractal analysis of the species structure of small mammal communities of the Volga-Ural paleocenosis, Dokl. Akademii nauk (DAN). 2008a. T. 421. No. 4. P. 562–566.
Gelashvili D. B. Iudin D. I. Rozenberg G. S. Solncev L. A. Fedyunin V. A. Yakimov V. N. Fractal characterization of the species structure of the communities of ichneummonid equestrians Urals, Dokl. Akademii nauk (DAN). 2010a. T. 434. No. 6. P. 838–841.
Gelashvili D. B. Iudin D. I. Rozenberg G. S. Solncev L. A. Ivanova I. O. Biodiversity and climate from the standpoint of the theory of fractals, Izv. SamNC RAN. 2007a. Spec. vypusk «ELPIT-2007». Ser. «Ekologiya». P. 139–150.
Gelashvili D. B. Iudin D. I. Rozenberg G. S. Solncev L. A. Ivanova I. O. Biodiversity and climate from the standpoint of the theory of fractals, Sb. tr. Pervogo mezhdunar. ekologicheskogo kongressa (tret'ey mezhdunar. nauchno-tehn. konf.) «Ekologiya i bezopasnost' zhiznedeyatel'nosti promyshlenno-transportnyh kompleksov» ELPIT 2007. (20–23 sentyabrya 2007 g., Tol'yatti, Rossiya). T. 1. Tol'yatti: TolGU, 2007b. P. 214–219.
Gelashvili D. B. Iudin D. I. Rozenberg G. S. Solncev L. A. Yakimov V. N. Multifractal structures in bioecology, Nelineynyy mir. 2008b. T. 6. No. 11–12. P. 697–703.
Gelashvili D. B. Iudin D. I. Rozenberg G. S. Yakimov V. N. Shurganova G. V. The power law and the principle of self-similarity in the description of the species structure of communities, Povolzh. ekol. zhurn. 2004. No. 3. P. 227–245.
Gelashvili D. B. Iudin D. I. Rozenberg G. S. Yakimov V. N. Solncev L. A. Varichev A. N. Fractal Aspects of Population Ecology, Vestn. Udmurt. un-ta. 2009a. Vyp. 1. P. 15–22.
Gelashvili D. B. Iudin D. I. Rozenberg G. S. Yakimov V. N. Solncev L. A. Basics of multifractal analysis of the species structure of a community, Uspehi sovrem. biol. 2008v. T. 128. No. 1. P. 21–34.
Gelashvili D. B. Iudin D. I. Rozenberg G. S. Yakimov V. N. Solncev L. A. Fractals and Multifractals in Bioecology. N. Novgorod: Izd-vo Nizhegorod. gop. un-ta, 2013a. 370 p.
Gelashvili D. B. Iudin D. I. Rozenberg G. S. Yakimov V. N. Elements of the fractal theory of the species structure of hydrobiocenoses, Izv. SamNC RAN. 2006b. T. 8. No. 1. P. 70–79.
Gelashvili D. B. Iudin D. I. Rozenberg G. S. Yakimov V. N. The power-lawn nature of accumulation of species richness as a manifestation of the fractal structure of a biocenosis, Zhurn. obsch. biol. 2007v. T. 68. No. 3. P. 170–179.
Gelashvili D. B. Iudin D. I. Rozenberg G. S. Fractal Structure of the Percolation Cluster and Spatial Distribution of Dominant Species, Dokl. Akademii nauk (DAN). 2006a. T. 408. No. 4. P. 560–563.
Gelashvili D. B. Iudin D. I. Solncev L. A. Rozenberg G. S. Evlanov I. A. Kirillova N. Yu. Kirillov A. A. Multifractal analysis of the species structure of helminth communities of small mammals of the Samara Luka, Dokl. Akademii nauk (DAN). 2009b. T. 427. No. 5. P. 703–706.
Gelashvili D. B. Iudin D. I. Yakimov V. N. Solncev L. A. Rozenberg G. S. Shurganova G. V. Ohapkin A. G. Starceva N. A. Puhnarevich D. A. Multifractal analysis of the species structure of freshwater hydrobiocenoses, Izv. RAN. Ser. biol. 2012. No. 3. P. 327–335.
Gelashvili D. B. Rozenberg G. S. Iudin D. I. Yakimov V. N. Solncev L. A. Fractal Aspects of the Structural Stability of Biotic Communities, Biosfera. 2013b. T. 5. No. 2. P. 143–159.
Gelashvili D. B. Rozenberg G. S. Saksonov S. V. Ivanova I. O. Vehnik V. P. Species structure of large mammal communities of Samarskaya Luka in connection with the problem of climate change, Izv. Samar. NC RAN. 2009v. T. 11. No. 1 (2). P. 28–41.
Gelashvili D. B. Saksonov S. V. Rozenberg G. S. Iudin D. I. Snegireva M. S. Solncev L. A. Yakimov V. N. The floristic phenomenon of Samara Luka: the fractal structure of taxonomic diversity, Samarskaya Luka: problemy regional'noy i global'noy ekologii: Byul. 2011. T. 20. No. 2 (36). P. 80–104.
Gelashvili D. B. Yakimov V. N. Iudin D. I. Rozenberg G. S. Solncev S. A. Saksonov S. V. Fractal Aspects of Taxonomic Diversity, Zhurn. obsch. biol. 2010b. T. 71. No. 2. P. 115–130.
Gelashvili D. B., Iudin D. I., Rozenberg G. S., Solntsev L. A., Ivanova I. O. Species structure of communities: explanation and forecast under "global warming" scenario (fractal theory approach), Climate Change and Possible Implications for the Volga Basin Ecosystem. The Volga River Basin in 50 Years: Perspectives and Forecast: Materials of the Conference. Togliatti: British Council, 2007. P. 11–13.
Gleason H. A. On the relation between species and area, Ecology. 1922. Vol. 3. P. 158–162.
Green D. M. Chaos, fractals and nonlinear dynamics in evolution and phylogeny, Trends in Ecology and Evolution. 1991. Vol. 6. P. 333–337.
Green J. L., Harte J., Ostling A. Species richness, endemism and abundance patterns: tests of two fractal models in a serpentine grassland, Ecol. Letters. 2003. Vol. 6. P. 919–928.
Harte J., Kinzig A. P., Green J. L. Self-similarity in the distribution and abundance of species, Science. 1999. Vol. 284. P. 334–336.
In his own words: BBM Interview by Anthony Barcellos, Mathematical People, Ed. by D. J. Albers & G. L. Alexanderson. Boston: Birkhäuser, 1985. P. 205–225.
Iudin D. I. Gelashvili D. B. Rozenberg G. S. Solncev L. A. Yakimov V. N. Biological and environmental aspects of the theory of percolation, Uspehi sovr. biol. 2010. T. 130. No. 5. P. 446–460.
Iudin D. I. Gelashvili D. B. Rozenberg G. S. Multi-fractal analysis of the species structure of biotic communities, Dokl. Akademii nauk (DAN). 2003. T. 389. No. 2. P. 279–282.
Iudin D. I. Gelashvili D. B. Application of multifractal analysis of the structure of biotic communities in environmental monitoring, Problemy regional'nogo ekologicheskogo monitoringa: Materialy nauch. konf. N. Novgorod: Izd-vo NNGU, 2002. P. 49–52.
Iudin D. I. Methodology of the principle of self-similarity in the study of the species structure of biotic communities: Dip. … d-ra biol. nauk. Tol'yatti, 2006. 273 p.
Iudin D. I., Gelashvili D. B. Multifractality in ecological monitoring, Nucl. Instr. Meth. Phys. Res. 2003. Vol. 502. P. 799–801.
Iudin D. I., Gelashvili D. B., Rozenberg G. S., Solntcev L. A., Yakimov V. N. Bases of the multifractal analysis of species structure of community, Types of Strategy and Not Only… (Materials of the Fourth Russian-Polish School of Young Ecologists; Togliatti, September, 6–12th, 2010), Editor-in-chief G. S. Rozenberg. Togliatti: Kassandra, 2010. P. 17–19.
Ivanova A. V. Rozenberg G. S. Saksonov S. V. Experience in a quantitative analysis of floristic diversity and floristic structure of Samara Luka, Ekologiya. 2006. No. 5. P. 332–339.
Kafanov A. I. Suhanov V. V. On the relationship between the number and volume of taxa, Zhurn. obsch. biol. 1981. T. 42. No. 3. P. 345–350.
Laurie H., Perrier E. A multi-fractal model for the species–area relationship. 2006. 18 p. URL: www.mth.uct.ac.za/~henri/multfrac6par.pdf (data obrascheniya: 12.12.2009).
Levich A. P. The structure of ecological communities. M.: Izd-vo MGU, 1980. 180 c.
Li B, L. Fractal geometry applications in description and analysis of patch patterns and patch dynamics, Ecol. Model. 2000. Vol. 132. P. 33–50.
Lukicheva A. N. Saburov D. N. Specific flora and landscape flora, Bot. zhurn. 1969. No. 12. P. 1911–1920.
MacArthur R. H. On the relative abundance of species, Amer. Nat. 1960. Vol. 94. P. 25–36.
Makarenko N. G. Fractals, attractors, neural networks and all that, Nauchnaya sessiya MIFI-2002. IV Vserossiyskaya nauchno-tehn. konf. «Neyroinformatika – 2002». Lekcii po neyroinformatike. Ch. 2. M.: MIFI, 2002. P. 121–169.
Mandel'brot B. B. Fractal Geometry of Nature. M.; Izhevsk: In-t komp'yut. issl., 2002. 656 p.
Mandel'brot B. B. Fractals and the revival of the theory of iterations, Paygen H, O., Rihter P. H. Krasota fraktalov. Obrazy kompleksnyh dinamicheskih sistem. M.: Mir, 1993. P. 131–140.
Mandelbrot B. B. Fractals: Form, Chance and Dimension. San Francisco (CA): W. H. Freeman and Co., 1977. 265 p.
Mandelbrot B. B. How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science. 1967. Vol. 156. No 3775. P. 636–638.
Margalef R. The appearance of the biosphere. M.: Nauka, 1992. 214 c.
Mayr E. Principles of zoological systematics. M.: Mir, 1971. 454 p.
Mel'chenko V. E. Landscapes of the Samara Luka, Samarskaya Luka: Byul. 1991. No. 1. P. 45–62.
Mirkin B. M. Naumova L. G. The science of vegetation (history and current state of basic concepts). Ufa: Gilem, 1998. 413 p.
Morozov A. D. Introduction to the theory of fractals. M.; Izhevsk: In-t komp'yut. issl., 2004. 160 p.
Morse D. R., Lawton J. H., Dodson M. M., Williamson M. H. Fractal dimension of vegetation and the distribution of arthropod body lengths, Nature. 1985. Vol. 314. No 6013. P. 731–733.
Motomura I. A statistical treatment of associations, Jap. J. Zool. 1932. Vol. 44. P. 379–383.
Odum Yu. Fundamentals of Ecology. M.: Mir, 1975. 740 p.
Pachepsky Ya. A., Gimenez D., Crawford J. W., Rawls W. J. Bibliography on applications of fractals in soil science, Fractals in Soil Science, Ed. by Ya. Pachepsky, J. Crawford, W. Rawls. Amsterdam; N. Y.: Elsevier, 2000b. P. 273–295.
Pachepsky Ya. A., Gimenez D., Crawford J. W., Rawls W. J. Conventional and fractal geometry in soil science, Fractals in Soil Science, Ed. by Ya. Pachepsky, J. Crawford, W. Rawls. Amsterdam; N. Y.: Elsevier, 2000a. P. 7–18.
Pachepsky Ya. A., Ritchie J. C. Seasonal changes in fractal landscape surface roughness estimated from airborne laser altimetry data, Int. J. Remote Sensing. 1998. Vol. 19. P. 2509–2516.
Pachepsky Ya. A., Timlin D. Water transport in soils as in fractal media, J. Hydrology. 1998. Vol. 204. No 1. P. 98–107.
Paygen H. Rihter P. H. The Beauty of Fractals. Images of complex dynamic systems. M.: Mir, 1993. 176 p. (Peitgen H, O., Richter P. H. The Beauty of Fractals. Images of Complex Dynamical Systems. Heidelberg; N. Y.: Springer-Verlag, 1986. 199 r.).
Pounds J. A., Puschendorf R. Clouded futures, Nature. 2004. Vol. 427. P. 107–109.
Pozdnyakov A. A. The Problem of Individuality in Taxonomy, Zhurn. obsch. biol. 2003. T. 64. No. 1. P. 55–64.
Pozdnyakov A. A. The Significance of the Willis Rule for Taxonomy, Zhurn. obsch. biol. 2005. T. 66. No. 4. P. 326–335.
Preston F. W. The canonical distribution of commonness and rarity: Part I, Ecology. 1962. Vol. 43. No 2. P. 185–215.
Puzachenko Yu. G. Puzachenko A. Yu. Semantic aspects of biodiversity, Zhurn. obsch. biol. 1996. T. 57. No. 1. P. 5–43.
Puzachenko Yu. G. Application of the theory of fractals to the study of the landscape structure, Izv. RAN. Ser. geogr. 1997. No. 2. P. 24–40.
Roschevskiy Yu. K. Preserves of the USSR. National parks and reserves, Zapovedniki SSSR. Nacional'nye parki i zakazniki. M.: ABF, 1996. P. 34–43.
Rosenzweig M. L. On continental steady states of species diversity, M. L. Cody, J. M. Diamond (Eds.). Ecology and Evolution of Communities. Cambridge (Mass.): Harvard Univ. Press, 1975. P. 121–140.
Rozenberg G. S. Gelashvili D. B. Iudin D. I. Fractal Organization of Ecosystems: cui prodest?, Ekologo-geograficheskie problemy prirodopol'zovaniya neftegazovyh regionov: Teoriya, metody, praktika: Materialy II Mezhdunar. nauchno-prakt. konf. Nizhnevartovsk: NGPI, 2003b. P. 342–347.
Rozenberg G. S. Gelashvili D. B. Iudin D. I. Fractals - a new language of theoretical ecology, Sb. tr. Pervogo mezhdunar. ekologicheskogo kongressa (tret'ey mezhdunar. nauchno-tehn. konf.) «Ekologiya i bezopasnost' zhiznedeyatel'nosti promyshlenno-transportnyh kompleksov» ELPIT 2007 (20–23 sentyabrya 2007 g., Tol'yatti, Rossiya). T. 1. Tol'yatti: TolGU, 2007. P. 4–9.
Rozenberg G. S. Gelashvili D. B. Iudin I. D. Fractal organization: do we need it?, Ekologicheskie problemy zapovednyh territoriy Rossii. Tol'yatti: IEVB RAN, 2003a. P. 61–68.
Rozenberg G. S. Mozgovoy D. P. Gelashvili D. B. Elements of theoretical constructions of modern ecology. Samara: SNC RAN, 1999. 396 p.
Rozenberg G. S. Information index and diversity: Boltzmann, Kotelnikov, Shannon, Weaver ..., Samarskaya Luka: problemy regional'noy i global'noy ekologii: Byul. 2010. T. 19. No. 2. P. 4–25.
Rozenberg G. S. Introduction to Theoretical Ecology: V 2 t. Tol'yatti: Kassandra, 2013. T. 1. 565 p.; T. 2. 445 p.
Rozenberg G. S. Statistical methods in phytocenology at the turn of the millennium (on the 50th anniversary of the publication of the monograph by P. Greig-Smith), Aktual'nye problemy geobotaniki: III Vserop. shkola-konf: Lekcii. Petrozavodsk: KarNC RAN, 2007. P. 72–116.
Ryabko B. Ya. Kudrin B. I. Zavalishin N. N. Kudrin A. I. Model of the formation of the statistical structure of a biocenosis, Izv. AN SSSR. Ser. biol. 1978. Vyp. 1. P. 121–127.
Saksonov S. V. Basics of large-scale floristic zoning of the Samara Luka (East of the Central part of the Volga Upland)., Samarskaya Luka: Byul. 1996. No. 7. P. 70–98.
Saksonov S. V. Samaraluksky floristic phenomenon. M.: Nauka, 2006. 263 p.
Scheuring I. The fractal nature of vegetation and the species-area relation, Theor. Populat. Biol. 1991. Vol. 39. P. 170–177.
Scheuring I., Riedi R. H. Application of multifractals to the analysis of vegetation patterns, J. Veg. Sci. 1994. Vol. 5. P. 489–496.
Seuront L. Fractals and Multifractals in Ecology and Aquatic Science. Boca Raton (FL): CRC Press, 2010. 344 p.
Seuront L., Schmitt F., Lagadeuc Y., Schertzer D., Lovejoy S., Frontier S. Multifractal analysis of phytoplankton biomass and temperature in the ocean, Geophys. Res. Lett. 1996. Vol. 23. P. 3591–3594.
Shatalkin A. I. Hierarchies in Systematics. Set-theoretic model, Zhurn. obsch. biol. 1995. T. 56. No. 3. P. 277–290.
Shitikov V. K. Rozenberg G. S. Zinchenko T. D. Quantitative hydroecology: methods, criteria, solutions: V 2 kn. M.: Nauka, 2005. Kn. 1. 281 p.; Kn. 2. 337 p.
Shreder M. Fractals, chaos, power laws. Miniatures from infinite paradise. M.; Izhevsk: In-t komp'yut. issl., 2001. 528 c.
Sugihara G., May R. M. Applications of fractals in ecology, Trends in Ecology and Evolution. 1990. Vol. 5. P. 79–86.
Turchin P. Fractal analysis of animal movement: a critique, Ecology. 1996. Vol. 77. P. 2086–2090.
Uilkoks B. Island Ecology and Nature Conservation, Biologiya ohrany prirody. M.: Mir, 1983. P. 117–142.
Wagensberg J. In memoriam. Benoit Mandelbrot y la fractalidad del mundo, El País. 19.10.2010. 44 p.
Willis J. C. The Course of Evolution by Differentiation or Divergent Mutation Rather than by Selection. London: Cambridge Univ. Press, 1940. 207 p.
With K. A., Cadaret S. J., Davis C. Movement responses to patch structure in experimental fractal landscapes, Ecology. 1999. Vol. 80. P. 1340–1353.
Yakimov B. N., Gelashvili D. B., Solntsev L. A., Iudin D. I., Rozenberg G. S. Nonconcavity of mass exponents’ spectrum in multifractal analysis of community spatial structure: The problem and possible solutions, Ecological Complexity. 2014. Vol. 20. P. 11–22.
Yakimov V. N. Gelashvili D. B. Iudin D. I. Rozenberg G. S. Markelov I. N. The universality of diversity scaling in neutral and niche models of ecological communities, Mezhdunar. nauch, issled. zhurn. 2016a. No. 12 (54). Ch. 1. P. 40–45.
Yakimov V. N. Gelashvili D. B. Rozenberg G. S. Bezel' V. S. Scaling of phylogenetic diversity in small mammals (on the example of the Nizhny Novgorod Volga region), Ekologiya. 2017. No. 3. P. 210–215.
Yakimov V. N. Gelashvili D. B. Rozenberg G. S. Krivonogov D. M. Modern methods of quantitative analysis of the phylogenetic diversity of ecological communities, Aspekty bioraznoobraziya: Sb. trudov Zoologicheskogo muzeya MGU im. M. V. Lomonosova. M.: Tov-vo nauch. izd. KMK, 2016b. T. 54 (1). Ch. 1. P. 72–98.
Yakimov V. N. Solncev L. A. Iudin D. I. Rozenberg G. S. Gelashvili D. B. Violations of the canonical form of the multifractal spectrum of the spatial structure of a community: causes and solutions, Izv. Samar. NC RAN. 2014a. T. 16. No. 1. P. 19–26.
Yakimov V. N. Solncev L. A. Rozenberg G. S. Iudin D. I. Gelashvili D. B. Large-scale invariance of biosystems: from the embryo to the community, Ontogenez. 2014b. T. 45. No. 3. P. 207–216.
Yakimov V. N. Solncev L. A. Rozenberg G. S. Iudin D. I. Shirokov A. I. Lokteva O. A. Gelashvili D. B. Local multifractal analysis of the spatial structure of meadow communities in small scales, Dokl. Akademii nauk (DAN). 2014v. T. 458. P. 613–617.
Yakimov V. N. Fractality of species and spatial structure of biological communities: concept development and verification: Dip. ... kand. biol. nauk. N. Novgorod, 2007. 125 p.
Zeide B. Fractal geometry in forestry applications, For. Ecol. Manage. 1991. Vol. 46. P. 179–188.
Zhang Y., Ma K., Anand M., Fu B. Do generalized scaling laws exist for species abundance distribution in mountains?, Oikos. 2006. Vol. 115. P. 81–88.
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