Considering the problem of taxonomic diversity as a fractal object is the aim of this article. The prerequisites for such an approach were articles with varying degrees of detail and argumentation that substantiate taxonomic diversity from the standpoint of fractal geometry. Common to these papers is that the authors in their theoretical constructs start from the Willis rule (law) describing the rank distribution of the relationship between the number of taxa and their volume. The flora of the Samara Bend (the bend of the Volga River in its middle reaches) has become an object of the research. The authors distinguish seven basic floristic areas on the Samara Bend, the boundaries of which coincide with the respective landscapes. The authors discuss the efficiency of the Willis rule (law), which approximates the relationship between the number of taxa and their volume by rank distribution. The multifractal spectrum (a generalized geometric image of generic structure) of the taxonomic diversity of vascular plants of the Samara Bend is presented.
Keywords: taxonomic diversity, fractal organization, Samara Ben

Gelashvili, D B

Iudin, D I

Rozenberg, G S

Saksonov, S V

Yakimov, V N

KnE Life Sciences

English

KnE Publishing Platform

Ecology and Geography of Plants and Plant CommunitiesThe fourth International Scientific Conference onEcology and Geography of Plants and Plant CommunitiesVolume 2018Conference PaperFloristic Phenomena of the Samara Bend:The Fractal Organization of TaxonomicDiversityG. S. Rozenberg1,3, D. B. Gelashvili2, D. I. Iudin2, S. V. Saksonov1,4, andV. N. Yakimov21Institute of Ecology of the Volga River Basin, Russian Academy of Sciences, 445003 Togliatti,Russia2N. I. Lobachevsky State University of Nizhni Novgorod, 603950 Nizhni Novgorod, Russia3Samara State University of Economics, 443090 Samara, Russia4UNESCO Chair ‘The study and conservation of biodiversity of the Volga Basin ecosystems’,Institute of Ecology of the Volga River Basin, Russian Academy of Sciences, 445003 Togliatti,RussiaAbstractConsidering the problem of taxonomic diversity as a fractal object is the aim of thisarticle. The prerequisites for such an approach were articles with varying degrees ofdetail and argumentation that substantiate taxonomic diversity from the standpointof fractal geometry. Common to these papers is that the authors in their theoreticalconstructs start from the Willis rule (law) describing the rank distribution of therelationship between the number of taxa and their volume. The flora of the SamaraBend (the bend of the Volga River in its middle reaches) has become an object ofthe research. The authors distinguish seven basic floristic areas on the Samara Bend,the boundaries of which coincide with the respective landscapes. The authors discussthe efficiency of the Willis rule (law), which approximates the relationship betweenthe number of taxa and their volume by rank distribution. The multifractal spectrum(a generalized geometric image of generic structure) of the taxonomic diversity ofvascular plants of the Samara Bend is presented.Keywords: taxonomic diversity, fractal organization, Samara Bend1. IntroductionThe term ‘diversity’ has a long history and has firmly established itself in the scientificbiological literature in the form of stable word combinations: ‘biological diversity’,‘species diversity’ and, finally, ‘taxonomic diversity’. Diversity can also be defined asHow to cite this article: G. S. Rozenberg, D. B. Gelashvili, D. I. Iudin, S. V. Saksonov, and V. N. Yakimov, (2018), “Floristic Phenomena of the SamaraBend: The Fractal Organization of Taxonomic Diversity” in The fourth International Scientific Conference on Ecology and Geography of Plants and PlantCommunities, KnE Life Sciences, pages 168–174. DOI 10.18502/kls.v4i7.3235Page 168Corresponding Author:G. S. Rozenberggenarozenberg@yandex.ruReceived: 12 September 2018Accepted: 15 October 2018Published: 29 October 2018Publishing services provided byKnowledge EG. S. Rozenberg et al. Thisarticle is distributed under theterms of the Creative CommonsAttribution License, whichpermits unrestricted use andredistribution provided that theoriginal author and source arecredited.Selection and Peer-reviewunder the responsibility of theEcology and Geography ofPlants and Plant CommunitiesConference Committee.Ecology and Geography of Plants and Plant Communitiesthe degree of difference in objects in a studied population. Consequently, the proce-dure for distinguishing biological objects from dissimilar characters with simultane-ous grouping according to the characteristics of similar characters is the essence ofa single process of cognition and ordering of the surrounding organic world. So, forexample, the term ‘biodiversity’, at present, is widely interpreted. It is understood asa set of specific parameters of communities, flora, fauna, richness of forms and theircorrelation. In addition, biodiversity is also a synthetic category, corresponding to thetasks of complex studies in ecology, biogeography and evolutionary theory. Thus, thestudy and quantification of biodiversity parameters is not only of academic interest,but of great practical importance, since it attempts to answer the most importantenvironmental problem – the causes and mechanisms of the stability of communities[1–3]. The aim of the article is to consider the problem of taxonomic diversity as a fractalobject, taking into account our experience of the fractal analysis of species diversity[2].2. MethodsThe prerequisites for writing the article were works with varying degrees of detail andreasoning declaring or justifying taxonomic diversity from the point of view of fractalgeometry. Common to these works is that the authors in their theoretical constructsstart from the Willis rule (law), describing the rank distribution of the relationshipbetween the number of taxa and their volume [4–6, etc.].The specific structure of biotic communities is a traditional subject of close attentionand lively discussion among environmentalists. In the first approximation, it is possibleto limit the consideration of the species structure by analyzing the species richness(the number of species) and the relative abundance of the species composing thecommunity, that is, species diversity. Taxonomic diversity, depending on the point ofview and ‘taste’ of the researcher, can be represented as the total number of speciesinhabiting the Earth, simply organisms or as a hierarchy of natural groups recognizedas objectively existing taxa of different ranks [7].As is well known, in the process of knowing the surrounding world, any scien-tific discipline passes through three main stages: descriptive or inventory, conceptual-theoretical (in which ideas about the structure and functioning of the systems understudy are advanced) and the mathematization of these representations (i.e., theirformalization in the language of mathematics, currently the most accurate and strictlanguage) [8, 9]. From this point of view, ecology is no exception: an environmentalistDOI 10.18502/kls.v4i7.3235 Page 169Ecology and Geography of Plants and Plant Communitieswants to know what ecosystems surround it, how many they are, how they function,how the ecosystem is limited in space, what are the ways of constructing artificialecosystems with specified properties and how to manage the productivity of ecosys-tems to maximum benefit. Of these three stages in the emergence of ecology as ascientific discipline, we can assume that the first – inventory – in the first approximationis basically completed, the second – the conceptual-theoretical – is in the floweringstage and the third – the formal and theoretical – is only in the process of formation. Inthis context, theoretical ecology can be defined as a biological discipline (a branch oftheoretical biology) that studies the formalization of the patterns of structure and thefunctioning of ecosystems [9].3. ResultsThe Samara Bend is a large hairpin bend of the Volga in the middle reaches of a totalarea of more than 1.5 thousand km2. On three sides, it is limited by the water areasof the Saratov (the Volga valley) and Kuibyshev (the Volga valley and its tributarythe Usa) reservoirs. The central and southern parts of the peninsula are a flat-slopingplateau with absolute elevations from 150 to 200 asl; the northern part is a mountainrange, known as the Zhiguli or the Zhiguli Mountains, where absolute elevations reach370 m asl.The flora of the Samara Bend is characterized not only by greater diversity in com-parison with the surrounding territories, but also by a significant number of endemicspecies (in the general list of endemics of different botanical-geographical ranks –102 species), including a subgroup of narrow-local endemics rare for such flat flora.Special value to the natural complex of the Samara Bend is given by species whoseclassical places of growth are located there. At present, 21 taxa have been identifiedand described according to data collection from this territory: 5 in the rank of variationsand 16 in the rank of species [10].Landscape zoning is fundamental to the development of the floristic zoning of theSamara Bend. In the territory of the Samara Bend, seven elementary floristic regionswere identified, the boundaries of which coincided with the corresponding landscapes:Zhigulevsky, Vinnovsky, Aleksandrovsky, Perevoloksko–Usinsky, Shelekhmetsky,Volzhsky and Rozhdestvensky (Table 1). The main criterion for isolating elementaryfloristic regions was the analysis of the species composition of the flora of the SamaraBend and the distribution of some of its elements – endemic, relict, rare and smallspecies [10].DOI 10.18502/kls.v4i7.3235 Page 170Ecology and Geography of Plants and Plant CommunitiesT 1: The number of genera and species of vascular plants in the elementary floristic regions of theSamara Bend.Elementary floristic region Number of genera (G) Number of species (S)Zhigulevsky 366 815Vinnovsky 375 778Aleksandrovsky 358 744Perevoloksko–Usinsky 323 654Shelekhmetsky 240 460Rozhdestvensky 355 731Volga district 276 531Samara Bend as a whole 505 1302Source: Authors’ own work.As mentioned earlier, most authors in the analysis of taxonomic diversity rely onWillis’ rule (law), which approximates the relationship between the number of taxaand their volume by rank distribution. Recall that rank distribution is a transformednumber vector: the most abundant group is assigned the first number, the next largestgroup – the second, and so on, to the least abundant group, whose number is the sameas the total number of groups analyzed.In the context of taxonomic diversity, the Willis rule relates the number of taxa andtheir volume: for example, a small number of genera is represented by a large numberof species, whereas most genera include one or two species [4–6]. The Willis rulein most cases is valid for such taxonomic categories as species, genera and orders,whereas for classes, types and kingdoms, due to their small number, such depen-dencies are statistically incorrect. In natural coordinates, the graphical relationshipbetween the number of taxa and their volume is represented by the so-called ‘concavecurve of taxonomists’, that is, the ‘Pareto or Zipf distribution’, which is synonymouswith what Mandelbrot [11] considers ‘asymptotically hyperbolic distribution’.Consider the results of applying the Willis rule to the analysis of the taxonomicdiversity of the flora of the Samara Bend, including 1,302 species of vascular plantsbelonging to 505 genera (Table 1). For the flora of the Samara Bend, the relationshipbetween the number of taxa and their volume is satisfactorily described by a powerfunction whose graph has the form of a ‘concave curve of taxonomists’ in ordinary(N𝐺 = 5,6*N𝑆−1,77) and bilogarithmic coordinates, confirming the validity of the Willisrule [1]. At the same time, the rank distributions and ‘concave curves of taxonomists’fix only the ‘static taxonomic portrait’ of the community and do not answer the ques-tion: howwill the number of genera change as new species appear in the community?Recall that within the extensional character of the taxon (the relation [similarity] isDOI 10.18502/kls.v4i7.3235 Page 171Ecology and Geography of Plants and Plant Communitiesprimary, the attribute is secondary [7]), its volume is, in fact, limited only to thosespecies that are really known to the taxonomist. At the same time, the elucidation ofthe regularity of the growth of the number of genera is subject to an increase in thenumber of species: this ‘dynamic taxonomic portrait’ may prove useful in analyzing thestructural and functional organization of the community, including in the prognosticplan.4. ConclusionWhen dealing with fractals, the main thing is the form-building idea of self-similarity,the essence of which is to replicate the fragment of the structure into its whole whilereproducing the properties of the whole in each fragment. Earlier, we developed themethodological foundations of a fractal theory of the species structure of a community[2], which allowed us to adapt the existing mathematical apparatus and the logic ofthe justifications to the analysis of the fractal structure of taxonomic diversity.In the application of multifractal formalism to the analysis of taxonomic diversityat the level of genera (‘generic diversity’), the totality of the taxa of generic rankis considered as a set consisting of separate fractal subsets that can be interpretedas aggregates of genera of a certain volume. For such subsets, one can calculatethe fractal dimension which will characterize the generic variety. This is precisely themeaning of the ordinate of points on a graph of the multifractal spectrum (Figure). Theabscissa of the points characterizes the relative volume of the genus of a particularsubset. If the presence on the graph of the spectrum of points lying along axis ‘a’(‘singularity index’) is close to zero, this means the presence of dominants (taxa ofgeneric rank with the highest species representation) in the studied taxonomic set:the lower the abscissa of the points, the stronger the dominance. If the presence ofpoints lying on the ‘a’ axis is close to one, this means the presence in the studiedtaxonomic set of genera with a small species representation (by analogy with rarespecies). The area under the curve of the spectrum can be interpreted as an indicatorinversely proportional to the leveling of genera over a relative volume: the larger it is,the less the alignment, and vice versa.The figure shows the multifractal spectrum of the taxonomic diversity of the vas-cular plants of the Samara Bend. Spectrum ‘1’ is a multifractal spectrum of the genericstructure of the plant community of the Samara Bend. The community is characterizedby a high proportion of one- and two-species genera (238 and 100, respectively).Spectrum ‘2’ is a theoretical spectrum with an increase in the general communityDOI 10.18502/kls.v4i7.3235 Page 172Ecology and Geography of Plants and Plant CommunitiesFigure 1: Multifractal spectrum of taxonomic diversity of the vascular plants of the Samara Bend; a – is the‘singularity index’; f(a) – is the function of the multifractal spectrum or the ‘singularity spectrum’ [1, 2].equalization and the ‘disappearance’ of genera with a much larger number of speciesthan the main mass. Spectrum ‘3’ is also a theoretical spectrum with an ‘increase’ ofdiversity in the community partly due to ‘getting rid’ of one- and two-species genera.This allows us to conclude that the patterns we identified earlier for the multifractalspectrum of the species structure of the community [1] to a certain extent persist forthe spectrum of the generic structure.We recall that within the framework of fractal methodology we were interestedin the question of how the structure of taxonomic diversity (in this case the genericstructure) is invariant with respect to the transformation of its scale, expressed throughthe growth of the number of species. Without going into the details of the discussionon the issue that ‘the increase in the number of species as a result of speciation doesnot yet mean the appearance of a new genus’, we note that, regardless of how theprocess of formation of new genera in time functions (on an evolutionary scale), anytaxonomic diversity should have a self-similar, or fractal, character.FundingThe authors are grateful to the Russian Foundation for Basic Research (RFBR), whichpartially supported this study (Grant 17-44-630113 p_povolzhye_a).DOI 10.18502/kls.v4i7.3235 Page 173Ecology and Geography of Plants and Plant CommunitiesReferences[1] Gelashvili, D. B., Iudin, D. I., Rozenberg, G. S., et al. (2013). Fractals and Multifractalsin Bioecology. Nizhnii Novgorod: Publishing House of Nizhny Novgorod StateUniversity.[2] Gelashvili, D. B., Saksonov, S. V., Rozenberg, G. S., et al. (2011). Floristic phenomenonof the Samara bend: Fractal structure of taxonomic diversity. Samarskaya Luka:Problems of Regional and Global Ecology, vol. 20, no. 2(36), pp. 80–104.[3] Chernov, Yu. I. (1991). Biological diversity: Essence and problems. Advances inModern Biology, vol. 111, no. 4, pp. 499–507.[4] Pozdnyakov, A. A. (2005). The value of the rule of Willis for taxonomy. Journal ofGeneral Biology, vol. 66, no. 4, pp. 326–335.[5] Chislenko, L. L. (1977). On the structure of taxa and taxonomic diversity. Journal ofGeneral Biology, vol. 38, no. 3, pp. 348–358.[6] Willis, J. C. (1940). The course of evolution by differentiation or divergent mutationrather than by selection. L.: Cambridge University Press.[7] Pavlinov, I. Ya. (2001). Concepts of systematics and biodiversity concept: Interactionproblem. Journal of General Biology, vol. 62, no. 4, pp. 362–366.[8] Rozenberg, G. S. (2005). On the ways of constructing theoretical ecology. Advancesin Modern Biology, vol. 125, no. 1, pp. 14–27.[9] Rozenberg, G. S. (2013). Introduction to Theoretical Ecology. In 2 volumes (secondedition). Togliatti: Kassandra.[10] Saksonov, S. V. (2006). The Samara Bend Floristic Phenomenon. Moscow: Nauka.[11] Mandelbrot, B. (2002). Fractal Geometry of Nature. Moscow: Institute of ComputerStudies.DOI 10.18502/kls.v4i7.3235 Page 174

Floristic Phenomena of the Samara Bend: The Fractal Organization of Taxonomic Diversity

Rozenberg, G. S. (G)

Gelashvili, D. B. (D)

Iudin, D. I. (D)

Saksonov, S. V. (S)

Yakimov, V. N. (V)

Neliti

Floristic Phenomena of the Samara Bend: the Fractal Organization of Taxonomic Diversity

Considering the problem of taxonomic diversity as a fractal object is the aim of this article. The prerequisites for such an approach were articles with varying degrees of detail and argumentation that substantiate taxonomic diversity from the standpoint of fractal geometry. Common to these papers is that the authors in their theoretical constructs start from the Willis rule (law) describing the rank distribution of the relationship between the number of taxa and their volume. The flora of the Samara Bend (the bend of the Volga River in its middle reaches) has become an object of the research. The authors distinguish seven basic floristic areas on the Samara Bend, the boundaries of which coincide with the respective landscapes. The authors discuss the efficiency of the Willis rule (law), which approximates the relationship between the number of taxa and their volume by rank distribution. The multifractal spectrum (a generalized geometric image of generic structure) of the taxonomic diversity of vascular plants of the Samara Bend is presented.The authors are grateful to the Russian Foundation for Basic Research (RFBR), which partially supported this study (Grant 17-44-630113 p_povolzhye_a

Rozenberg, G. S.

Gelashvili, D. B.

Iudin, D. I.

Saksonov, S. V.

Yakimov, V. N.

Institutional repository of Ural Federal University named after the first President of Russia B.N.Yeltsin

Ecology and Geography of Plants and Plant CommunitiesThe fourth International Scientiﬁc Conference onEcology and Geography of Plants and Plant CommunitiesVolume 2018Conference PaperFloristic Phenomena of the Samara Bend:The Fractal Organization of TaxonomicDiversityG. S. Rozenberg1,3, D. B. Gelashvili2, D. I. Iudin2, S. V. Saksonov1,4, andV. N. Yakimov21Institute of Ecology of the Volga River Basin, Russian Academy of Sciences, 445003 Togliatti,Russia2N. I. Lobachevsky State University of Nizhni Novgorod, 603950 Nizhni Novgorod, Russia3Samara State University of Economics, 443090 Samara, Russia4UNESCO Chair ‘The study and conservation of biodiversity of the Volga Basin ecosystems’,Institute of Ecology of the Volga River Basin, Russian Academy of Sciences, 445003 Togliatti,RussiaAbstractConsidering the problem of taxonomic diversity as a fractal object is the aim of thisarticle. The prerequisites for such an approach were articles with varying degrees ofdetail and argumentation that substantiate taxonomic diversity from the standpointof fractal geometry. Common to these papers is that the authors in their theoreticalconstructs start from the Willis rule (law) describing the rank distribution of therelationship between the number of taxa and their volume. The ﬂora of the SamaraBend (the bend of the Volga River in its middle reaches) has become an object ofthe research. The authors distinguish seven basic ﬂoristic areas on the Samara Bend,the boundaries of which coincide with the respective landscapes. The authors discussthe efﬁciency of the Willis rule (law), which approximates the relationship betweenthe number of taxa and their volume by rank distribution. The multifractal spectrum(a generalized geometric image of generic structure) of the taxonomic diversity ofvascular plants of the Samara Bend is presented.Keywords: taxonomic diversity, fractal organization, Samara Bend1. IntroductionThe term ‘diversity’ has a long history and has ﬁrmly established itself in the scientiﬁcbiological literature in the form of stable word combinations: ‘biological diversity’,‘species diversity’ and, ﬁnally, ‘taxonomic diversity’. Diversity can also be deﬁned asHow to cite this article: G. S. Rozenberg, D. B. Gelashvili, D. I. Iudin, S. V. Saksonov, and V. N. Yakimov, (2018), “Floristic Phenomena of the SamaraBend: The Fractal Organization of Taxonomic Diversity” in The fourth International Scientific Conference on Ecology and Geography of Plants and PlantCommunities, KnE Life Sciences, pages 168–174. DOI 10.18502/kls.v4i7.3235Page 168Corresponding Author:G. S. Rozenberggenarozenberg@yandex.ruReceived: 12 September 2018Accepted: 15 October 2018Published: 29 October 2018Publishing services provided byKnowledge EG. S. Rozenberg et al. Thisarticle is distributed under theterms of the Creative CommonsAttribution License, whichpermits unrestricted use andredistribution provided that theoriginal author and source arecredited.Selection and Peer-reviewunder the responsibility of theEcology and Geography ofPlants and Plant CommunitiesConference Committee.Ecology and Geography of Plants and Plant Communitiesthe degree of difference in objects in a studied population. Consequently, the proce-dure for distinguishing biological objects from dissimilar characters with simultane-ous grouping according to the characteristics of similar characters is the essence ofa single process of cognition and ordering of the surrounding organic world. So, forexample, the term ‘biodiversity’, at present, is widely interpreted. It is understood asa set of speciﬁc parameters of communities, ﬂora, fauna, richness of forms and theircorrelation. In addition, biodiversity is also a synthetic category, corresponding to thetasks of complex studies in ecology, biogeography and evolutionary theory. Thus, thestudy and quantiﬁcation of biodiversity parameters is not only of academic interest,but of great practical importance, since it attempts to answer the most importantenvironmental problem – the causes and mechanisms of the stability of communities[1–3]. The aim of the article is to consider the problem of taxonomic diversity as a fractalobject, taking into account our experience of the fractal analysis of species diversity[2].2. MethodsThe prerequisites for writing the article were works with varying degrees of detail andreasoning declaring or justifying taxonomic diversity from the point of view of fractalgeometry. Common to these works is that the authors in their theoretical constructsstart from the Willis rule (law), describing the rank distribution of the relationshipbetween the number of taxa and their volume [4–6, etc.].The speciﬁc structure of biotic communities is a traditional subject of close attentionand lively discussion among environmentalists. In the ﬁrst approximation, it is possibleto limit the consideration of the species structure by analyzing the species richness(the number of species) and the relative abundance of the species composing thecommunity, that is, species diversity. Taxonomic diversity, depending on the point ofview and ‘taste’ of the researcher, can be represented as the total number of speciesinhabiting the Earth, simply organisms or as a hierarchy of natural groups recognizedas objectively existing taxa of different ranks [7].As is well known, in the process of knowing the surrounding world, any scien-tiﬁc discipline passes through three main stages: descriptive or inventory, conceptual-theoretical (in which ideas about the structure and functioning of the systems understudy are advanced) and the mathematization of these representations (i.e., theirformalization in the language of mathematics, currently the most accurate and strictlanguage) [8, 9]. From this point of view, ecology is no exception: an environmentalistDOI 10.18502/kls.v4i7.3235 Page 169Ecology and Geography of Plants and Plant Communitieswants to know what ecosystems surround it, how many they are, how they function,how the ecosystem is limited in space, what are the ways of constructing artiﬁcialecosystems with speciﬁed properties and how to manage the productivity of ecosys-tems to maximum beneﬁt. Of these three stages in the emergence of ecology as ascientiﬁc discipline, we can assume that the ﬁrst – inventory – in the ﬁrst approximationis basically completed, the second – the conceptual-theoretical – is in the ﬂoweringstage and the third – the formal and theoretical – is only in the process of formation. Inthis context, theoretical ecology can be deﬁned as a biological discipline (a branch oftheoretical biology) that studies the formalization of the patterns of structure and thefunctioning of ecosystems [9].3. ResultsThe Samara Bend is a large hairpin bend of the Volga in the middle reaches of a totalarea of more than 1.5 thousand km2. On three sides, it is limited by the water areasof the Saratov (the Volga valley) and Kuibyshev (the Volga valley and its tributarythe Usa) reservoirs. The central and southern parts of the peninsula are a ﬂat-slopingplateau with absolute elevations from 150 to 200 asl; the northern part is a mountainrange, known as the Zhiguli or the Zhiguli Mountains, where absolute elevations reach370 m asl.The ﬂora of the Samara Bend is characterized not only by greater diversity in com-parison with the surrounding territories, but also by a signiﬁcant number of endemicspecies (in the general list of endemics of different botanical-geographical ranks –102 species), including a subgroup of narrow-local endemics rare for such ﬂat ﬂora.Special value to the natural complex of the Samara Bend is given by species whoseclassical places of growth are located there. At present, 21 taxa have been identiﬁedand described according to data collection from this territory: 5 in the rank of variationsand 16 in the rank of species [10].Landscape zoning is fundamental to the development of the ﬂoristic zoning of theSamara Bend. In the territory of the Samara Bend, seven elementary ﬂoristic regionswere identiﬁed, the boundaries of which coincided with the corresponding landscapes:Zhigulevsky, Vinnovsky, Aleksandrovsky, Perevoloksko–Usinsky, Shelekhmetsky,Volzhsky and Rozhdestvensky (Table 1). The main criterion for isolating elementaryﬂoristic regions was the analysis of the species composition of the ﬂora of the SamaraBend and the distribution of some of its elements – endemic, relict, rare and smallspecies [10].DOI 10.18502/kls.v4i7.3235 Page 170Ecology and Geography of Plants and Plant CommunitiesT˔˕˟˘ 1: The number of genera and species of vascular plants in the elementary ﬂoristic regions of theSamara Bend.Elementary ﬂoristic region Number of genera (G) Number of species (S)Zhigulevsky 366 815Vinnovsky 375 778Aleksandrovsky 358 744Perevoloksko–Usinsky 323 654Shelekhmetsky 240 460Rozhdestvensky 355 731Volga district 276 531Samara Bend as a whole 505 1302Source: Authors’ own work.As mentioned earlier, most authors in the analysis of taxonomic diversity rely onWillis’ rule (law), which approximates the relationship between the number of taxaand their volume by rank distribution. Recall that rank distribution is a transformednumber vector: the most abundant group is assigned the ﬁrst number, the next largestgroup – the second, and so on, to the least abundant group, whose number is the sameas the total number of groups analyzed.In the context of taxonomic diversity, the Willis rule relates the number of taxa andtheir volume: for example, a small number of genera is represented by a large numberof species, whereas most genera include one or two species [4–6]. The Willis rulein most cases is valid for such taxonomic categories as species, genera and orders,whereas for classes, types and kingdoms, due to their small number, such depen-dencies are statistically incorrect. In natural coordinates, the graphical relationshipbetween the number of taxa and their volume is represented by the so-called ‘concavecurve of taxonomists’, that is, the ‘Pareto or Zipf distribution’, which is synonymouswith what Mandelbrot [11] considers ‘asymptotically hyperbolic distribution’.Consider the results of applying the Willis rule to the analysis of the taxonomicdiversity of the ﬂora of the Samara Bend, including 1,302 species of vascular plantsbelonging to 505 genera (Table 1). For the ﬂora of the Samara Bend, the relationshipbetween the number of taxa and their volume is satisfactorily described by a powerfunction whose graph has the form of a ‘concave curve of taxonomists’ in ordinary(N𝐺 = 5,6*N𝑆−1,77) and bilogarithmic coordinates, conﬁrming the validity of the Willisrule [1]. At the same time, the rank distributions and ‘concave curves of taxonomists’ﬁx only the ‘static taxonomic portrait’ of the community and do not answer the ques-tion: howwill the number of genera change as new species appear in the community?Recall that within the extensional character of the taxon (the relation [similarity] isDOI 10.18502/kls.v4i7.3235 Page 171Ecology and Geography of Plants and Plant Communitiesprimary, the attribute is secondary [7]), its volume is, in fact, limited only to thosespecies that are really known to the taxonomist. At the same time, the elucidation ofthe regularity of the growth of the number of genera is subject to an increase in thenumber of species: this ‘dynamic taxonomic portrait’ may prove useful in analyzing thestructural and functional organization of the community, including in the prognosticplan.4. ConclusionWhen dealing with fractals, the main thing is the form-building idea of self-similarity,the essence of which is to replicate the fragment of the structure into its whole whilereproducing the properties of the whole in each fragment. Earlier, we developed themethodological foundations of a fractal theory of the species structure of a community[2], which allowed us to adapt the existing mathematical apparatus and the logic ofthe justiﬁcations to the analysis of the fractal structure of taxonomic diversity.In the application of multifractal formalism to the analysis of taxonomic diversityat the level of genera (‘generic diversity’), the totality of the taxa of generic rankis considered as a set consisting of separate fractal subsets that can be interpretedas aggregates of genera of a certain volume. For such subsets, one can calculatethe fractal dimension which will characterize the generic variety. This is precisely themeaning of the ordinate of points on a graph of the multifractal spectrum (Figure). Theabscissa of the points characterizes the relative volume of the genus of a particularsubset. If the presence on the graph of the spectrum of points lying along axis ‘a’(‘singularity index’) is close to zero, this means the presence of dominants (taxa ofgeneric rank with the highest species representation) in the studied taxonomic set:the lower the abscissa of the points, the stronger the dominance. If the presence ofpoints lying on the ‘a’ axis is close to one, this means the presence in the studiedtaxonomic set of genera with a small species representation (by analogy with rarespecies). The area under the curve of the spectrum can be interpreted as an indicatorinversely proportional to the leveling of genera over a relative volume: the larger it is,the less the alignment, and vice versa.The ﬁgure shows the multifractal spectrum of the taxonomic diversity of the vas-cular plants of the Samara Bend. Spectrum ‘1’ is a multifractal spectrum of the genericstructure of the plant community of the Samara Bend. The community is characterizedby a high proportion of one- and two-species genera (238 and 100, respectively).Spectrum ‘2’ is a theoretical spectrum with an increase in the general communityDOI 10.18502/kls.v4i7.3235 Page 172Ecology and Geography of Plants and Plant CommunitiesFigure 1: Multifractal spectrum of taxonomic diversity of the vascular plants of the Samara Bend; a – is the‘singularity index’; f(a) – is the function of the multifractal spectrum or the ‘singularity spectrum’ [1, 2].equalization and the ‘disappearance’ of genera with a much larger number of speciesthan the main mass. Spectrum ‘3’ is also a theoretical spectrum with an ‘increase’ ofdiversity in the community partly due to ‘getting rid’ of one- and two-species genera.This allows us to conclude that the patterns we identiﬁed earlier for the multifractalspectrum of the species structure of the community [1] to a certain extent persist forthe spectrum of the generic structure.We recall that within the framework of fractal methodology we were interestedin the question of how the structure of taxonomic diversity (in this case the genericstructure) is invariant with respect to the transformation of its scale, expressed throughthe growth of the number of species. Without going into the details of the discussionon the issue that ‘the increase in the number of species as a result of speciation doesnot yet mean the appearance of a new genus’, we note that, regardless of how theprocess of formation of new genera in time functions (on an evolutionary scale), anytaxonomic diversity should have a self-similar, or fractal, character.FundingThe authors are grateful to the Russian Foundation for Basic Research (RFBR), whichpartially supported this study (Grant 17-44-630113 p_povolzhye_a).DOI 10.18502/kls.v4i7.3235 Page 173Ecology and Geography of Plants and Plant CommunitiesReferences[1] Gelashvili, D. B., Iudin, D. I., Rozenberg, G. S., et al. (2013). Fractals and Multifractalsin Bioecology. Nizhnii Novgorod: Publishing House of Nizhny Novgorod StateUniversity.[2] Gelashvili, D. B., Saksonov, S. V., Rozenberg, G. S., et al. (2011). Floristic phenomenonof the Samara bend: Fractal structure of taxonomic diversity. 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In 2 volumes (secondedition). Togliatti: Kassandra.[10] Saksonov, S. V. (2006). The Samara Bend Floristic Phenomenon. Moscow: Nauka.[11] Mandelbrot, B. (2002). Fractal Geometry of Nature. Moscow: Institute of ComputerStudies.DOI 10.18502/kls.v4i7.3235 Page 174

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Floristic Phenomena of the Samara Bend: The Fractal Organization of Taxonomic Diversity

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