Quantification of non-power-law diversity scaling with local multifractal analysis

Abstract Description of spatial structure is in the focus of community ecology. Multifractal analysis is a prospective method because it connects fractal geometry tools with the conventional methodology of diversity analysis. The current version is based on a power-law scaling in the form of species-area relation. It is well suited for the description of self-similar communities. However, it is well known that many natural communities and theoretical models deviate from power-law diversity scaling. We propose to relax the requirement of self-similarity and introduce local multifractal analysis. This new method is based on a local approximation of the relation of effective diversity to scale. It is a fractal analysis in the sense of dealing with power-law approximations and estimation of scaling exponents within the framework of multifractal analysis. However, the multifractal spectra that resulted from this analysis do not characterize global scale-invariant properties but reflect details of diversity scaling specific for a given scale. In that sense, the analysis proposed is local. We present the results of analysis in theoretical and empirical contexts. Classical Fisher model of species abundance distribution deviate from power-law scaling and may be described with local multifractal spectra. We illustrate an empirical application in a case study of spatial structure of two temperate deciduous forests in different landscapes. Our approach provides new perspectives in the investigation of community spatial structure because it widens the scope of fractal analysis beyond self-similar communities. It may be adapted for other versions of multifractal analysis applied in community ecology.

[1]  Stephen P. Hubbell,et al.  Species‐Area Curves, Diversity Indices, and Species Abundance Distributions: A Multifractal Analysis , 2002, The American Naturalist.

[2]  E. Tjørve.  Shapes and functions of species–area curves: a review of possible models , 2003 .

[3]  M. Hill.  Diversity and Evenness: A Unifying Notation and Its Consequences , 1973 .

[4]  E. Tjørve,et al.  Scale‐dependence in species‐area relationships , 2005 .

[5]  L. Saravia.  A new method to analyse species abundances in space using generalized dimensions , 2015 .

[6]  G. Rozenberg,et al.  Nonconcavity of mass exponents’ spectrum in multifractal analysis of community spatial structure: The problem and possible solutions , 2014 .

[7]  L. Solntsev,et al.  Local multifractal analysis of the spatial structure of meadow comminities at small scale , 2014, Doklady Biological Sciences.

[8]  D. Storch.  The theory of the nested species–area relationship: geometric foundations of biodiversity scaling , 2016 .

[9]  M. Crawley,et al.  Scale dependence in plant biodiversity. , 2001, Science.

[10]  B. Bossuyt,et al.  Multifractal diversity‐area relationship at small scales in dune slack plant communities , 2008 .

[11]  Keming Ma,et al.  Do generalized scaling laws exist for species abundance distribution in mountains , 2006 .

[12]  J. Harte,et al.  Biodiversity scales from plots to biomes with a universal species-area curve. , 2009, Ecology letters.

[13]  J. Dengler,et al.  Which function describes the species–area relationship best? A review and empirical evaluation , 2009 .

[14]  E. Tjørve.  Shapes and functions of species–area curves (II): a review of new models and parameterizations , 2009 .

[15]  J. Chave.  The problem of pattern and scale in ecology: what have we learned in 20 years? , 2013, Ecology letters.

[16]  G. Rozenberg,et al.  Multifractal Analysis of the Species Structure of Biotic Communities , 2003, Doklady Biological Sciences.

[17]  L. Saravia,et al.  Multifractal Spatial Patterns and Diversity in an Ecological Succession , 2012, PloS one.

[18]  Wenzhi Zhao,et al.  Scale dependence in desert plant diversity , 2006, Biodiversity & Conservation.

[19]  B. Fu,et al.  Scale dependence of the beta diversity-scale relationship , 2015 .

[20]  Kinzig,et al.  Self-similarity in the distribution and abundance of species , 1999, Science.

[21]  S. Levin.  The problem of pattern and scale in ecology , 1992 .

[22]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[23]  A. Rosas,et al.  Effect of Landscape Structure on Species Diversity , 2013, PloS one.

[24]  A. Chao,et al.  Proposing a resolution to debates on diversity partitioning. , 2012, Ecology.

[25]  Stephen J Cornell,et al.  Species-area relationships from a spatially explicit neutral model in an infinite landscape. , 2007, Ecology letters.

[26]  S. Hubbell,et al.  The unified neutral theory of biodiversity and biogeography at age ten. , 2011, Trends in ecology & evolution.

[27]  E. Maggi,et al.  Multifractal spatial distribution of epilithic microphytobenthos on a Mediterranean rocky shore , 2015 .

[28]  L. Jost.  Entropy and diversity , 2006 .

[29]  E. Perrier,et al.  Beyond species area curves: application of a scale-free measure for spatial variability of species richness , 2011 .

[30]  S. Cornell,et al.  Universal scaling of species‐abundance distributions across multiple scales , 2013 .

[31]  P. White,et al.  Scale Dependence and the Species-Area Relationship , 1994, The American Naturalist.

[32]  L. Seuront.  Fractals and Multifractals in Ecology and Aquatic Science , 2009 .

[33]  M. A. Muñoz,et al.  What Ecological Factors Shape Species-Area Curves in Neutral Models? , 2012, PloS one.

[34]  David R. Anderson,et al.  Model Based Inference in the Life Sciences: A Primer on Evidence , 1998 .

[35]  B. N. Yakimov,et al.  Multifractal analysis of neutral community spatial structure. , 2014, Journal of theoretical biology.

[36]  Anne Chao,et al.  Estimating diversity and entropy profiles via discovery rates of new species , 2015 .

[37]  Michael Breuer,et al.  Scale Dependency in the Species-Area Relationship of Plant Communities , 2008, Folia Geobotanica.

[38]  Zhongliang Huang,et al.  Multifractal analysis of diversity scaling laws in a subtropical forest , 2013 .

[39]  Multifractal analysis of the species structure of freshwater hydrobiocenoses , 2012, Biology Bulletin.

[40]  S. Hartley,et al.  Uses and abuses of fractal methodology in ecology , 2004 .

[41]  D. Harte.  Multifractals: Theory and Applications , 2001 .

[42]  S. Scheiner.  Six types of species‐area curves , 2003 .

[43]  Lindsay A. Turnbull,et al.  Identification of 100 fundamental ecological questions , 2013 .