Spatial complexity of ecological communities: Bridging the gap between probabilistic and non-probabilistic uncertainty measures

Recently, it has become well appreciated that disorder-based measures of biological diversity, such as Shannon's entropy, fail to adequately capture the structural complexity of an ecological community. The contributions of spatial complexity to community structure are however quantifiable if we consider, for example, the degree of spatial co-occurrence between species. The larger and more intricate these correlations, the more structurally complex the community. We suggest that Juhasz-Nagy information-theoretical functions offer an adequate basis for the measurement of structural complexity of plant communities. However, whereas Juhasz-Nagy's developed his model solely in terms of classical probabilistic uncertainty, we show that these functions are based both on traditional probabilistic concepts and on non-probabilistic elements borrowed from fuzzy set theory. Therefore, the proposed representation of community structure offers an interesting way for linking probabilistic uncertainty and fuzzy uncertainty. It also turns out that Juhasz-Nagy information-theoretical functions fit previously established theoretical definitions of ecological complexity. We illustrate the utility of the proposed functions to the multi-scale analysis of disturbed and undisturbed plant communities.

[1]  P. Greig-Smith,et al.  QUANTITATIVE PLANT ECOLOGY , 1959 .

[2]  H. Gleason.  The individualistic concept of the plant association , 1926 .

[3]  David W. Roberts,et al.  Ordination on the basis of fuzzy set theory , 1986, Vegetatio.

[4]  S. Redner,et al.  Introduction To Percolation Theory , 2018 .

[5]  P.F.M. van Gaans,et al.  Continuous classification in soil survey: spatial correlation, confusion and boundaries , 1997 .

[6]  Madhur Anand,et al.  The use of matrix models to detect natural and pollution-induced forest gradients , 2003 .

[7]  B. Tóthmérész.  On the characterization of scale-dependent diversity , 1998 .

[8]  M. Sahini,et al.  Applications of Percolation Theory , 2023, Applied Mathematical Sciences.

[9]  P. Juhász-Nagy,et al.  Information theory methods for the study of spatial processes and succession , 1983, Vegetatio.

[10]  Cristian S. Calude.  Information and Randomness , 1994, Monographs in Theoretical Computer Science An EATCS Series.

[11]  M. Palmer,et al.  VARIANCE IN SPECIES RICHNESS, SPECIES ASSOCIATION, AND NICHE LIMITATION , 1995 .

[12]  P. White,et al.  On the existence of ecological communities , 1994 .

[13]  Madhur Anand,et al.  Quantifying the Components of Biocomplexity Along Ecological Perturbation Gradients , 2005, Biodiversity & Conservation.

[14]  D. Tilman.  Competition and Biodiversity in Spatially Structured Habitats , 1994 .

[15]  Madhur Anand,et al.  Complexity in Plant Communities: the Notion and Quantification , 1996 .

[16]  A. Kolmogorov.  Three approaches to the quantitative definition of information , 1968 .

[17]  S. Naeem,et al.  Disentangling biodiversity effects on ecosystem functioning: deriving solutions to a seemingly insurmountable problem , 2003 .

[18]  D. Dubois,et al.  Fundamentals of fuzzy sets , 2000 .

[19]  Solomon Kullback,et al.  Information Theory and Statistics , 1960 .

[20]  S. Bartha.  Preliminary scaling for multi-species coalitions in primary succession , 1992 .

[21]  F. E. Egler.  Arid Southeast Oahu Vegetation, Hawaii , 1947 .

[22]  J. Wilson,et al.  Does Vegetation Science exist , 1991 .

[23]  David W. Roberts,et al.  Modelling forest dynamics with vital attributes and fuzzy systems theory , 1996 .

[24]  Settimo Termini,et al.  A Definition of a Nonprobabilistic Entropy in the Setting of Fuzzy Sets Theory , 1972, Inf. Control..

[25]  Daniel Goodman,et al.  The Theory of Diversity-Stability Relationships in Ecology , 1975, The Quarterly Review of Biology.

[26]  T. Czárán,et al.  Exploring plant community dynamics in abstract coenostate spaces , 1998 .

[27]  R. Piasecki,et al.  Inhomogeneity and complexity measures for spatial patterns , 2002 .

[28]  H. Gleason,et al.  The individualistic concept of the plant association , 1939 .

[29]  Gunter Ochs,et al.  Complexity and demographic stability in population models. , 2004, Theoretical population biology.

[30]  James C. Bezdek,et al.  Quantifying Different Facets of Fuzzy Uncertainty , 2000 .

[31]  Wolfgang Sander.  On measures of fuzziness , 1989 .

[32]  László Orlóci,et al.  On hierarchical partitioning of an ecological complexity function , 2000 .

[33]  Alessandro Ludovisi,et al.  Investigating beta diversity by Kullback–Leibler information measures , 2006 .

[34]  G. N. Lance,et al.  Studies in the Numerical Analysis of Complex Rain-Forest Communities: IV. A Method for the Elucidation of Small-Scale Forest Pattern , 1969 .

[35]  F. Papentin.  On order and complexity. I. General considerations , 1980 .

[36]  R. Yager.  ON THE MEASURE OF FUZZINESS AND NEGATION Part I: Membership in the Unit Interval , 1979 .

[37]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[38]  Stefano Marsili-Libelli,et al.  Fuzzy Clustering of Ecological Data , 1991 .

[39]  Moshe Koppel,et al.  Complexity, Depth, and Sophistication , 1987, Complex Syst..

[40]  Enrico Feoli,et al.  Fuzzy set analysis of the Ethiopian rift valley vegetation in relation to anthropogenic influences , 2004, Plant Ecology.

[41]  Thomas Manke,et al.  Robustness and network evolution--an entropic principle , 2005 .

[42]  Pál Juhász-Nagy,et al.  Notes on compositional diversity , 2004, Hydrobiologia.

[43]  B. Ebanks.  On measures of fuzziness and their representations , 1983 .

[44]  R. Macarthur.  Fluctuations of Animal Populations and a Measure of Community Stability , 1955 .

[45]  J. Crutchfield,et al.  Measures of statistical complexity: Why? , 1998 .

[46]  M. Minozzo,et al.  Modelling the horizontal spatial structure of planktonic community in Lake Trasimeno (Umbria, Italy) using multivariate geostatistical methods , 2005 .

[47]  Helen M. Regan,et al.  A TAXONOMY AND TREATMENT OF UNCERTAINTY FOR ECOLOGY AND CONSERVATION BIOLOGY , 2002 .

[48]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[49]  J. Podani,et al.  Space series analysis of vegetation: processes reconsidered , 1992 .

[50]  G. Klir,et al.  Uncertainty-based information: Elements of generalized information theory (studies in fuzziness and soft computing). , 1998 .

[51]  K. McCann.  The diversity–stability debate , 2000, Nature.

[52]  T. Czárán,et al.  Pattern, area and diversity: the importance of spatial scale in species assemblages , 1993 .

[53]  Madhur Anand,et al.  SPATIOTEMPORAL DYNAMICS IN A TRANSITION ZONE: PATCHINESS, SCALE, AND AN EMERGENT PROPERTY , 2001 .

[54]  An information-theoretical measure of β-diversity , 2003 .