gtag('config', 'UA-114241270-1');
Your browser is out-of-date!

Update your browser to view this website correctly. Update my browser now

×

Discipline
Biological
Keywords
Species Abundance Distribution
Stretched Exponential
Observation Type
Standalone
Nature
Resources / Big Data
Submitted
Nov 29th, 2016
Published
Jan 26th, 2017
  • Abstract

    A central object in community ecology is species abundance distribution. We are interested in the power law and its allies for ranked species abundance data. We collected 12 large data sets consisting of many samples. The preliminary fitting result makes a robust impression (12 systems at three scales of integration) that the stretched exponential is an interesting alternative for the power law. For further work, advanced statistics are required. Not only ‘our’ data but, quite often, other data as well consist of sample×species cross-tables. With cross-tables, also ‘within species over samples’ characteristics can be studied. An integrated view on data patterns in multi-sample sets may help to identify generative processes for and the formulation of a relatively simple model for species abundance data.

  • Figure
  • Introduction

    A central object in community ecology is species abundance distribution (SAD). It has been studied for over a century since Raunkiaer (in). The review of an expert group is a benchmark for properties and generative theory. The edited volume on biological diversity is, among others, an update. For applications of the SAD, like the measurement of biodiversity, we refer to.

    The expert group described a law for species abundance data: “When plotted as a histogram of number (or percent) of species on the y-axis vs. abundance on an arithmetic x-axis, the classic hyperbolic, ‘lazy J-curve’ or ‘hollow curve’ is produced, indicating a few very abundant species and many rare species (Fig. 1A). In this form, the law appears to be universal; we know of no multispecies community, ranging from the marine benthos to the Amazonian rainforest, that violates it” (in our Supplement A, we make some remarks to the meaning of ‘community’ and ‘sample’). Several terms are used for a ‘hollow curve’: the distribution is (right)-skewed, long-tailed, has extreme values, shows rare events. Is there a simple meaningful equation for ‘the’ species abundance distribution? One is inclined to think so if there is a ‘universal law’. Review is steeped with the idea of a relatively simple equation for SADs, but it presents the opposite too, that different communities have rather different SADs and that groups of species within a community have different SADs, making the community’s SAD a mixed one. The SAD is mostly treated as a histogram, based on the binning of data into frequency classes (for a probability mass or density function). However, the SAD can be illustrated as a rank abundance or Whittaker plot (see, their Fig. 1c; see also). Ranked data are used for exploratory data analysis. Rank-size plots and (cumulative) probability plots are strongly related (and see our Supplement B). SADs bear similarity to distributions in other fields of science.

    Long-tailed distributions of natural and manmade phenomena, in rank-size form (where ‘size’ can be read as ‘abundance’), often show power law behaviour (and see Wikipedia headword ‘Power law’). The ubiquitous power law has been considered for species abundance data. However, ideal power law behaviour is absent or rare: data points do not lie in a straight line in a log/log plot. For this reason, the interest in the power law for species abundance data seems to have vanished. However, the imperfect power law behaviour in other fields of science is well documented. Paper is of particular interest. It revisits the data and analysis in a seminal paper and concludes that 9 of 24 data sets conjectured to follow a power law actually do not.

  • Objective

    The direct aim is to generate interest again in the power law, especially in its allies like the stretched exponential, for species abundance data. Further reaching aims, to be tackled in the future and for which we introduce a frame, are (i) to generate the interest of community ecologists in the generative processes of the power law and allies (that have been studied in other fields of science) and (ii) to complete the quest for a simple yet meaningful equation/model for the SAD.

  • Results & Discussion

    Rank  abundance plots were made, shown in figure 1. Data points in the log/log plots do not form straight lines, ideal power law behaviour is absent. The data points indicate curved lines, concave in almost all cases. As an alternative to the power law, the stretched exponential function was fitted. The parameter values of the fitted function, for the composite samples of the complete sets only, are given in our supplemental table B. No advanced statistics was applied. No comparison with other functions/models was made. The actual data show some deviations from the fitted curves, but the overall result is visually satisfying.

    For further work, advanced statistics are required; we refer to. For model comparison and selection, we refer to. For allies of the power law, we refer to (especially chapter 4).

    A challenge lies in a remark in the review of the expert group: “Starting in the 1970s and running unabated to the present day, mechanistic models (models attempting to explain the causes of the hollow curve SAD) and alternative interpretations and extensions of prior theories have proliferated to an extraordinary degree”. The power law and its allies are often considered for the degree or connectivity distribution of networks. Species abundance data are retrieved from ecological communities that are networks. However, a network topology behind species abundance data is not immediately clear. Species abundance data are reminiscent to data of food webs. For instance, the interactions between fruits and frugivorous birds can be presented in a cross-table of fruit species×bird species (data of in). From such a table, one can summarize the number of connections for the fruiting species with the bird species and vice versa: two connectivity distributions. The one dealing with the number of interactions for fruiting species over bird species is reminiscent of an assemblage of fruiting species, ‘sampled’ by birds. Networks can be generated by a process called preferential attachment (assortative mixing, assortativity (his Fig. 1, as well as a video in its Supplement S3), and see also and Wikipedia headword ‘Preferential attachment’). We suggest to link the quest for a simple distribution equation for the SAD with network research.

    Not only ‘our’ data but, quite often, other data as well consist of sample×species cross-tables. Such tables provide for the opportunity to merge samples into a composite sample for a subset, or the whole set, as we did. Another opportunity is to study ‘within species over samples’ characteristics. We point to the abundance-occupancy relation, to Taylor’s law (fluctuation scaling) and to sampling theory. Hopefully, all patterns can be integrated and applied for analysis with resampling statistics (see also Wikipedia headwords ‘Nonparametric statistics’ and ‘Resampling (statistics)’) to obtain robust results, especially on the SAD.

    Some of ‘our’ data sets provide for a time or a spatial series (fine scale: Mushrooms, Fish, Crustaceans, Fish+Crustaceans, Trees and Rodents). This makes them eligible to study the process of accumulation (‘sample’ growth, collectors curve, and species area relation (see also) and to look for autocorrelation). Seasonality aspects have already been described for the Mushrooms set, for Fish, and for Rodents and Annual plants. Spatial patterns for the Trees set have been described in .

  • Conclusions

    Our result makes a robust impression (12 systems at three scales of integration) that the stretched exponential is a possible alternative for the power law. This result may stimulate others to pick up again the power law, its allies, and their generative properties for species abundance data.

  • Limitations

    Our statistics are traditional and limited in scope. Neither advanced goodness-of-fit testing is done nor statistical comparison with other models is made. We did not study, just considered, generative processes.

  • Conjectures

    Ideal power law behaviour is absent in the data sets. Data points in log/log plots show curvature, concave in almost all cases. Fitted stretched exponentials meet this curvature. Advanced goodness-of-fit testing, model comparison/selection, and generative processes need to be done, expertise that we do not master and for which we seek collaboration.

  • Methods

    12 data sets were studied. [I] A data set on Mushrooms, the property of the Swiss Federal Institute for Forest, Snow and Landscape Research WSL, managed by Simon Egli. Data sets on [II] Fish and [III] Crustaceans, the property of Pisces Conservation Ltd., managed by Peter A. Henderson (see also). Fish and Crustaceans were enumerated from the same physical samples. We also studied the ‘whole’ samples: [IV] Fish+Crustaceans. We consider this an integration of 2 (sub) assemblages into a (new) assemblage (see our Supplement A). [V] A data set on tropical rainforest Trees from the Smithsonian Tropical Research Institute’s Center for Tropical Forest Science, managed by Condit et al. (see also). We also used data sets on four different desert assemblages of [VI] Rodents, [VII] Winter annuals, [VIII] Summer annuals and [IX] Ant colonies in the Chihuahuan desert, near Portal, Arizona. [X] A data set on weed Seedlings managed by the Centre for Ecology and Hydrology. [XI] A data set on Brachiopod fossils obtained from Thomas D. Olszewski. He re-enumerated material that had been deposited at the National Museum of Natural History, Washington DC. The material was sampled from Permian deposits spanning a period of approx. 10 Myr in a mountain range of approx. 40 km. The set of 187 samples was presented as consisting of 4 composite assemblages representing four geological formations. We consider the data as 1 composite set on our account. [XII] A data set on cow patty Flies. Characteristics of the data sets are given in the supplemental table. The sets, IV and XI excepted, were collected and studied previously for a characteristic of SADs as histograms, with data binned into frequency classes. Some additional information on the data sets can be found there.

    Most sets have samples that were collected in different years (Mushrooms, Fish, Crustaceans, Rodents, Winter and Summer Annuals, Ants, Flies). Within-years sampling was done in different weeks (Mushrooms), in different months (Fish, Crustaceans), or at different locations (Rodents, Winter and Summer Annuals, Ants, Flies). Thus, samples can be assigned to subsets (terminology of set theory: the many samples are objects that form different subsets that form the set (Wikipedia headword ‘Set theory’)). In the other sets (Trees, Seedlings, Brachiopods) a similar structure can be applied. Within the subsets and the set, the samples can be merged, abundances adding up over species, forming composited ‘samples’. We studied (i) samples, (ii) composite samples of subsets and (iii) composite samples of sets, representing 3 scales of integration. Total abundance and species richness values, n and S, of samples and of composite samples of subsets were rank-transformed. The ranks over both parameters were averaged and their median was used to select ‘average’ samples among the primary and the composited samples of subsets, for the figure.

    For the stretched exponential, we followed. The equation is y = (b+a×ln(x))^(1/c), with y for abundance and x for rank (rank 1 assigned to highest abundance value). It has three parameters: a, b, and c. The function can be rewritten to y^c = b+a×ln(x). This linear function can be used in simple fitting, using least squares. First, in an iterative process, the correlation between ln(x) and y^c is maximized by varying c, resulting in the best fitting value for c. Additionally, a linear regression is performed of y^c on ln(x), resulting in fitted values for a and b. For what they call the intuitive interpretation of the three parameters a, b and c, we refer to.

  • Ethics statement

    Not applicable.

  • References
  • 1
    Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum

    Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum ipsum

    Lorem ipsum Lorem ipsum Lorem ipsum
    2
    Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum

    Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum ipsum

    Lorem ipsum Lorem ipsum Lorem ipsum
    3
    Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum

    Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum ipsum

    Lorem ipsum Lorem ipsum Lorem ipsum
    4
    Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum

    Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum ipsum

    Lorem ipsum Lorem ipsum Lorem ipsum
    5
    Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum

    Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum ipsum

    Lorem ipsum Lorem ipsum Lorem ipsum
    Matters10/20

    Stretched exponential as one of the alternatives for the power law to fit ranked species abundance data

    Affiliation listing not available.
    Abstractlink

    A central object in community ecology is species abundance distribution. We are interested in the power law and its allies for ranked species abundance data. We collected 12 large data sets consisting of many samples. The preliminary fitting result makes a robust impression (12 systems at three scales of integration) that the stretched exponential is an interesting alternative for the power law. For further work, advanced statistics are required. Not only ‘our’ data but, quite often, other data as well consist of sample×species cross-tables. With cross-tables, also ‘within species over samples’ characteristics can be studied. An integrated view on data patterns in multi-sample sets may help to identify generative processes for and the formulation of a relatively simple model for species abundance data.

    Figurelink

    Figure 1. Rank abundance plots for 12 data sets.

    For each set, 3 samples at different scales of integration are shown: an average primary sample, an average composite sample of a subset, and the composite sample of the whole set (full-resolution file available as raw data). Data points in the log/log plots do not form straight lines; thus, ideal power law behaviour is absent. Stretched exponentials were fitted (drawn lines).

    Introductionlink

    A central object in community ecology is species abundance distribution (SAD). It has been studied for over a century since Raunkiaer (in[1]). The review of an expert group[2] is a benchmark for properties and generative theory. The edited volume[3] on biological diversity is, among others, an update. For applications of the SAD, like the measurement of biodiversity, we refer to[4].

    The expert group[2] described a law for species abundance data: “When plotted as a histogram of number (or percent) of species on the y-axis vs. abundance on an arithmetic x-axis, the classic hyperbolic, ‘lazy J-curve’ or ‘hollow curve’ is produced, indicating a few very abundant species and many rare species (Fig. 1A). In this form, the law appears to be universal; we know of no multispecies community, ranging from the marine benthos to the Amazonian rainforest, that violates it” (in our Supplement A, we make some remarks to the meaning of ‘community’ and ‘sample’). Several terms are used for a ‘hollow curve’: the distribution is (right)-skewed, long-tailed, has extreme values, shows rare events. Is there a simple meaningful equation for ‘the’ species abundance distribution? One is inclined to think so if there is a ‘universal law’. Review[2] is steeped with the idea of a relatively simple equation for SADs, but it presents the opposite too, that different communities have rather different SADs and that groups of species within a community have different SADs, making the community’s SAD a mixed one[5]. The SAD is mostly treated as a histogram, based on the binning of data into frequency classes (for a probability mass or density function). However, the SAD can be illustrated as a rank abundance or Whittaker plot (see[2], their Fig. 1c; see also[6][7]). Ranked data are used for exploratory data analysis. Rank-size plots and (cumulative) probability plots are strongly related[8][9] (and see our Supplement B). SADs bear similarity to distributions in other fields of science[10][11][12].

    Long-tailed distributions of natural and manmade phenomena, in rank-size form (where ‘size’ can be read as ‘abundance’), often show power law behaviour[13] (and see Wikipedia headword ‘Power law’). The ubiquitous power law has been considered for species abundance data. However, ideal power law behaviour is absent or rare: data points do not lie in a straight line in a log/log plot[14][15][16]. For this reason, the interest in the power law for species abundance data seems to have vanished. However, the imperfect power law behaviour in other fields of science is well documented[8][17]. Paper[17] is of particular interest. It revisits the data and analysis in a seminal paper[13] and concludes that 9 of 24 data sets conjectured to follow a power law actually do not.

    Objectivelink

    The direct aim is to generate interest again in the power law, especially in its allies like the stretched exponential, for species abundance data. Further reaching aims, to be tackled in the future and for which we introduce a frame, are (i) to generate the interest of community ecologists in the generative processes of the power law and allies (that have been studied in other fields of science) and (ii) to complete the quest for a simple yet meaningful equation/model for the SAD.

    Results & Discussionlink

    Rank  abundance plots were made, shown in figure 1. Data points in the log/log plots do not form straight lines, ideal power law behaviour is absent. The data points indicate curved lines, concave in almost all cases. As an alternative to the power law, the stretched exponential function was fitted. The parameter values of the fitted function, for the composite samples of the complete sets only, are given in our supplemental table B. No advanced statistics was applied. No comparison with other functions/models was made. The actual data show some deviations from the fitted curves, but the overall result is visually satisfying.

    For further work, advanced statistics are required; we refer to[17][18]. For model comparison and selection, we refer to[19][18][20]. For allies of the power law, we refer to[8][21][22] (especially chapter 4)[17][23][24][25].

    A challenge lies in a remark in the review of the expert group[2]: “Starting in the 1970s and running unabated to the present day, mechanistic models (models attempting to explain the causes of the hollow curve SAD) and alternative interpretations and extensions of prior theories have proliferated to an extraordinary degree”. The power law and its allies are often considered for the degree or connectivity distribution of networks. Species abundance data are retrieved from ecological communities that are networks[26]. However, a network topology behind species abundance data is not immediately clear. Species abundance data are reminiscent to data of food webs. For instance, the interactions between fruits and frugivorous birds can be presented in a cross-table of fruit species×bird species (data of[27] in[28]). From such a table, one can summarize the number of connections for the fruiting species with the bird species and vice versa: two connectivity distributions. The one dealing with the number of interactions for fruiting species over bird species is reminiscent of an assemblage of fruiting species, ‘sampled’ by birds. Networks can be generated by a process called preferential attachment (assortative mixing, assortativity[13][29] (his Fig. 1, as well as a video in its Supplement S3), and see also[30] and Wikipedia headword ‘Preferential attachment’). We suggest to link the quest for a simple distribution equation for the SAD with network research[31][32].

    Not only ‘our’ data but, quite often, other data as well consist of sample×species cross-tables. Such tables provide for the opportunity to merge samples into a composite sample for a subset, or the whole set, as we did. Another opportunity is to study ‘within species over samples’ characteristics. We point to the abundance-occupancy relation[33], to Taylor’s law (fluctuation scaling[34][35]) and to sampling theory[36]. Hopefully, all patterns can be integrated and applied for analysis with resampling statistics[37] (see also Wikipedia headwords ‘Nonparametric statistics’ and ‘Resampling (statistics)’) to obtain robust results, especially on the SAD.

    Some of ‘our’ data sets provide for a time or a spatial series (fine scale: Mushrooms, Fish, Crustaceans, Fish+Crustaceans, Trees and Rodents). This makes them eligible to study the process of accumulation (‘sample’ growth, collectors curve, and species area relation[38] (see also[32]) and to look for autocorrelation). Seasonality aspects have already been described for the Mushrooms set[39], for Fish[40], and for Rodents and Annual plants[41][42][43]. Spatial patterns for the Trees set have been described in [44].

    Conclusionslink

    Our result makes a robust impression (12 systems at three scales of integration) that the stretched exponential[8][17] is a possible alternative for the power law. This result may stimulate others to pick up again the power law, its allies, and their generative properties for species abundance data.

    Limitationslink

    Our statistics are traditional and limited in scope. Neither advanced goodness-of-fit testing is done nor statistical comparison with other models is made. We did not study, just considered, generative processes.

    Conjectureslink

    Ideal power law behaviour is absent in the data sets. Data points in log/log plots show curvature, concave in almost all cases. Fitted stretched exponentials meet this curvature. Advanced goodness-of-fit testing, model comparison/selection, and generative processes need to be done, expertise that we do not master and for which we seek collaboration.

    Methodslink

    12 data sets were studied. [I] A data set on Mushrooms, the property of the Swiss Federal Institute for Forest, Snow and Landscape Research WSL, managed by Simon Egli[45]. Data sets on [II] Fish and [III] Crustaceans, the property of Pisces Conservation Ltd., managed by Peter A. Henderson (see also[46]). Fish and Crustaceans were enumerated from the same physical samples. We also studied the ‘whole’ samples: [IV] Fish+Crustaceans. We consider this an integration of 2 (sub) assemblages into a (new) assemblage (see our Supplement A). [V] A data set on tropical rainforest Trees from the Smithsonian Tropical Research Institute’s Center for Tropical Forest Science, managed by Condit et al.[47] (see also[48]). We also used data sets on four different desert assemblages of [VI] Rodents, [VII] Winter annuals, [VIII] Summer annuals and [IX] Ant colonies in the Chihuahuan desert, near Portal, Arizona[49][50]. [X] A data set on weed Seedlings managed by the Centre for Ecology and Hydrology[51][52]. [XI] A data set on Brachiopod fossils obtained from Thomas D. Olszewski. He re-enumerated material[53] that had been deposited at the National Museum of Natural History, Washington DC[54]. The material was sampled from Permian deposits spanning a period of approx. 10 Myr in a mountain range of approx. 40 km. The set of 187 samples was presented as consisting of 4 composite assemblages representing four geological formations[54]. We consider the data as 1 composite set on our account. [XII] A data set on cow patty Flies[55]. Characteristics of the data sets are given in the supplemental table. The sets, IV and XI excepted, were collected and studied previously[56] for a characteristic of SADs as histograms, with data binned into frequency classes. Some additional information on the data sets can be found there[56].

    Most sets have samples that were collected in different years (Mushrooms, Fish, Crustaceans, Rodents, Winter and Summer Annuals, Ants, Flies). Within-years sampling was done in different weeks (Mushrooms), in different months (Fish, Crustaceans), or at different locations (Rodents, Winter and Summer Annuals, Ants, Flies). Thus, samples can be assigned to subsets (terminology of set theory: the many samples are objects that form different subsets that form the set (Wikipedia headword ‘Set theory’)). In the other sets (Trees, Seedlings, Brachiopods) a similar structure can be applied. Within the subsets and the set, the samples can be merged, abundances adding up over species, forming composited ‘samples’. We studied (i) samples, (ii) composite samples of subsets and (iii) composite samples of sets, representing 3 scales of integration. Total abundance and species richness values, n and S, of samples and of composite samples of subsets were rank-transformed. The ranks over both parameters were averaged and their median was used to select ‘average’ samples among the primary and the composited samples of subsets, for the figure.

    For the stretched exponential, we followed[8]. The equation is y = (b+a×ln(x))^(1/c), with y for abundance and x for rank (rank 1 assigned to highest abundance value). It has three parameters: a, b, and c. The function can be rewritten to y^c = b+a×ln(x). This linear function can be used in simple fitting, using least squares. First, in an iterative process, the correlation between ln(x) and y^c is maximized by varying c, resulting in the best fitting value for c. Additionally, a linear regression is performed of y^c on ln(x), resulting in fitted values for a and b. For what they call the intuitive interpretation of the three parameters a, b and c, we refer to[8].

    Conflict of interestlink

    The authors declare no conflicts of interest.

    Ethics Statementlink

    Not applicable.

    No fraudulence is committed in performing these experiments or during processing of the data. We understand that in the case of fraudulence, the study can be retracted by ScienceMatters.

    Referenceslink
    1. Leslie A. Kenoyer
      A Study of Raunkaier's Law of Frequence
      Ecology, 8/1927, pages 341-349 DOI: 10.2307/1929336chrome_reader_mode
    2. McGill Brian J., Etienne Rampal S., Gray John S.,more_horiz, White Ethan P.
      Species abundance distributions: moving beyond single prediction theories to integration within an ecological framework
      Ecology Letters, 10/2007, pages 995-1015 DOI: 10.1111/j.1461-0248.2007.01094.xchrome_reader_mode
    3. Anne E. Magurran, Brian J. McGill
      Biological Diversity: Frontiers in Measurement and Assessment
      Oxford University Press, 2011, page 345 chrome_reader_mode
    4. Matthews Thomas J., Whittaker Robert J.
      REVIEW: On the species abundance distribution in applied ecology and biodiversity management
      Journal of Applied Ecology, 52/2015, pages 443-454 DOI: 10.1111/1365-2664.12380chrome_reader_mode
    5. Henriques Antão Laura, Connolly Sean R., Magurran Anne E.,more_horiz, Dornelas Maria
      Prevalence of multimodal species abundance distributions is linked to spatial and taxonomic breadth
      Global Ecology and Biogeography, 26/2017, pages 203-215 DOI: 10.1111/geb.12532chrome_reader_mode
    6. Robert H. Macarthur
      ON THE RELATIVE ABUNDANCE OF BIRD SPECIES
      Proceedings of the National Academy of Sciences, 43/1957, pages 293-295 DOI: 10.1073/pnas.43.3.293chrome_reader_mode
    7. Whittaker R. H.
      Dominance and Diversity in Land Plant Communities: Numerical relations of species express the importance of competition in community function and evolution
    8. Laherrère J., Sornette D.
      Stretched exponential distributions in nature and economy: “fat tails” with characteristic scales
      The European Physical Journal B - Condensed Matter and Complex Systems, 2/1998, pages 525-539 DOI: 10.1007/s100510050276chrome_reader_mode
    9. Anne E. Magurran, Brian J. McGill
      Species abundance distributions: Biological diversity and frontiers in measurement and assessment.
      Oxford University Press, 2011, pages 105-122 chrome_reader_mode
    10. Frank W. Preston
      Gas Laws and Wealth Laws
      The Scientific Monthly, 71/1950, pages 309-311 chrome_reader_mode
    11. Nekola Jeffrey C., Brown James H.
      The wealth of species: ecological communities, complex systems and the legacy of Frank Preston
      Ecology Letters, 10/2007, pages 188-196 DOI: 10.1111/j.1461-0248.2006.01003.xchrome_reader_mode
    12. Bowler Michael G., Kelly Colleen K.
      On the statistical mechanics of species abundance distributions
      Theoretical Population Biology, 82/2012, pages 85-91 DOI: 10.1016/j.tpb.2012.05.006chrome_reader_mode
    13. Newman Mej
      Power laws, Pareto distributions and Zipf's law
      Contemporary Physics, 46/2005, pages 323-351 DOI: 10.1080/00107510500052444chrome_reader_mode
    14. Tokeshi M.
      Species Abundance Patterns and Community Structure
      Advances in Ecological Research, 24/1993, pages 111-186 DOI: 10.1016/s0065-2504(08)60042-2chrome_reader_mode
    15. Mouillot David, Lepretre Alain
      Introduction of relative abundance distribution (RAD) indices, estimated from the rank-frequency diagrams (RFD), to assess changes in community diversity
      Environmental Monitoring and Assessment, 63/2000, pages 279-295 DOI: 10.1023/a:1006297211561chrome_reader_mode
    16. Pueyo Salvador
      Diversity: between neutrality and structure
    17. Clauset Aaron, Shalizi Cosma Rohilla, Newman M. E. J.
      Power-Law Distributions in Empirical Data
      SIAM Review, 51/2009, pages 661-703 DOI: 10.1137/070710111chrome_reader_mode
    18. Sean R. Connolly, M. Aaron Macneil, M. Julian Caley, Nancy Knowlton, Ed Cripps, Mizue Hisano, Loïc M. Thibaut, Bhaskar D. Bhattacharya, Lisandro Benedetti-Cecchi, Russell E. Brainard, Angelika Brandt, Fabio Bulleri, Kari E. Ellingsen, Stefanie Kaiser, Ingrid Kröncke, Katrin Linse, Elena Maggi, Timothy D. O’hara, Laetitia Plaisance, Gary C. B. Poore, Santosh K. Sarkar, Kamala K. Satpathy, Ulrike Schückel, Alan Williams, Robin S. Wilson
      Commonness and rarity in the marine biosphere
      Proceedings of the National Academy of Sciences, 111/2014, pages 8524-8529 DOI: 10.1073/pnas.1406664111chrome_reader_mode
    19. Ulrich Werner, Ollik Marcin, Ugland Karl Inne
      A meta-analysis of species-abundance distributions
    20. Matthews, T.J., Whittaker, R.J.
      Fitting and comparing competing models of the species abundance distribution: assessment and prospect
      Frontiers of Biogeography, 6/2014, pages 67-82 chrome_reader_mode
    21. Benguigui Lucien, Blumenfeld-Lieberthal Efrat
      A dynamic model for city size distribution beyond Zipf's law
      Physica A: Statistical Mechanics and its Applications, 384/2007, pages 613-627 DOI: 10.1016/j.physa.2007.05.059chrome_reader_mode
    22. Didier Sornette
      Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools
      Springer Series in Synergetics, Springer Berlin Heidelberg, 2006, page 528 chrome_reader_mode
    23. Martínez-Mekler Gustavo, Martínez Roberto Alvarez, Del Río Manuel Beltrán,more_horiz, Cocho Germinal
      Universality of Rank-Ordering Distributions in the Arts and Sciences
    24. Maruvka Yosef E., Kessler David A., Shnerb Nadav M.
      The Birth-Death-Mutation Process: A New Paradigm for Fat Tailed Distributions
    25. Finley Benjamin J., Kilkki Kalevi
      Exploring Empirical Rank-Frequency Distributions Longitudinally through a Simple Stochastic Process
    26. Parker V. Thomas
      The community of an individual: implications for the community concept
    27. Bruce Beehler
      Frugivory and Polygamy in Birds of Paradise
      The Auk, 100/1983, pages 1-12 chrome_reader_mode
    28. Jordano Pedro, Bascompte Jordi, Olesen Jens M.
      Invariant properties in coevolutionary networks of plant-animal interactions
      Ecology Letters, 6/2002, pages 69-81 DOI: 10.1046/j.1461-0248.2003.00403.xchrome_reader_mode
    29. Albert-László Barabási
      Scale-Free Networks: A Decade and Beyond
      Science, 325/2009, pages 412-413 DOI: 10.1126/science.1173299chrome_reader_mode
    30. Olesen Jens M., Bascompte Jordi, Elberling Heidi, Jordano Pedro
      TEMPORAL DYNAMICS IN A POLLINATION NETWORK
      Ecology, 89/2008, pages 1573-1582 DOI: 10.1890/07-0451.1chrome_reader_mode
    31. Ings Thomas C., Montoya José M., Bascompte Jordi,more_horiz, Woodward Guy
      Review: Ecological networks - beyond food webs
      Journal of Animal Ecology, 78/2009, pages 253-269 DOI: 10.1111/j.1365-2656.2008.01460.xchrome_reader_mode
    32. Lee Sang Hoon, Kim Pan-Jun, Jeong Hawoong
      Statistical properties of sampled networks
    33. Holt Alison R., Gaston Kevin J., He Fangliang
      Occupancy-abundance relationships and spatial distribution: A review
      Basic and Applied Ecology, 3/2002, pages 1-13 DOI: 10.1078/1439-1791-00083chrome_reader_mode
    34. Taylor L. R.
      Aggregation, Variance and the Mean
      Nature, 189/1961, pages 732-735 DOI: 10.1038/189732a0chrome_reader_mode
    35. Xiao Xiao, Locey Kenneth J., White Ethan P.
      A Process-Independent Explanation for the General Form of Taylor’s Law
      The American Naturalist, 186/2015, pages E51-E60 DOI: 10.1086/682050chrome_reader_mode
    36. Wang Jin-Feng, Stein A., Gao Bin-Bo, Ge Yong
      A review of spatial sampling
      Spatial Statistics, 2/2012, pages 1-14 DOI: 10.1016/j.spasta.2012.08.001chrome_reader_mode
    37. Clarke K. Robert, Somerfield Paul J., Gorley Raymond N.
      Testing of null hypotheses in exploratory community analyses: similarity profiles and biota-environment linkage
      Journal of Experimental Marine Biology and Ecology, 366/2008, pages 56-69 DOI: 10.1016/j.jembe.2008.07.009chrome_reader_mode
    38. Pueyo Salvador
      Self-similarity in species-area relationship and in species abundance distribution
    39. Straatsma Gerben, Ayer François, Egli Simon
      Species richness, abundance, and phenology of fungal fruit bodies over 21 years in a Swiss forest plot
      Mycological Research, 105/2001, pages 515-523 DOI: 10.1017/s0953756201004154chrome_reader_mode
    40. Shimadzu Hideyasu, Dornelas Maria, Henderson Peter A, Magurran Anne E
      Diversity is maintained by seasonal variation in species abundance
    41. Brown James H., Heske Edward J.
      Temporal Changes in a Chihuahuan Desert Rodent Community
      Oikos, 59/1990, pages 290-302 DOI: 10.2307/3545139chrome_reader_mode
    42. Guo Qinfeng, Brown James H.
      Temporal fluctuations and experimental effects in desert plant communities
      Oecologia, 107/1996, pages 568-577 DOI: 10.1007/bf00333950chrome_reader_mode
    43. Ernest S. K. Morgan, Brown James H., Parmenter Robert R.
      Rodents, plants, and precipitation: spatial and temporal dynamics of consumers and resources
    44. Richard Condit, Peter S. Ashton, Patrick Baker, Sarayudh Bunyavejchewin, Savithri Gunatilleke, Nimal Gunatilleke, Stephen P. Hubbell, Robin B. Foster, Akira Itoh, James V. Lafrankie, Hua Seng Lee, Elizabeth Losos, N. Manokaran, R. Sukumar, Takuo Yamakura
      Spatial Patterns in the Distribution of Tropical Tree Species
      Science, 288/2000, pages 1414-1418 DOI: 10.1126/science.288.5470.1414chrome_reader_mode
    45. Egli, S., Ayer,more_horiz, F.
      Die Beschreibung der Diversitaet von Makromyzeten. Erfahrungen aus pilzoekologischen Langzeitstudien im Pilzreservat la Chaneaz
      Mycologia Helvetica, 9/1977, pages 19-32 chrome_reader_mode
    46. Henderson P.A., Bird D.J.
      Fish and macro-crustacean communities and their dynamics in the Severn Estuary
      Marine Pollution Bulletin, 61/2010, pages 100-114 DOI: 10.1016/j.marpolbul.2009.12.017chrome_reader_mode
    47. Condit, R., Hubbell,more_horiz, R.B.
      Barro Colorado forest census plot data. Smithsonian Tropical Research Institute, Center for tropical forest science
      http://www.ctfs.si.edu/group/Resources/Data, 2005 chrome_reader_mode
    48. Condit Richard, Hubbell Stephen P., Lafrankie James V.,more_horiz, Ashton Peter S.
      Species-Area and Species-Individual Relationships for Tropical Trees: A Comparison of Three 50-ha Plots
      The Journal of Ecology, 84/1996, page 549 DOI: 10.2307/2261477chrome_reader_mode
    49. James H. Brown, Thomas G. Whitham, S. K. Morgan Ernest, Catherine A. Gehring
      Complex Species Interactions and the Dynamics of Ecological Systems: Long-Term Experiments
    50. Ernest, S.K.M., Valone,more_horiz, J.H.
      Long-term monitoring and experimental manipulation of a Chihuahuan Desert ecosystem near Portal, Arizona, USA
      Ecology; Ecological Archives. Available from: http://www.esapubs.org/archive/ecol/E090/118/, 90/2009, page 1708 chrome_reader_mode
    51. Centre For Ecology, Hydrology
      Information Gateway. Farm Scale Evaluations. Available after registration from: https://gateway.ceh.ac.uk/
    52. Firbank, L. G., Heard, M. S., Woiwod, I. P.,more_horiz, Perry, J. N.
      An introduction to the Farm-Scale Evaluations of genetically modified herbicide-tolerant crops
      Journal of Applied Ecology, 40/2003, pages 2-16 DOI: 10.1046/j.1365-2664.2003.00787.xchrome_reader_mode
    53. Cooper, G. A., Grant, R. E.
      Permian Brachiopods of West Texas, VI
      Smithsonian Contributions to Paleaobiology 32. Washington: Smithsonian Institution Press., 1977 chrome_reader_mode
    54. Olszewski Thomas D., Erwin Douglas H.
      Dynamic response of Permian brachiopod communities to long-term environmental change
      Nature, 428/2004, pages 738-741 DOI: 10.1038/nature02464chrome_reader_mode
    55. Papp, L.
      A study of the cow pat Diptera on the Hortobágy, Hungary
      Folia entomologica hungarica, 68/2007, pages 123-135 chrome_reader_mode
    56. Straatsma Gerben, Egli Simon
      Rarity in large data sets: Singletons, modal values and the location of the species abundance distribution
      Basic and Applied Ecology, 13/2012, pages 380-389 DOI: 10.1016/j.baae.2012.03.011chrome_reader_mode
    Commentslink

    Create a Matters account to leave a comment.