^{1}

^{[C]}

^{2}

The authors declare no conflicts of interest.

Conceptualization: Gerben Straatsma

Data curation: Gerben Straatsma, Simon Egli

Investigation: Gerben Straatsma

Original draft: Gerben Straatsma

Review & editing: Simon Egli

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 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).

A central object in community ecology is species abundance distribution (SAD). It has been studied for over a century since Raunkiaer (in^{[1]}^{[2]}^{[3]}^{[4]}

The expert group^{[2]}^{[2]}^{[5]}^{[2]}^{[6]}^{[7]}^{[8]}^{[9]}^{[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]}^{[14]}^{[15]}^{[16]}^{[8]}^{[17]}^{[17]}^{[13]}

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.

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]}^{[19]}^{[18]}^{[20]}^{[8]}^{[21]}^{[22]}^{[17]}^{[23]}^{[24]}^{[25]}

A challenge lies in a remark in the review of the expert group^{[2]}^{[26]}^{[27]}^{[28]}^{[13]}^{[29]}^{[30]}^{[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]}^{[34]}^{[35]}^{[36]}^{[37]}

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]}^{[32]}^{[39]}^{[40]}^{[41]}^{[42]}^{[43]}^{[44]}

Our result makes a robust impression (12 systems at three scales of integration) that the stretched exponential^{[8]}^{[17]}

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.

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.

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]}^{[46]}^{[47]}^{[48]}^{[49]}^{[50]}^{[51]}^{[52]}^{[53]}^{[54]}^{[54]}^{[55]}^{[56]}^{[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]}^{[8]}