Jaccard dissimilarity in stochastic community models based on the species-independence assumption

A fundamental problem in ecology is understanding the changes in species composition among sites (i.e. beta-diversity). It is unclear how spatial heterogeneity in species occupancy across sites shapes patterns of beta-diversity. To address this question, we develop probabilistic models that consider two spatial or temporal sites, where presence probabilities vary both among species and between the sites. We derive analytical and approximate formulae for the expectation of pairwise beta-diversity. Using a graphical tool, stochastic incidence plots (SIPs), which depict the presence probabilities in two sites along species labels, we develop a means to conceptualize the heterogeneity in presence probabilities: the steepness or unevenness of SIPs reflects species-level heterogeneity, while the degree of overlap between SIPs indicates site-level heterogeneity. We find that when SIPs completely overlap (i.e. two sites have the same presence probability for e...,
Data for the stochastic simulation in Figure 2 (testing the accuracy of the heuristic approximation).
Numerical recipe for Figure 3 (transfer-principle for beta).
Numerical recipe for Figure 4 (2-by-2 incidence matrices).

Section 1 contains a PDF of a Mathematica Notebook file (Wolfram). For more details, see README., # Jaccard dissimilarity in stochastic community models based on the species-independence assumption

[https://doi.org/10.5061/dryad.h18931zxv](https://doi.org/10.5061/dryad.h18931zxv)

Codes to reproduce the results of (i) Figure 2, the stochastic simulations checking the numerical accuracy of the approximation of the expected Jaccard dissimilarity (E[J]) by the heuristic approximation Jheur in Iritani et al. 2025, Ecography; (ii) Figure 3, the transfer-principle for beta or the Schur-concavity of E[J]; and (iii) Figure 4, the results of two-by-two matrices and their relation to E[J].

By running the Mathematica Notebook file, all the figures will be produced and exported to the directory where the `**.nb` is located (e.g., `DFig2` subdirectory for `Fig2.nb`).

## Structure of the data directory

The `Data` directory contains three subdirectories each containing one `.nb` file:

* `Fig2.nb` in `DFig2` for Figure 2
* `Fig3.nb` in `DFig3` for Figure 3
* `Fig4.nb` in `DFig4` for Figure 4
...,