DOI: 10.5061/dryad.h18931zxv
title: Jaccard dissimilarity in stochastic community models based on the species-independence assumption

description: 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
 each species), flat SIPs - with all species having the same presence
 probability - maximize the expected beta-diversity. We refer to this
 prediction as the ‘transfer principle for beta’. Second, using
 SIPs and the probabilistic method in a two-species scenario, we
 demonstrate that beta-diversity is lower when SIPs are
 parallel compared to when they are anti-parallel. We also find
 that this prediction is consistent with the well-known
 checkerboard pattern in incidence matrices. Finally, we apply
 the method to the species distribution models for five woodpecker
 species in Switzerland, showing that their spatial distributions
 will change significantly. Overall, this work improves our
 understanding of how pairwise beta-diversity responds to
 occupancy heterogeneity.
description: 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.
description: # 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 Each mathematica `**.nb` file produces the corresponding figure, each converted into `**.pdf` file. For other files therein, see the following table. | subdirectory | files | note | | :----------: | :---------------------------------------: | :-----------------------------------------------------------------------------------------------------------------------------------: | | `DFig2` | `Error4.pdf`, `Error8.pdf`, `Error16.pdf` | Panels (A)-(C), with the variable number of species S ranging from 4, 8, and 16. | | `DFig3` | `Fig3A.pdf`, `Fig3B.pdf` | Panel A depicts the dependence of SIPs on the unevenness parameter, whereas panel B depicts the dependence of E\[J] on the parameter. | | `DFig4` | N/A | N/A | # Software version Mathematica Japanese Edition for Sites, ``` 13.3.0 ```