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
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.