Material and Methods
The model
To analyse change within a plot we use the partitioning approach from
Godsoe et al. (2021). Within a given plot the absolute abundance of each
species is \(n_{i}\) and the relative abundance of each species is\(p_{i}=n_{i}/\sum_{i}n_{i}\). For many diversity indices, the
contribution that each individual makes to diversity can be defined as
zi, the rarity of the species to which it belongs
(Equation 1). For Shannon entropy, the measure of rarity is
zi=–log(pi). Because we are interested
in change between one period and another we can define rarity scores for
Gini-Simpson’s diversity as zi = -pi(Godsoe et al. 2021). Diversity is an average measurement of rarity
across all individuals in a plot (Patil and Taillie 1982); For Shannon
entropy, the arithmetic mean of rarity scores are weighted by the
relative abundance of each species (Jost 2006, 2007, Cover and Thomas
2012).
\(D=\sum_{i}{p_{i}z_{i}}\).
Total change in diversity over time is the difference between a
measurement of diversity in the present and diversity in the past
(Equation 2). Here and elsewhere, describes change over time and
the prime superscript denotes present measurements (Frank 2012).
\(D=D^{{}^{\prime}}-D\) .
To understand the effects of immigrants on diversity change, we further
divide the present community is into two components:
\(D^{{}^{\prime}}=\text{\ ω}\sum_{i}{p_{i}^{\prime}z_{i}^{\prime}}+\mu\sum_{j}{a_{j}z_{j}^{*}}\).
The first term represents what we will call “descendants”, individuals
that are not immigrants. They may represent individuals that were in the
community in the first survey, or their offspring. Here \(\omega\) is
the proportion of individuals in the present community that descended
from the past community. Among descendants the proportion belonging to
species i is pi\({}^{\prime}\), and the rarity score of
species i is zi \({}^{\prime}\) (note this is rarity
as a proportion of total individuals in both time steps). The second
term describes the contribution of immigrants, where \(\mu\) is the
proportion of individuals in the present community which immigrated to
the community since the initial measurement. Among immigrants, the
proportion belonging to species i is aj\({}^{\prime}\), and
the rarity score of species j is zj* (note this is
rarity as a proportion of all individuals, not just immigrants).
Overall diversity change in equation 2 can then be partitioned using
extensions of the Price equation (Price 1970, Kerr and Godfrey‐Smith
2009). This leads to explicit definitions for the effects of selection,
immigration and rarity shifts on diversity (Equation 4).
(4) \(D=\underset{\par
\begin{matrix}\text{Selection}\\
(+\ drift)\\
\end{matrix}}{}+\underset{\text{immigration}}{}+\underset{\par
\begin{matrix}\text{rarity\ }\\
\text{shifts}\\
\end{matrix}}{}\)
In Equation 4, the first term describes the effect of selection, where
the tendency of species i to leave more descendants increases its
frequency relative to other descendants (\(p_{i}=p_{i}^{{}^{\prime}}-p_{i})\).
This term implicitly includes effects of drift because both mechanisms
change species’ relative abundances (Rice 2004). In this framework,
selection on species identity emerges when one species tends to leave
more descendants than another either by leaving more offspring or having
higher survival (Vellend 2016). Drift emerges when one species increases
in relative abundance due to stochastic sampling. The second term
describes immigration, which changes diversity when diversity among
immigrants \(\sum_{j}{a_{j}z_{j}^{*}}\) is different from the
diversity among resident ancestors. The final term denotes rarity shifts
where the rarity score for descendants is different from the rarity
scores of ancestors \(z_{i}\), as in (Figure 1 C).
We have presented diversity indices that are familiar and easy to
partition on a linear scale. More complexities emerge when analysing so
called “Hill numbers”, a re-scaled version of diversity indices
expressed in common units. Both Shannon entropy and Simpsons diversity
indices can be converted into common units the equivalent number of
uniformly distributed species needed to produce the observed diversity
index (Jost 2006, 2007). While not used in this paper, the numbers
equivalent Shannon entropy can be partitioned by exponentiation Equation
4 (Godsoe et al. 2022). This conversion changes the scale of measurement
from additive to multiplicative but preserves many of the qualitative
patterns. At present, techniques exist to other Hill numbers into
selection and transmission terms, in the absence of immigration (Frank
and Godsoe 2020), but this condition is unrealistic for most
observations in our study.
The Data
We analysed the causes of biodiversity change from plots in 15 studies
compiled in the BioTIME database (Dornelas et al. 2018). As of December
2021, the BioTIME database contained 361 studies, 201 of which were
terrestrial surveys. We selected vegetation studies with multiple plots
of area greater than 1 m2, including observations from
3 or more time periods, 5 or more species, and counts of individuals (as
opposed to biomass, presence or vegetation cover). This resulted in a
database of 3341 observations of diversity change, with each observation
representing the difference in diversity between one sample and a
previous sample.
To compare the effect of rarity shifts with other mechanisms, we
calculated the change in Shannon entropy and Gini-Simpson’s diversity
within each plot across each time step using Equations 1 and 2. This
change in diversity was partitioned using Equation 4. We used histograms
to compare the prevalence of rarity shifts relative to other sources of
diversity change.
To determine how the strength of rarity shifts changes with other
mechanisms, we used a generalized additive model (GAM). This was done
using the mgcv package (Wood and Wood 2015) where the strength of rarity
shifts is modelled as a smoothed non-linear function of selection and
immigration, with study and plot treated as random effects. Increasing
effective degrees of freedom (edf) indicate greater model complexity,
where edf = 1 indicates a straight line. The model was fit using
Restricted Maximum Likelihood (REML). Rarity shifts had an extremely
long tail and so to improve the interpretability of Figure 4,
~3% of the data with the lowest rarity shifts were
removed from the analysis of Shannon entropy. We also removed one
unusually large estimate of selection.