The base parameter values were set as follows: species pool (i.e.
metacommunity) size (J M) = 1,000,000 individuals;
fundamental biodiversity number (θ) = 100; local community size
(J ) = 1,000; mortality rate (d ) = 0.1; immigration rate
(m ) = 0.1. Based on the base parameter values of this model, the
parameter-dependency patterns were studied by changing the four targeted
parameters (θ, J , d , and m ). For the fundamental
biodiversity number (θ), 100 numbers were randomly selected between 1
and 1,000, without allowing duplicates. For the local community size
(J ), 25 numbers were randomly selected for each order of
magnitude (102–105); thus, 100
numbers were selected in total. For both mortality rate (d ) and
immigration rate (m ), 100 numbers were determined between
0.01–1.00, in 0.01 increments.
At the beginning of each simulation, local community members were chosen
depending on the set number of local community size, and 1,100 time
steps, including the initial state, were conducted for each simulation
(Fig. 1b). After the simulation, the information of local community
dynamics between time step 1 (T1) and time step 99 (T99) were excluded
from the following analysis for testing parameter dependency because the
randomness of the initial sampling could affect the community dynamics
at the beginning phase. The dynamics from T100 to T1100 were targeted
for the main analysis, and thus, a 1001 time-series dataset was used for
each simulation (Fig. 1b). The Bray–Curtis dissimilarity index (Odum,
1950) and Sørensen dissimilarity index (Sørensen, 1948) were calculated
for each simulation.
The preliminary analysis revealed that the temporal distance-decay curve
rarely reached one (i.e. completely dissimilar); thus, upper limits
exist at less than one in temporal beta diversity under neutral
dynamics. Therefore, to describe the form of the simulated temporal
distance-decay patterns of both the Bray–Curtis and Sørensen
dissimilarity indices, a negative exponential curve was fitted for each
dataset (Fig. 1c). The equation for the negative exponential function is
as follows:
\(TBD=-\delta\times e^{-\varepsilon\ \times\ td}+\zeta\),
where TBD is the temporal beta diversity value, e is Napier’s
constant, td is the temporal distance between a pair of communities, and
δ, ε, and ζ are the parameters of the negative exponential function
(Table 1). The changes in the curve shape with parameter changes are
summarised in Fig. 2. The parameter ζ determines the position of the
upper limit of the curve; thus, a larger ζ indicates higher upper limits
in the temporal distance-decay of temporal beta diversity (Fig. 2c).
Parameter δ determines the relative position of the intercept against
parameter ζ (i.e. the upper limit). The ζ – δ value indicates the
intercept of the curve; thus, a larger δ indicates a lower position of
the intercept (Fig. 2a). Parameter ε determines the curvature of the
curve; thus, a larger ε indicates a higher curvature of the curve (Fig.
2b). All estimated parameter values without errors were used in the
analysis to check parameter dependency in the present study. Although
Martín-Devasa, Martínez‐Santalla, Gómez‐Rodríguez, Crujeiras, &
Baselga, (2022) recently developed a method to compare the fit of three
different functions on distance-decay curves, judging the best-fitted
function for a distance-decay curve was the main use of the method and
ecologically interpreting the changes in parameter values was difficult
based on their methods. Therefore, the three-parameter negative
exponential function was used in this study owing to the differences in
purpose.