Statistical Analysis
Resident biodiversity was calculated using the effective number of
species (Jost 2006)\(\exp\left(-\sum_{i=1}^{5}{p_{i}\text{\ ln\ }p_{i}}\right)\) wherepi is the proportion of theith species. Diversity was then scaled to the
proportion of maximum diversity possible in this experiment (i.e., 5
species). Previous work in a similar laboratory system has shown that
disturbance has quadratic effects on diversity (Buckling et al.2000). We therefore tested the effect of pulse frequency (quantified as,
e.g., 1/16 for every 16 days), its quadratic form,invasion (0 for uninvaded and 1 for invaded), and the interactions
between invasion and pulse frequency on resident diversity using a
logistic linear regression model:
Equation 1
\(logit(\text{diversity})\ \sim\ (pulse\ +\ \text{pulse}^{2})\ *\ invasion\).
We performed stepwise model selection on this model. We also performed
the same analysis using linear regressions on Gini-Simpson index
(Simpson 1949) for diversity (see Supplemental Information, Fig. S1).
Additionally, generalised linear models were used to test treatment
effects on total resident density and total density (resident +
invader).
To test the effect of treatments on the density of each of the five
resident species, we used a generalised linear latent variable model
with the R package ‘gllvm’ (Niku et al. 2019), as this
multivariate approach allows us to separate treatment effects from the
latent interactions among the resident species (i.e., it models the
species interactions by adding additional parameters as well as
treatment effects).
To quantify invader success, we calculated the proportion of the invader
in the community (invader CFU / (invader CFU + resident CFU)) at day 16.
We then used a binomial regression to test how this responds to pulse
frequency.
Equation 2
\(logit(invader\ proportion)\ \sim\ (pulse\ +\ \text{pulse}^{2})\).
Additionally, we used a negative binomial regression to test the effects
of pulse frequency on the final density of the invader (Ripley et
al. 2013).
All statistical tests were carried out in R version 3.4.0 (R Core Team).