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).