Many ecological systems exhibit critical transitions. Bifurcations, small smooth changes in underlying drivers that induce sudden shifts in system behavior, are a mechanism for critical transitions that are of considerable interest. A bifurcation of a system may be anticipated because prior to reaching the dynamical threshold, the system gradually loses stability (`critical slowing down'). Signatures of critical slowing down may be detectable through summary statistics, but how environmental and demographic stochasticity influence statistical patterns prior to a transition is unclear. To determine how different noise processes drive trends in summary statistics as a critical transition is approached, we considered a range of stochastic models that exhibit transcritical, saddle-node and pitchfork bifurcations. Models were written in normal form to investigate generality of trends across models. Noise was assumed to be either environmental or demographic in nature. We derived expressions for the stationary variance, autocorrelation and power spectrum for all cases.
Trends in summary statistics signaling the approach of each bifurcation depend on the interplay between the exponential return rate and noise type. For example, models with mechanistic environmental noise may predict an increase in, or a decline in variance as the critical threshold is approached, whereas models with additive noise predict an increase in variance. To investigate the robustness of the theory, we examined epidemiological models undergoing a supercritical transcritical bifurcation. Predictions for summary statistics trends depend on how the noise is modeled. The ability to classify trends in summary statistics for a broad class of models enhances our understanding of how critical slowing down manifests in ecological systems approaching a transition.