Measuring stability in ecological systems without static equilibria

Ecological stability refers to a range of concepts used to quantify how species and environments change over time and in response to disturbances. Most empirically tractable ecological stability metrics assume that systems have simple dynamics and static equilibria. However, ecological systems are typically complex and often lack static equilibria (e.g., predator–prey oscillations, transient dynamics, chaos). Failing to account for these factors can lead to biased estimates of stability, in particular, by conflating effects of observation error, process noise, and underlying deterministic dynamics. To distinguish among these processes, we combine three existing approaches: state space models; delay embedding methods; and particle filtering. Jointly, these provide something akin to a deterministically “detrended” version of the coefficient of variation, separately tracking variability due to deterministic dynamics versus stochastic perturbations. Moreover, these variability estimates can be used to forecast dynamics, classify underlying sources of stochastic dynamics, and estimate the “exit time” before a state change takes place (e.g., local extinction events). Importantly, the time-delay embedding methods that we employ make very few assumptions about the functions governing deterministic dynamics, which facilitates applications in systems with limited data and a priori biological knowledge. To demonstrate how complex dynamics without static equilibria can bias ecological stability estimates, we analyze simulated time series of abundance dynamics in a system with time-varying carrying capacity and empirically observed abundance dynamics of the green algae Chlamydomonas terricola grown in a diverse microcosm mixture under variable temperature conditions. We show that stability estimates based on raw observations greatly overestimate temporal variability and fail to accurately forecast time to extinction. In contrast, joint application of state space modeling, delay embedding, and particle filters were able to: (1) correctly quantify the contributions of deterministic versus stochastic variability; (2) successfully estimate “true” abundance dynamics; and (3) correctly forecast time to extinction. Our results therefore demonstrate the importance of accounting for effects of complex, nonstatic dynamics in studies of ecological stability and provide an empirically tractable and flexible toolkit for conducting these measurements