Measuring Ecological Stability in Systems Without Static Equilibria

Background/Question/Methods

Most empirically tractable ecological stability metrics assume that systems have simple dynamics and static equilibria (e.g. logistic growth with a fixed carrying capacity). However, ecological systems are typically complex, and often lack static equilibria (e.g. predator-prey oscillations, or transient responses to environmental changes). 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; nonparametric delay embedding methods; and particle filtering. Jointly, these provide something akin to a “detrended” version of the classic coefficient of variation, separately tracking variability in system state due to deterministic dynamics vs. stochastic perturbations. Moreover, these methods make very few assumptions about the mathematical functions governing deterministic dynamics, and can be applied in systems with limited data and a priori biological knowledge.

Results/Conclusions

To demonstrate how complex dynamics can bias classic ecological stability estimates, we analyze simulated time series of population dynamics in a system with non-static carrying capacity, and empirically observed abundance dynamics of the green algae Chlamydomonas terricola grown under fluctuating temperature conditions. We find that classic ecological stability estimates based on raw observations greatly overestimate temporal variability, and failed to accurately forecast stability metrics such as time to extinction. In contrast, the joint application of state space modelling, delay embedding, and particle filters were able to correctly quantify the contributions of deterministic vs. stochastic variability, successfully forecast “true” abundance dynamics (i.e. after controlling for effects of observation error), and correctly estimated stability metrics related to resilience, resistance, and time to extinction. Although accuracy dropped as a function of higher observation error, estimates remained unbiased for systems with up to 30% observation error. Our results therefore demonstrate the importance of accounting for effects of complex, non-static dynamics in studies of ecological stability, and provide an empirically tractable and flexible toolkit for conducting these measurements, available in the accompanying pttstability (Particle-Takens Stability) package for the R programming language.