Noise can create or erase long transient dynamics

Abstract

Recent theoretical work has highlighted several mechanisms giving rise to so-called long transient dynamics. These long transients tantalizingly appear to replicate dynamics seen in real systems—with one critical difference: ecological data is noisy, a reality theoretical work often ignores. In general, stochasticity is known to have important consequences: it can qualitatively alter model dynamics as well as impact our ability to infer underlying processes through statistical analysis. To explore the effect of stochasticity on qualitative model behavior and the implications for our ability to infer underlying mechanisms, we generated time series from a simple model of long transient behavior with multiplicative noise. We then examined whether noise qualitatively changes the expected dynamics of the system and the insights that four different statistical methods could provide about the underlying dynamics. We found that the expected duration of the long transient was significantly reduced in the stochastic model compared to the deterministic model. These transient dynamics arise for parameterizations very near to a bifurcation point in the deterministic model, and we also found that as we varied parameterizations to include two alternative stable states, stochasticity caused the population to jump from one basin of attraction to another, resulting in time series that suggest long transient dynamics. Despite challenges estimating the underlying model parameters, we illustrate that statistical inference on a single realization may still provide insight into the presence of a ghost attractor. Further, we highlight that inference improves, across parameterizations, for an increasing number of realizations of the process.

Appendix

Effective sample size and integrated mean squared error

We computed median ESS using the effectiveSize() function from the coda package for the R statistical programming environment (Plummer 2006). We first computed the ESS of $\frac{\mathrm {d}\mu (t)}{\mathrm {d}t}$ over a grid of 100 equally-spaced values between $x(t) = 0.2$ and 1.8. We then took the median value across the grid.

Let $\frac{\mathrm {d}\hat{\mu }_\mathrm {P}(t)}{\mathrm {d}t}$ represent the functional $\frac{\mathrm {d}\mu (t)}{\mathrm {d}t}$ in Eq. (2) evaluated at the values $\hat{\theta }= (\hat{r}, \hat{K}, \hat{a}, \hat{h}, \text {and } \hat{Q})$. Analogously, let $\frac{\mathrm {d}\hat{\mu }_\mathrm {NP}(t)}{\mathrm {d}t}$ represent the functional in the nonparametric model given by Eq. (8) evaluated at $\hat{\beta }$. We defined the IMSE for either the parametric (P) or nonparametric (NP) approach as

$$\begin{aligned} \mathrm {IMSE}_\mathrm {M} = \int _{0.2}^{1.8} \mathrm {E}_{\hat{\theta }} \left( \frac{\mathrm {d}\hat{\mu }_\mathrm {M}(t)}{\mathrm {d}t} - \frac{\mathrm {d}\mu (t)}{\mathrm {d}t}\right) ^2 \mathrm {d}x(t), \quad \mathrm {M} \in \{\mathrm {P}, \mathrm {NP}\}. \end{aligned}$$

(9)

The expectation was approximated using a Monte Carlo approximation via samples from the joint posterior distribution of $\hat{\theta }$, and the integral was approximated numerically using a simple Riemannian quadrature.