## 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.

This is a preview of subscription content, access via your institution.

## Data Availability

No unpublished data were used in this study.

## Code availability

Code and simulated data used to generate all figures and analysis is freely available at https://doi.org/10.5281/zenodo.3897393.

## References

Bartlett MS (1960) Stochastic population models in ecology and epidemiology. Methuen and Wiley, London

Beaulieu C, Chen J, Sarmiento JL (2012) Change-point analysis as a tool to detect abrupt climate variations. Philos Trans R Soc A Math Phys Eng Sci 370(1962):1228–1249

Beisner BE, Haydon DT, Cuddington K (2003) Alternative stable states in ecology. Front Ecol Environ 1(7):376–382

Beygelzimer A, Kakadet S, Langford J, Arya S, Mount D, Li S (2019) FNN: Fast Nearest Neighbor Search Algorithms and Applications. https://CRAN.R-project.org/package=FNN, r package version 1.1.3

Biggs R, Carpenter SR, Brock WA (2009) Turning back from the brink: detecting an impending regime shift in time to avert it. Proc Natl Acad Sci 106(3):826–831

Boettiger C (2018) From noise to knowledge: how randomness generates novel phenomena and reveals information. Ecol Lett. http://doi.wiley.com/10.1111/ele.13085

Chen P, Liu R, Li Y, Chen L (2016) Detecting critical state before phase transition of complex biological systems by hidden markov model. Bioinformatics 32(14):2143–2150

Côté SD, Rooney TP, Tremblay JP, Dussault C, Waller DM (2004) Ecological impacts of deer overabundance. Annu Rev Ecol Evol Syst 35:113–147

Dakos V, Scheffer M, van Nes EH, Brovkin V, Petoukhov V, Held H (2008) Slowing down as an early warning signal for abrupt climate change. Proc Natl Acad Sci 105(38):14308–14312

Dulvy NK, Freckleton RP, Polunin NV (2004) Coral reef cascades and the indirect effects of predator removal by exploitation. Ecol Lett 7(5):410–416

Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB (2013) Bayesian data analysis. CRC Press

Gelman A, Vehtari A, Simpson D, Margossian CC, Carpenter B, Yao Y, Kennedy L, Gabry J, Bürkner PC, Modrák M (2020) Bayesian workflow. arXiv preprint arXiv:201101808

Gleeson SK, Tilman D (1990) Allocation and the transient dynamics of succession on poor soils. Ecology 71(3):1144–1155

Hastings A (2001) Transient dynamics and persistence of ecological systems. Ecol Lett 4(3):215–220. https://doi.org/10.1046/j.1461-0248.2001.00220.x

Hastings A, Higgins K (1994) Persistence of transients in spatially structured ecological models. Science 263(5150):1133–1136. https://doi.org/10.1126/science.263.5150.1133, http://www.ncbi.nlm.nih.gov/pubmed/17831627

Hastings A, Abbott KC, Cuddington K, Francis T, Gellner G, Lai YC, Morozov A, Petrovskii S, Scranton K, Zeeman ML (2018) Transient phenomena in ecology. Science 361(6406). https://doi.org/10.1126/science.aat6412

Itti L, Baldi PF (2006) Bayesian surprise attracts human attention. In: Advances in neural information processing systems, pp 547–554

James NA, Matteson DS (2014) ecp: An R package for nonparametric multiple change point analysis of multivariate data. J Stat Softw 62(7):1–25. http://www.jstatsoft.org/v62/i07/

Kullback S, Leibler RA (1951) On information and sufficiency. Ann Math Stat 22(1):79–86

Ling S, Scheibling R, Rassweiler A, Johnson C, Shears N, Connell S, Salomon A, Norderhaug K, Pérez-Matus A, Hernández J et al (2015) Global regime shift dynamics of catastrophic sea urchin overgrazing. Philos Trans R Soc B 370(1659):20130269

Ludwig D, Jones DD, Holling CS et al (1978) Qualitative analysis of insect outbreak systems: the spruce budworm and forest. J Anim Ecol 47(1):315–332

May RM (1977) Thresholds and breakpoints in ecosystems with a multiplicity of stable states. Nature 269(5628):471

Mitchell TJ, Beauchamp JJ (1988) Bayesian variable selection in linear regression. J Am Stat Assoc 83(404):1023–1032

Nolting BC, Abbott KC (2016) Balls, cups, and quasi-potentials: quantifying stability in stochastic systems. Ecology

Plummer M, Best N, Cowles K, Vines K (2006) CODA: convergence diagnosis and output analysis for MCMC.

*R news*,*6*(1), 7-11R Core Team (2019) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/

Rietkerk M, van de Koppel J (1997) Alternate stable states and threshold effects in semi-arid grazing systems. Oikos 69–76

Scheffer M, Bascompte J, Brock WA, Brovkin V, Carpenter SR, Dakos V, Held H, Van Nes EH, Rietkerk M, Sugihara G (2009) Early-warning signals for critical transitions. Nature 461(7260):53

de Valpine P, Turek D, Paciorek CJ, Lang DT, Bodik R (2017) Programming with models: Writing statistical algorithms for general model structures with NIMBLE. J Comput Graph Stat 26:403–417 arXiv:1505.05093v1

Van Geest G, Coops H, Scheffer M, Van Nes E (2007) Long transients near the ghost of a stable state in eutrophic shallow lakes with fluctuating water levels. Ecosystems 10(1):37–47

Visser I, Speekenbrink M (2010) depmixS4: An R package for hidden markov models. J Stat Softw 36(7):1–21. http://www.jstatsoft.org/v36/i07/

Wissel C (1984) A universal law of the characteristic return time near thresholds. Oecologia 65(1):101–107

## Acknowledgements

The authors would like to thank the organizers of the Transients in Biological Systems workshop in May 2019 in NIMBioS at the University of Tennessee, Knoxville for the space to develop this article.

## Funding

KZ acknowledges support from the US National Science Foundation, DEB1926438 and the University of California, Santa Cruz, Committee on Research, Faculty Research Grant.

## Author information

### Affiliations

### Corresponding author

## Ethics declarations

### Ethics approval

Ethics approval was not required for this study.

### Consent to participate and publication

Consent to participate and consent for publication were not required for this study.

### Conflict of interest

The authors declare that they have no conflict of interest.

## Supplementary Information

Below is the link to the electronic supplementary material.

## Appendix

### 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

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.

## Rights and permissions

## About this article

### Cite this article

Reimer, J.R., Arroyo-Esquivel, J., Jiang, J. *et al.* Noise can create or erase long transient dynamics.
*Theor Ecol* (2021). https://doi.org/10.1007/s12080-021-00518-6

Received:

Accepted:

Published:

### Keywords

- Inference
- Model fitting
- Stochasticity
- Transient dynamics
- Nonparametric models