# Long-term cyclic persistence in an experimental predator–prey system

## Abstract

Predator–prey cycles rank among the most fundamental concepts in ecology, are predicted by the simplest ecological models and enable, theoretically, the indefinite persistence of predator and prey1,2,3,4. However, it remains an open question for how long cyclic dynamics can be self-sustained in real communities. Field observations have been restricted to a few cycle periods5,6,7,8 and experimental studies indicate that oscillations may be short-lived without external stabilizing factors9,10,11,12,13,14,15,16,17,18,19. Here we performed microcosm experiments with a planktonic predator–prey system and repeatedly observed oscillatory time series of unprecedented length that persisted for up to around 50 cycles or approximately 300 predator generations. The dominant type of dynamics was characterized by regular, coherent oscillations with a nearly constant predator–prey phase difference. Despite constant experimental conditions, we also observed shorter episodes of irregular, non-coherent oscillations without any significant phase relationship. However, the predator–prey system showed a strong tendency to return to the dominant dynamical regime with a defined phase relationship. A mathematical model suggests that stochasticity is probably responsible for the reversible shift from coherent to non-coherent oscillations, a notion that was supported by experiments with external forcing by pulsed nutrient supply. Our findings empirically demonstrate the potential for infinite persistence of predator and prey populations in a cyclic dynamic regime that shows resilience in the presence of stochastic events.

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## Data availability

Experimental data are publicly available on Figshare (https://doi.org/10.6084/m9.figshare.10045976.v1).

## Code availability

Data analysis and the model were implemented in the language Julia v.1.141. The source code for the wavelet analysis is publicly available on GitHub (https://github.com/berndblasius/WaveletAnalysis).

## References

1. 1.

Volterra, V. Fluctuations in the abundance of a species considered mathematically. Nature 118, 558–560 (1926).

2. 2.

May, R. M. Limit cycles in predator–prey communities. Science 177, 900–902 (1972).

3. 3.

Kendall, B. E. et al. Why do populations cycle? A synthesis of statistical and mechanistic modeling approaches. Ecology 80, 1789–1805 (1999).

4. 4.

Berryman, A. A. (ed.) Population Cycles: The Case for Trophic Interactions (Oxford Univ. Press, 2002).

5. 5.

Elton, C. & Nicholson, M. The ten-year cycle in numbers of the lynx in Canada. J. Anim. Ecol. 11, 215–244 (1942).

6. 6.

Blasius, B., Huppert, A. & Stone, L. Complex dynamics and phase synchronization in spatially extended ecological systems. Nature 399, 354–359 (1999).

7. 7.

Gilg, O., Hanski, I. & Sittler, B. Cyclic dynamics in a simple vertebrate predator–prey community. Science 302, 866–868 (2003).

8. 8.

Ryabov, A. B., de Roos, A. M., Meyer, B., Kawaguchi, S. & Blasius, B. Competition-induced starvation drives large-scale population cycles in Antarctic krill. Nat. Ecol. Evol. 1, 0177 (2017).

9. 9.

Gause, G. F., Smaragdova, N. P. & Witt, A. A. Further studies of interaction between predators and prey. J. Anim. Ecol. 5, 1–18 (1936).

10. 10.

Utida, S. Cyclic fluctuations of population density intrinsic to the host–parasite system. Ecology 38, 442–449 (1957).

11. 11.

Huffaker, C. B. Experimental studies on predation: dispersion factors and predator–prey oscillations. Hilgardia 27, 343–383 (1958).

12. 12.

Luckinbill, L. S. Coexistence in laboratory populations of Paramecium aurelia and its predator Didinium nasutum. Ecology 54, 1320–1327 (1973).

13. 13.

Krebs, C. J. et al. Impact of food and predation on the snowshoe hare cycle. Science 269, 1112–1115 (1995).

14. 14.

Hudson, P. J., Dobson, A. P. & Newborn, D. Prevention of population cycles by parasite removal. Science 282, 2256–2258 (1998).

15. 15.

McCauley, E., Nisbet, R. M., Murdoch, W. W., de Roos, A. M. & Gurney, W. S. C. Large-amplitude cycles of Daphnia and its algal prey in enriched environments. Nature 402, 653–656 (1999).

16. 16.

Fussmann, G. F., Ellner, S. P., Shertzer, K. W. & Hairston, N. G. Jr. Crossing the Hopf bifurcation in a live predator–prey system. Science 290, 1358–1360 (2000).

17. 17.

Ellner, S. P. et al. Habitat structure and population persistence in an experimental community. Nature 412, 538–543 (2001).

18. 18.

Becks, L., Hilker, F. M., Malchow, H., Jürgens, K. & Arndt, H. Experimental demonstration of chaos in a microbial food web. Nature 435, 1226–1229 (2005).

19. 19.

Massie, T. M., Blasius, B., Weithoff, G., Gaedke, U. & Fussmann, G. F. Cycles, phase synchronization, and entrainment in single-species phytoplankton populations. Proc. Natl Acad. Sci. USA 107, 4236–4241 (2010).

20. 20.

Radchuk, V., Ims, R. A. & Andreassen, H. P. From individuals to population cycles: the role of extrinsic and intrinsic factors in rodent populations. Ecology 97, 720–732 (2016).

21. 21.

Benincà, E., Jöhnk, K. D., Heerkloss, R. & Huisman, J. Coupled predator–prey oscillations in a chaotic food web. Ecol. Lett. 12, 1367–1378 (2009).

22. 22.

Holyoak, M. & Lawler, S. P. Persistence of an extinction-prone predator–prey interaction through metapopulation dynamics. Ecology 77, 1867–1879 (1996).

23. 23.

Stenseth, N. C. et al. The effect of climatic forcing on population synchrony and genetic structuring of the Canadian lynx. Proc. Natl Acad. Sci. USA 101, 6056–6061 (2004).

24. 24.

Torrence, C. & Compo, G. P. A practical guide to wavelet analysis. Bull. Am. Meteorol. Soc. 79, 61–78 (1998).

25. 25.

Cazelles, B. et al. Wavelet analysis of ecological time series. Oecologia 156, 287–304 (2008).

26. 26.

Keitt, T. H. Coherent ecological dynamics induced by large-scale disturbance. Nature 454, 331–334 (2008).

27. 27.

Paraskevopoulou, S., Tiedemann, R. & Weithoff, G. Differential response to heat stress among evolutionary lineages of an aquatic invertebrate species complex. Biol. Lett. 14, 20180498 (2018).

28. 28.

Guillard, R. R. L. & Lorenzen, C. J. Yellow-green algae with chlorophyllide c. J. Phycol. 8, 10–14 (1972).

29. 29.

Torrence, C. & Webster, P. J. Interdecadal changes in the ENSO–monsoon system. J. Clim. 12, 2679–2690 (1999).

30. 30.

Maraun, D. & Kurths, J. Cross wavelet analysis: significance testing and pitfalls. Nonlin. Processes Geophys. 11, 505–514 (2004).

31. 31.

Si, B. C. Spatial scaling analyses of soil physical properties: a review of spectral and wavelet methods. Vadose Zone J. 7, 547–562 (2008).

32. 32.

Bandrivskyy, A., Bernjak, A., McClintock, P. & Stefanovska, A. Wavelet phase coherence analysis: application to skin temperature and blood flow. Cardiovasc. Eng. 4, 89–93 (2004).

33. 33.

Batschelet, E. Circular Statistics in Biology (Academic, 1981).

34. 34.

McCauley, E., Nisbet, R. M., de Roos, A. M., Murdoch, W. W. & Gurney, W. S. C. Structured population models of herbivorous zooplankton. Ecol. Monogr. 66, 479–501 (1996).

35. 35.

de Roos, A. M. & Persson, L. Population and Community Ecology of Ontogenetic Development 59 (Princeton Univ. Press, 2013).

36. 36.

Massie, T. M., Weithoff, G., Kuckländer, N., Gaedke, U. & Blasius, B. Enhanced Moran effect by spatial variation in environmental autocorrelation. Nat. Commun. 6, 5993 (2015).

37. 37.

Mourelatos, S., Pourriot, R. & Rougier, C. Taux de filtration du rotifère Brachionus calyciflorus: comparaison des méthodes de mesure; influence de l’âge. Vie Milieu 40, 39–43 (1990).

38. 38.

Halbach, U. Einfluß der Temperatur auf die Populationsdynamik des planktischen Rädertieres Brachionus calyciflorus Pallas. Oecologia 4, 176–207 (1970).

39. 39.

Fussmann, G. E., Weithoff, G. & Yoshida, T. A direct, experimental test of resource vs. consumer dependence. Ecology 86, 2924–2930 (2005).

40. 40.

Seifert, L. I. et al. Heated relations: temperature-mediated shifts in consumption across trophic levels. PLoS ONE 9, e95046 (2014).

41. 41.

Bezanson, J., Edelman, A., Karpinski, S. & Shah, V. B. Julia: a fresh approach to numerical computing. SIAM Rev. 59, 65–98 (2017).

## Acknowledgements

This work was supported by German VW-Stiftung. We thank A. Bandrivskyy, T. Massie, E. Denzin and all other technical staff of the Department of Ecology & Ecosystem Modelling; and R. Ceulemans, C. Feenders, A. Jamieson-Lane, A. Ryabov and E. van Velzen for comments on the manuscript.

## Author information

B.B., G.F.F., G.W. and U.G. designed the research, L.R., G.F.F. and G.W. performed the experiments, B.B. and L.R. performed the numerical simulations and data analysis, B.B. and G.F.F. wrote the paper. All authors discussed the results and commented on the manuscript.

Correspondence to Bernd Blasius.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Alan Hastings and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Detailed dynamics and phase relationship in experiment C1.

a, Time series of the predator, B. calyciflorus (red), and its prey, M. minutum (green), (normalized to the range of 0 and 1). b, Phase portrait of the bandpass-filtered predator–prey time series between days 90 and 130. The arrow indicates the direction of oscillation. c, Local WPS for the prey, ranging from 0 (blue) to maximum (red), and the instantaneous oscillation period $$\tilde{s}(t)$$ (black line) of the highest WCO within the prefixed period length band (horizontal black lines). The cone of influence is also indicated (black). d, Global WPS P(s) for the prey (green). Horizontal lines as in c. e, f, As in c, d, but for the predator. g, WCS between predator and prey, colour-coding and other lines as in c. h, Amplitude of the global WCS for the predator and the prey. i, WCO between predator and prey, ranging from 0 (blue) to 1 (red). Significant areas (WCO > 0.83) are enclosed by thin solid lines; the thick black line segments show the instantaneous oscillation period $$\tilde{s}(t)$$ in these segments. j, Histogram of period lengths within coherent oscillation regions as detected in i. k, Phase difference θ(t) to prey for the predator (red), the predator egg ratio (blue), the number of eggs (black) and dead animals (yellow). Significant relations between respective signals are marked as a solid line. l, Circular distribution of the mutual phase difference to prey (indicated in green) for the predator (red), the predator egg ratio (blue), the number of eggs (black) and dead animals (yellow).

### Extended Data Fig. 2 Dynamics and phase relationships in three further experimental time series in a constant environment.

ad, Analysis of experiment C2. a, Predator–prey time series. b, Phase signature. c, WCO. d, Global WPS. eh, As in ad, for experiment C3. il, As in ad, for experiment C4. Details and colours as in Fig. 1.

### Extended Data Fig. 3 Dynamics and phase relationships in measured time series with a different algal species (C. vulgaris).

ad, Analysis of experiment C6. a, Predator–prey time series. b, Phase signature. c, WCO. d, Global WPS. eh, As in ad, for experiment C7. Details and colours as in Fig. 1.

### Extended Data Fig. 4 Dynamics and phase relationship in surrogate data and length distribution of coherent oscillation regimes.

af, As in Fig. 1, but for surrogate data, obtained by time scrambling—that is, randomly drawing 360 values with replacement from the predator and prey time series in experiment C1—yielding a time series without temporal correlation (white noise) but with the same abundance distribution as the experimental data. The figure shows a typically outcome. Occasionally, for short bursts the randomized predator and prey time series oscillate with a common frequency (a), yielding areas of high coherence (c; WCO > 0.83; spurious coherent oscillation regimes) with random phase differences (e). Whereas most time intervals with coherent predator–prey cycles in the surrogate data have a short duration, in this example (by chance) there is one regime (days 187–209) with a duration of 23 days. g, h, Box plots of oscillation regime length. Boxes show the interquartile range, orange lines indicate the median, whiskers range from the lowest to the highest data point within 1.5× the interquartile range and markers indicate outliers. g, Length distribution of coherent oscillation regimes for the experimental data under free-running conditions (C1–C5) and for a large ensemble of surrogate data. This shows that time intervals with coherent oscillations are significantly longer in the experiments (n = 20; median, 16 days; mean ± s.d., 29 ±28 days) than in the surrogate data (n = 13,719; median, 8 days; mean ± s.d., 9 ± 7 days). The two outliers in the left box plot correspond to the two long coherent oscillation regimes of 111 days and 85 days in C1, which (even though they are outliers in the length distribution) effectively dominate the dynamic behaviour in C1. Consequently, we calculate the typical duration $$\tilde{T}$$ of coherent oscillations as the expected regime duration for a randomly chosen time instance taken from a coherent oscillation regime (Methods) ($$\tilde{T}$$ = 58 days in C1–C5; $$\tilde{T}$$ = 15 days in the surrogate data). h, As in g, but for durations of non-coherent oscillation regimes, showing that time intervals of non-coherent oscillations are significantly shorter in the experiments (n = 15; median, 17 days; mean ± s.d., 16 ± 11 days; typical duration $$\tilde{T}$$ = 23 days) than in the surrogate data (n = 12,719; median, 43 days, mean ± s.d., 60 ± 11 days; typical duration $$\tilde{T}$$ = 113 days).

### Extended Data Fig. 5 Detailed dynamics and phase relationships in a modelled time series.

al, As in Extended Data Fig. 1, but for a simulated time series of the stochastic time-delayed model in a constant environment. The simulation time was 300 days. A typical model outcome is shown. In this example, we obtained five coherent oscillation regimes (indicated as thick lines in i, k). Similar to the experimental systems (C1–C7), these regimes are interspersed by short non-coherent oscillation regimes without a significant phase relationship.

### Extended Data Fig. 6 Dynamics and phase relationships in a model without juvenile maturation delay.

a, Simulated time series of algae (green), rotifers (red), eggs (black), egg ratio (blue) and dead animals (yellow) in a model without juvenile maturation delay, τ = 0, under constant conditions with all stochasticity removed (mean centred, relative units). b, Phase signature obtained from stochastic simulations (free-running conditions, τ = 0). c, d, As in a, b, but for an externally driven system (external nutrient concentration shown in magenta). The figure shows that in a scenario without juvenile maturation delay (τ = 0), the phase of the egg ratio coincides with that of the algae, both in a free-running and in an externally driven model. This is in contrast to the experimental observations (Fig. 3, rows 1 and 3) and simulations with non-vanishing juvenile maturation delay (τ = 1.8 days) (Fig. 3 rows 2 and 4, Extended Data Fig. 9d, f), in which the egg ratio is significantly preceding the algae signal. Thus, the phase signature helps to disentangle life-history mechanisms, as—by comparison of observed and simulated phase signatures—a delay of juvenile maturation is essential to attain the experimentally observed phase relations.

### Extended Data Fig. 7 Dynamics and phase relationships in externally driven systems.

ad, Analysis of experiment C8. a, Predator–prey time series. b, Phase signature. c, WCO. d, Global WPS. eh, As in ad, for experiment C9. il, As in ad, for simulation results of an externally driven system. Details and colours as in Fig. 1. In a, e, i, the blue dotted line shows the input concentration in the external medium in normalized units. In b, f, j, the phase difference distribution between the external medium and the prey is shown in magenta.

### Extended Data Fig. 8 Dynamics and phase relationships in experiments and simulations with changed dilution rate and press perturbations.

ad, Analysis of experiment C5, which has an increased dilution rate (δ = 0.66 per day). a, Predator–prey time series. b, Phase signature. c, WCO. d, Global WPS. eh, As in ad, for experiment C10, which undergoes press perturbations (vertical black lines) at day 84 (dilution rate was increased from δ = 0.55 per day to δ = 1.2 per day) and day 123 (dilution rate increased further to δ = 1.35 per day). il, As in ad, with simulation results of a system undergoing a press perturbation in which the dilution rate was increased to δ = 0.75 per day at day 100 (vertical black line in i, k). Under conditions of the increased dilution rate, the oscillations are suppressed (Extended Data Fig. 9a). Details and colours as in Fig. 1.

### Extended Data Fig. 9 Bifurcation diagrams.

a, Simulated values of predator (red) and prey (green) for different values of the dilution rate. Thick solid lines show maximal and minimal values obtained in the unforced model without stochasticity. Shaded areas show range of values in the stochastic model. Sustained predator–prey oscillations appear for 0.47 < δ < 0.71 per day. For large values of the dilution rate, δ > 0.87 per day, the predator goes extinct. b, As in a, but for the externally forced system. The model shows oscillations for all parameters without predator extinction δ < 0.8 per day, it exhibits a bistability regime (small- versus large-amplitude oscillations) for 0.23 < δ < 0.31 per day and a period-2 oscillation regime for 0.42 < δ < 0.57 per day. c, d, Bifurcation diagrams of the unforced model as a function of the juvenile maturation delay τ. c, As in a, but for the different bifurcation parameter. d, Phase difference to the algae for the predator (red), eggs (black) and the egg ratio (blue). The shaded areas indicate the range of phase differences (±1 s.d.) from the experiments in a constant environment (C1–C5). For large values of τ, the simulated phase signature aligns with the experimental data, whereas for small values of τ, the simulated phase difference of the egg ratio approaches zero. e, f, As in c, d, but for the externally forced model. e, For small delay times, 0.4 < τ < 1 day, the model exhibits a chaotic regime. f, Additionally the phase difference between algae and external nutrients is shown (magenta). Vertical solid lines show the actually used parameter values δ = 0.55 per day and τ = 1.8 days.

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Blasius, B., Rudolf, L., Weithoff, G. et al. Long-term cyclic persistence in an experimental predator–prey system. Nature 577, 226–230 (2020) doi:10.1038/s41586-019-1857-0

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