Small-amplitude cycles emerge from stage-structured interactions in Daphnia–algal systems


A long-standing issue in ecology is reconciling the apparent stability of many populations with robust predictions of large-amplitude population cycles from general theory on consumer–resource interactions1. Even when consumers are decoupled from dynamic resources, large-amplitude cycles can theoretically emerge from delayed feedback processes found in many consumers2,3. Here we show that resource-dependent mortality and a dynamic developmental delay in consumers produces a new type of small-amplitude cycle that coexists with large-amplitude fluctuations in coupled consumer–resource systems. A distinctive characteristic of the small-amplitude cycles is slow juvenile development for consumers, leading to a developmental delay that is longer than the cycle period. By contrast, the period exceeds the delay in large-amplitude cycles. These theoretical predictions may explain previous empirical results on coexisting attractors found in Daphnia–algal systems4,5. To test this, we used bioassay experiments that measure the growth rates of individuals in populations exhibiting each type of cycle. The results were consistent with predictions. Together, the new theory and experiments establish that two very general features of consumers—a resource-dependent juvenile stage duration and resource-dependent mortality—combine to produce small-amplitude resource–consumer cycles. This phenomenon may contribute to the prevalence of small-amplitude fluctuations in many other consumer–resource populations6,7.

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Figure 1: Multiple limit-cycle attractors in the structured predator–prey model.
Figure 2: Bioassay experiments for resource–consumer systems.
Figure 3: Egg density dynamics during cycles.
Figure 4: Comparison of individual growth rates in large- and small-amplitude cycles.


  1. 1

    Murdoch, W. W. Population regulation in theory and practise. Ecology 75, 271–287 (1994)

    Article  Google Scholar 

  2. 2

    Murdoch, W. W., Briggs, C. J. & Nisbet, R. M. Consumer-Resource Dynamics (Princeton Univ. Press, 2003)

    Google Scholar 

  3. 3

    de Roos, A. M. & Persson, L. Competition in size-structured populations, mechanisms inducing cohort formation and population cycles. Theor. Popul. Biol. 63, 1–16 (2003)

    Article  Google Scholar 

  4. 4

    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)

    ADS  CAS  Article  Google Scholar 

  5. 5

    Bjornstad, O. N. & Grenfell, B. T. Noisy clockwork: Time-series analysis of population fluctuations in animals. Science 293, 638–643 (2001)

    CAS  Article  Google Scholar 

  6. 6

    McCauley, E. & Murdoch, W. W. Cyclic and stable populations: Plankton as paradigm. Am. Nat. 129, 97–121 (1987)

    Article  Google Scholar 

  7. 7

    McCauley, E. & Murdoch, W. W. Predator–prey dynamics in environments rich and poor in nutrients. Nature 343, 455–457 (1990)

    ADS  Article  Google Scholar 

  8. 8

    Claessen, D., de Roos, A. M. & Persson, L. Population dynamic theory of size-dependent cannibalism. Proc. R. Soc. Lond. B 271, 333–340 (2004)

    Article  Google Scholar 

  9. 9

    Medvinsky, A. B., Tikhonova, I. A., Li, B. L. & Malchow, H. Time delay as a key factor in model plankton dynamics. C. R. Biologies 327, 277–282 (2004)

    Article  Google Scholar 

  10. 10

    Hastings, A. & Wollkind, D. Age structure in predator-prey systems. Theor. Pop. Biol. 21, 44–56 (1982)

    Article  Google Scholar 

  11. 11

    De Roos, A. M., Metz, J. A. J., Evers, E. & Leipoldt, A. A size-dependent predator-prey interaction: who pursues whom? J. Math. Biol. 28, 609–643 (1990)

    MathSciNet  Article  Google Scholar 

  12. 12

    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)

    Article  Google Scholar 

  13. 13

    Nelson, W. A., McCauley, E. & Nisbet, R. M. Stage-structured cycles generate strong fitness-equalizing mechanisms. Evol. Ecol. 21, 499–515 (2007)

    Article  Google Scholar 

  14. 14

    Kretzschmar, M., Nisbet, R. M. & McCauley, E. A predator-prey model for zooplankton grazing on competing algal populations. Theor. Pop. Biol. 44, 32–66 (1993)

    Article  Google Scholar 

  15. 15

    Grover, J. P. Assembly rules for communities of nutrient-limited plants and specialist herbivores. Am. Nat. 143, 258–282 (1994)

    Article  Google Scholar 

  16. 16

    Abrams, P. A. & Holt, R. D. The impact of consumer-resource cycles on the coexistence of competing consumers. Theor. Pop. Biol. 62, 281–295 (2002)

    Article  Google Scholar 

  17. 17

    Murdoch, W. W. & McCauley, E. Three distinct types of dynamic behaviour shown by a single planktonic system. Nature 316, 628–630 (1985)

    ADS  Article  Google Scholar 

  18. 18

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

    Article  Google Scholar 

  19. 19

    Murdoch, W. W. et al. Single-species models for many-species food webs. Nature 417, 541–543 (2002)

    ADS  CAS  Article  Google Scholar 

  20. 20

    Benton, T. G., Plaistow, S. J. & Coulson, T. N. Complex population dynamics and complex causation: devils, details and demography. Proc. R. Soc. B 273, 1173–1181 (2006)

    Article  Google Scholar 

  21. 21

    Dennis, B., Desharnais, R. A., Cushing, J. M., Henson, S. M. & Constantino, R. F. Estimating chaos and complex dynamics in an insect population. Ecol. Monogr. 71, 277–303 (2001)

    Article  Google Scholar 

  22. 22

    Yoshida, T., Jones, L. E., Ellner, S. P., Fussmann, G. F. & Hairston, N. G. Rapid evolution drives ecological dynamics in a predator–prey system. Nature 424, 303–306 (2003)

    ADS  CAS  Article  Google Scholar 

  23. 23

    Persson, L. et al. Culling prey promotes predator recovery-alternative states in a whole-lake experiment. Science 316, 1743–1746 (2007)

    ADS  CAS  Article  Google Scholar 

  24. 24

    Nisbet, R. M., McCauley, E., Gurney, W. S. C., Murdoch, W. W. & Wood, S. N. Formulating and testing a partially specified Dynamic Energy Budget Model. Ecology 85, 3132–3139 (2004)

    Article  Google Scholar 

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We acknowledge feedback from A. de Roos, J. Fox, J. Casas, L. Persson and C. Briggs. A. Potapov provided advice on the bifurcation analysis. Experiments and theoretical analysis were supported by NSERC (Discovery Grants and Accelerator Award), Canada Foundation for Innovation, and the Canada Research Chairs Program to E.M. W.A.N. acknowledges support from Alberta Ingenuity. R.M.N. acknowledges support from the US National Science Foundation (Grant DEB-0717259).

Author Contributions All authors contributed to the planning, execution and analysis of theory and experiments.

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Correspondence to Edward McCauley.

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McCauley, E., Nelson, W. & Nisbet, R. Small-amplitude cycles emerge from stage-structured interactions in Daphnia–algal systems. Nature 455, 1240–1243 (2008).

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