Effects of rapid prey evolution on predator–prey cycles


We study the qualitative properties of population cycles in a predator–prey system where genetic variability allows contemporary rapid evolution of the prey. Previous numerical studies have found that prey evolution in response to changing predation risk can have major quantitative and qualitative effects on predator–prey cycles, including: (1) large increases in cycle period, (2) changes in phase relations (so that predator and prey are cycling exactly out of phase, rather than the classical quarter-period phase lag), and (3) “cryptic” cycles in which total prey density remains nearly constant while predator density and prey traits cycle. Here we focus on a chemostat model motivated by our experimental system (Fussmann et al. in Science 290:1358–1360, 2000; Yoshida et al. in Proc roy Soc Lond B 424:303–306, 2003) with algae (prey) and rotifers (predators), in which the prey exhibit rapid evolution in their level of defense against predation. We show that the effects of rapid prey evolution are robust and general, and furthermore that they occur in a specific but biologically relevant region of parameter space: when traits that greatly reduce predation risk are relatively cheap (in terms of reductions in other fitness components), when there is coexistence between the two prey types and the predator, and when the interaction between predators and undefended prey alone would produce cycles. Because defense has been shown to be inexpensive, even cost-free, in a number of systems (Andersson et al. in Curr Opin Microbiol 2:489–493, 1999: Gagneux et al. in Science 312:1944–1946, 2006; Yoshida et al. in Proc Roy Soc Lond B 271:1947–1953, 2004), our discoveries may well be reproduced in other model systems, and in nature. Finally, some of our key results are extended to a general model in which functional forms for the predation rate and prey birth rate are not specified.

This is a preview of subscription content, log in to check access.


  1. 1.

    Abrams P. and Matsuda H. (1997). Prey adaptation as a cause of predator–prey cycles. Evolution 51: 1742–1750

    Article  Google Scholar 

  2. 2.

    Abrams P. (1999). Is predator-mediated coexistence possible in unstable systems?. Ecology 80: 608–621

    Google Scholar 

  3. 3.

    Andersson D.I. and Levin B.R. (1999). The biological cost of antibiotic resistance. Curr. Opin. Microbiol. 2: 489–493

    Article  Google Scholar 

  4. 4.

    Antonovics J., Bradshaw A.D. and Turner R.G. (1971). Heavy metal tolerance in plants. Adv. Ecol. Res. 71: 1–85

    Article  Google Scholar 

  5. 5.

    Arino J., Pilyugin S. and Wolkowicz G.S.K. (2003). Considerations on yield, nutrient uptake, cellular growth, and competition in chemostat models. Can. Appl. Math. Quart. 11: 107–142

    MATH  MathSciNet  Google Scholar 

  6. 6.

    Ashley M.V., Willson M.F., Pergams O.R.W., O’Dowd D.J., Gende S.M. and Brown J.S. (2003). Evolutionarily enlightened management. Biol. Conserv. 111: 115–123

    Article  Google Scholar 

  7. 7.

    Barry M. (1994). The costs of crest induction for Daphnia carinata. Oecologia 97: 278–288

    Article  Google Scholar 

  8. 8.

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

    Article  Google Scholar 

  9. 9.

    Bergelson J. and Purrington C.B. (1996). Surveying patterns in the cost of resistance in plants. Am. Nat. 148: 536–558

    Article  Google Scholar 

  10. 10.

    Bohannan B.J.M. and Lenski R. (1997). Effect of resource enrichment on a chemostat community of bacteria and bacteriophage. Ecology 78: 2303–2315

    Article  Google Scholar 

  11. 11.

    Bohannan B.J.M. and Lenski R. (1999). Effect of prey heterogeneity on the response of a model food chain to resource enrichment. Am. Nat. 153: 73–82

    Article  Google Scholar 

  12. 12.

    Butler G.J. and Wolkowicz G.S.K. (1986). Predator-mediated competition in the chemostat. J. Math. Biol. 24: 167–191

    MATH  Article  MathSciNet  Google Scholar 

  13. 13.

    Coltman D.W., O’Donoghue P., Jorgenson J.T., Hogg J.T., Strobeck C. and Festa-Blanchet M. (2003). Undesirable evolutionary consequences of trophy-hunting. Nature 426: 655–658

    Article  Google Scholar 

  14. 14.

    Conover D.O. and Munch S.B. (2002). Sustaining fisheries yields over evolutionary time scales. Science 297: 94–96

    Article  Google Scholar 

  15. 15.

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

    Article  Google Scholar 

  16. 16.

    Fussmann G.F., Ellner S.P. and Hairston N.G. (2003). Evolution as a critical component of plankton dynamics. Proc. Roy. Soc. Lond. Ser B 270: 1015–1022

    Article  Google Scholar 

  17. 17.

    Gagneux S., Long C.D., Small P.M., Van T., Schoolnik G.K. and Bohannan B.J.M. (2006). The competitive cost of antibiotic resistance in Mycobacterium tuberculosis. Science 312: 1944–1946

    Article  Google Scholar 

  18. 18.

    Grant P.R. and Grant B.R. (2002). Unpredictable evolution in a thirty year study of Darwin’s finches. Science 296: 707–710

    Article  Google Scholar 

  19. 19.

    Guckenheimer J., Myers M. and Sturmfels B. (1997). Computing Hopf bifurcations I. SIAM J. Numer. Anal. 34: 1–27

    MATH  Article  MathSciNet  Google Scholar 

  20. 20.

    Hairston N.G. and Walton W.E. (1986). Rapid evolution of a life-history trait. Proc. Natl. Acad. Sci. USA 83: 4831–4833

    Article  Google Scholar 

  21. 21.

    Hairston N.G., Ellner S.P., Geber M.A., Yoshida T. and Fox J.A. (2005). Rapid evolution and the convergence of ecological and evolutionary time. Ecol. Lett. 8: 1114–1127

    Article  Google Scholar 

  22. 22.

    Heath D.D., Heath J.W., Bryden C.A., Johnson R.M. and Fox C.W. (2003). Rapid evolution of egg size in captive salmon. Science 299: 1738–1740

    Article  Google Scholar 

  23. 23.

    Hendry A.P. and Kinnison M.T. (1999). The pace of modern life: measuring rates of contemporary microevolution. Evolution 53: 1637–1653

    Article  Google Scholar 

  24. 24.

    Jones L.E. and Ellner S.P. (2004). Evolutionary tradeoff and equilibrium in a predator–prey system. Bull. Math. Biol. 66: 1547–1573

    Article  MathSciNet  Google Scholar 

  25. 25.

    Kinnison, M.T., Hairston, N.G. Jr.: Eco-evolutionary conservation biology: contemporary evolution and the dynamics of persistence. Funct. Ecol. (submitted) (2006)

  26. 26.

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

    MATH  Article  Google Scholar 

  27. 27.

    Kuznetsov, Y.A.: Elements of applied bifurcation theory. Applied Mathematical Sciences, vol. 112, Chap. 8. Springer, New York (1994)

    Google Scholar 

  28. 28.

    May, R.M.: Stability and complexity in model ecosystems. Princeton University Press, Princeton, New York (1974)

    Google Scholar 

  29. 29.

    Meyer J., Ellner S.P., Jones L.E., Yoshida T. and Hairston N.G. (2006). Prey evolution of the time scale of predator–prey dynamics revealed by allele-specific quantitative PCR. Proc. Natl. Acad. Sci. 103: 10690–10695

    Article  Google Scholar 

  30. 30.

    Olsen E.M., Heino M., Lilly G.R., Morgan M.J., Brattey J. and Dieckmann U. (2004). Maturation trends indicative of rapid evolution preceded the collapse of northern cod. Nature 428: 932–935

    Article  Google Scholar 

  31. 31.

    Palumbi S. (2001). The evolution explosion: how humans cause rapid evolutionary change. Norton W.W., New York

    Google Scholar 

  32. 32.

    Pickett-Heaps J.D. (1975). Green Algae: Structure, Reproduction and Evolution in Selected Genera. Sinauer Associates, Sunderland

    Google Scholar 

  33. 33.

    Press W.H., Flannery B.P., Teukolsky S.A. and Vetterling W.T. (1988). Numerical Recipes in C. Cambridge University Press, Cambridge

    Google Scholar 

  34. 34.

    Preisser E.L., Bolnick D.J. and Benard M.F. (2005). Scared to Death? The effects of intimidation and consumption in predator–prey interactions. Ecology 86: 501–509

    Article  Google Scholar 

  35. 35.

    Reznick D.N., Shaw F.H., Rodd F.H. and Shaw R.G. (1997). Evaluation of the rate of evolution in natural populations of guppies (Poecilia reticulata). Science 275: 1934–1937

    Article  Google Scholar 

  36. 36.

    Ruan S. and Wolkowicz G.S.K. (1996). Bifurcation of a chemostat model with a distributed delay. J. Math. Anal. Appl. 204: 786–812

    MATH  Article  MathSciNet  Google Scholar 

  37. 37.

    Saccheri I. and Hanski I. (2006). Natural selection and population dynamics. Trends Ecol. Evolut. 21: 341–347

    Article  Google Scholar 

  38. 38.

    Shertzer K.W., Ellner S.P., Fussmann G.F. and Hairston N.G. (2002). Predator–prey cycles in an aquatic microcosm: testing hypotheses of mechanism. J. Anim. Ecol. 71: 802–815

    Article  Google Scholar 

  39. 39.

    Searle S.R. (1982). Matrix Algebra Useful for Statistics. Wiley, New York

    Google Scholar 

  40. 40.

    Smith H.L. and Waltman P. (1995). The Theory of the Chemostat. Cambridge University Press, Cambridge

    Google Scholar 

  41. 41.

    Strauss S.Y., Rudgers J.A., Lau J.A. and Irwin R.E. (2002). Direct and ecological costs of resistance to herbivory. Trends Ecol. Evol. 17: 278–285

    Article  Google Scholar 

  42. 42.

    Thompson J.N. (1998). Rapid evolution as an ecological process. Trends Ecol. Evol. 13: 329–332

    Article  Google Scholar 

  43. 43.

    Toth D. and Kot M. (2006). Limit cycles in a chemostat model for a single species with age structure. Math. Biosci. 202: 194–217

    MATH  MathSciNet  Google Scholar 

  44. 44.

    Vayenis D.V. and Pavlou S. (1999). Chaotic dynamics of a food web in a chemostat. Math. Biosci. 162: 69–84

    Article  Google Scholar 

  45. 45.

    Xia H., Wolkowicz G.S.K. and Wang L. (2005). Transient oscillation induced by delayed growth response in the chemostat. J. Math. Biol. 50: 489–530

    MATH  Article  MathSciNet  Google Scholar 

  46. 46.

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

    Article  Google Scholar 

  47. 47.

    Yoshida T., Ellner S.P. and Hairston N.G. (2004). Evolutionary tradeoff between defense against grazing and competitive ability in a simple unicellular alga, Chlorella vulgaris. Proc. Roy. Soc. Lond. B. 271: 1947–1953

    Article  Google Scholar 

  48. 48.

    Yoshida, T., Ellner, S.P., Jones, L.E., Hairston, N.G. Jr.: Cryptic population dynamics: rapid evolution masks trophic interaction. PLOS Biol. (submitted) (2007)

  49. 49.

    Zimmer C. (2003). Rapid evolution can foil even the best-laid plans. Science 300: 895

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Laura E. Jones.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Jones, L.E., Ellner, S.P. Effects of rapid prey evolution on predator–prey cycles. J. Math. Biol. 55, 541–573 (2007). https://doi.org/10.1007/s00285-007-0094-6

Download citation


  • Predator–prey
  • Consumer-resource
  • Cycles
  • Chemostat
  • Evolution

Mathematics Subject Classification (2000)

  • 92D25
  • 92D40
  • 92D15
  • 34C15