Journal of Mathematical Biology

, Volume 64, Issue 1–2, pp 341–360 | Cite as

The hydra effect in predator–prey models

  • Michael Sieber
  • Frank M. Hilker


The seemingly paradoxical increase of a species population size in response to an increase in its mortality rate has been observed in several continuous-time and discrete-time models. This phenomenon has been termed the “hydra effect”. In light of the fact that there is almost no empirical evidence yet for hydra effects in natural and laboratory populations, we address the question whether the examples that have been put forward are exceptions, or whether hydra effects are in fact a common feature of a wide range of models. We first propose a rigorous definition of the hydra effect in population models. Our results show that hydra effects typically occur in the well-known Gause-type models whenever the system dynamics are cyclic. We discuss the apparent discrepancy between the lack of hydra effects in natural populations and their occurrence in this standard class of predator–prey models.


Consumer–resource models Gause-type model Population cycles Allee effect Mean population density Population extinction 

Mathematics Subject Classification (2000)

92D25 92D40 34C60 34D05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abrams PA (2009) When does greater mortality increase population size? The long history and diverse mechanisms underlying the hydra effect. Ecol Lett 12: 462–474CrossRefGoogle Scholar
  2. Abrams PA, Brassil CE, Holt RD (2003) Dynamics and responses to mortality rates of competing predators undergoing predator–prey cycles. Theor Popul Biol 64: 163–176CrossRefzbMATHGoogle Scholar
  3. Abrams PA, Ginzburg LR (2000) The nature of predation: prey dependent, ratio dependent or neither?. Trends Ecol Evol 15(8): 337–341CrossRefGoogle Scholar
  4. Abrams PA, Matsuda H (2005) The effect of adaptive change in the prey on the dynamics of an exploited predator population. Can J Fish Aquat Sci 62: 758–766CrossRefGoogle Scholar
  5. Allee WC (1931) Animal aggregations: a study in general sociology. University of Chicago Press, ChicagoCrossRefGoogle Scholar
  6. Armstrong RA, McGehee R (1980) Competitive exclusion. Am Nat 115(2): 151–170CrossRefMathSciNetGoogle Scholar
  7. Bauer F (1979) Boundedness of solutions of predator–prey systems. Theor Popul Biol 15(2): 268–273CrossRefGoogle Scholar
  8. Bazykin AD (1998) Nonlinear dynamics of interacting populations. World Scientific, SingaporeCrossRefGoogle Scholar
  9. Beddington JR (1975) Mutual interference between parasites or predators and its effect on searching efficiency. J Anim Ecol 44: 331–340CrossRefGoogle Scholar
  10. Conway ED, Smoller JA (1986) Global analysis of a system of predator–prey equations. SIAM J Appl Math 46(4): 630–642CrossRefzbMATHMathSciNetGoogle Scholar
  11. Courchamp F, Berec L, Gascoigne J (2008) Allee effects in ecology and conservation. Oxford University Press, New YorkCrossRefGoogle Scholar
  12. Dattani J, Blake JCH, Hilker FM (2011) Target-oriented chaos control (in review)Google Scholar
  13. De Angelis DL, Goldstein RA, O’Neill RV (1975) A model for trophic interaction. Ecology 56: 881–892CrossRefGoogle Scholar
  14. de Feo O, Rinaldi S (1997) Yield and dynamics of tritrophic food chains. Am Nat 150: 328–345CrossRefGoogle Scholar
  15. Dercole F, Ferrière R, Gragnani A, Rinaldi S (2006) Coevolution of slow-fast populations: evolutionary sliding, evolutionary pseudo-equilibria and complex Red Queen dynamics. Proc R Soc B 273: 983–990CrossRefGoogle Scholar
  16. Eckmann JP, Ruelle D (1985) Ergodic theory of chaos and strange attractors. Rev Mod Phys 57: 617–656CrossRefMathSciNetGoogle Scholar
  17. Gaunersdorfer A (1992) Time averages for heteroclinic attractors. SIAM J Appl Math 52(5): 1476–1489CrossRefzbMATHMathSciNetGoogle Scholar
  18. Gause GF (1934) The struggle for existence. Williams and Wilkins, BaltimoreCrossRefGoogle Scholar
  19. Gragnani A, de Feo O, Rinaldi S (1998) Food chains in the chemostat: relationships between mean yield and complex dynamics. Bull Math Biol 60: 703–719CrossRefzbMATHGoogle Scholar
  20. Hilker FM, Westerhoff FH (2006) Paradox of simple limiter control. Phys Rev E 73: 052901CrossRefGoogle Scholar
  21. Kuznetsov YA (1995) Elements of applied bifurcation theory. Springer, New YorkzbMATHGoogle Scholar
  22. Liou LP, Cheng KS (1988) On the uniqueness of a limit cycle for a predator–prey system. SIAM J Math Anal 88: 67–84MathSciNetGoogle Scholar
  23. Liu Y (2005) Geometric criteria for the nonexistence of cycles in Gause-type predator–prey systems. Proc Am Math Soc 133: 3619–3626CrossRefzbMATHGoogle Scholar
  24. Liz E (2010) How to control chaotic behaviour and population size with proportional feedback. Phys Lett A 374: 725–728CrossRefMathSciNetGoogle Scholar
  25. Matsuda H, Abrams PA (2004) Effects of adaptive change and predator–prey cycles on sustainable yield. Can J Fish Aquat Sci 61: 175–184CrossRefGoogle Scholar
  26. May RM (1976) Theoretical ecology: principles and applications. Blackwell, OxfordGoogle Scholar
  27. May RM, Leonard W (1975) Nonlinear aspects of competition between three species. SIAM J Appl Math 29: 243–252CrossRefzbMATHMathSciNetGoogle Scholar
  28. McGehee R, Armstrong RA (1977) Some mathematical problems concerning the ecological principle of competitive exclusion. J Differ Equ 23: 30–52CrossRefzbMATHMathSciNetGoogle Scholar
  29. Ricker WE (1954) and recruitment. J Fish Res Board Can 11: 559–623CrossRefGoogle Scholar
  30. Rosenzweig ML, MacArthur RH (1963) Graphical representation and stability conditions of predator–prey interactions. Am Nat 97: 209–223CrossRefGoogle Scholar
  31. Schreiber SJ (2003) Allee effects, extinctions, and chaotic transients in simple population models. Theor Popul Biol 64: 201–209CrossRefzbMATHGoogle Scholar
  32. Schreiber SJ, Rudolf VHW (2008) Crossing habitat boundaries: coupling dynamics of ecosystems through complex life cycles. Ecol Lett 11: 576–587CrossRefGoogle Scholar
  33. Seno H (2008) A paradox in discrete single-species population dynamics with harvesting/thinning. Math Biosci 214: 63–69CrossRefzbMATHMathSciNetGoogle Scholar
  34. Sieber M, Hilker FM (2011) Prey, predators, parasites: intraguild predation or simpler community modules in disguise?. J Anim Ecol 80: 414–421CrossRefGoogle Scholar
  35. Sinha S, Parthasarathy S (1996) Unusual dynamics of extinction in a simple ecological model. Proc Natl Acad Sci 93: 1504–1508CrossRefzbMATHGoogle Scholar
  36. Terry AJ, Gourley SA (2010) Perverse consequences of infrequently culling a pest. Bull Math Biol 72: 1666–1695CrossRefzbMATHMathSciNetGoogle Scholar
  37. Turchin P (2003) Complex population dynamics: a theoretical/empirical synthesis. Princeton University Press, PrincetonzbMATHGoogle Scholar
  38. van Voorn GAK, Hemerik L, Boer MP, Kooi BW (2007) Heteroclinic orbits indicate overexploitation in predator–prey systems with a strong Allee effect. Math Biosci 209: 451–469CrossRefzbMATHMathSciNetGoogle Scholar
  39. Volterra V (1931) Leçons sur la théorie mathématique de la lutte pour la vie. Gauthier-Villars, ParisGoogle Scholar
  40. Wang J, Shi J, Wei J (2011) Predator–prey system with strong Allee effect in prey. J Math Biol 62: 291–331CrossRefMathSciNetGoogle Scholar
  41. Yodzis P (1989) Introduction to theoretical ecology. Harper & Row, New YorkzbMATHGoogle Scholar
  42. Zicarelli J (1975) Mathematical analysis of a population model with several predators on a single prey. Ph.D. Thesis. University of MinnesotaGoogle Scholar
  43. Zipkin EF, Kraft CE, Cooch EG, Sullivan PJ (2009) When can efforts to control nuisance and invasive species backfire?. Ecol Appl 19: 1585–1595CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science, Institute of Environmental Systems ResearchUniversity of OsnabrückOsnabrückGermany
  2. 2.Department of Mathematical Sciences, Centre for Mathematical BiologyUniversity of BathBathUK

Personalised recommendations