Theoretical Ecology

, Volume 2, Issue 3, pp 149–160 | Cite as

Simple testing procedures for the Holling type II model

  • Sorana FrodaEmail author
  • Ashkan Zahedi
Original paper


In this paper, we consider predator–prey data that can be viewed as solutions to a planar system of ordinary differential equations (ODE) observed with random error. The ODE system admits a limit cycle, while the random error is supposed to act additively in the log-scale. One of the oldest such systems is Holling’s type II model. In spite of its simplicity, it is still very popular in data analyses, although more sophisticated models have been introduced in the literature. We propose a simple way of deciding whether a set of predator–prey pairs is indicative or not of a departure from this basic model by exploiting the geometric properties of the solution in the phase plane. To illustrate our method, we use simulated and real data.


Holling type II model Observational error Predator–prey system of differential equations Phase plane Isocline Linear regression slope Two-sample tests Nonparametric trend test 



This research received financial support from the Natural Sciences and Engineering Research Council (Canada). We are indebted to C. Jost for providing various data sets; to R. Ferland, C. Gravel, and F. Larribe for advice and support; and to an anonymous referee for careful reading and very useful suggestions. The first author is thankful to F. Rousset and I. Olivieri and her team at the Institut des sciences de l’évolution de Montpellier for their hospitality and very stimulating research exchanges.


  1. Abrams PA, Ginzburg L (2000) The nature of predation: prey dependent, ratio dependent or neither? Tree 15:337–341PubMedGoogle Scholar
  2. Akçakaya R, Arditi R, Ginzburg L (1995) Ratio-dependent predation: an abstraction that works. Ecology 76:995–1004CrossRefGoogle Scholar
  3. Albrecht F, Gatzke H, Haddad A, Wax N (1974) The dynamics of two interacting populations. J Math Anal Appl 46:658–670CrossRefGoogle Scholar
  4. Arditi R, Ginzburg L (1989) Coupling in predator–prey dynamics: ratio-dependence. J Theor Biol 139:311–326CrossRefGoogle Scholar
  5. Arditi R, Callois J-M, Tyutyunov Y, Jost C (2004) Does mutual interference always stabilize predator–prey dynamics? A comparison of models. C R Biol 327:1037–1057PubMedCrossRefGoogle Scholar
  6. Berezovskaya F, Karev G, Arditi R (2001) Parametric analysis of the ratio-dependent predator–prey model. J Math Biol 43:221–246PubMedCrossRefGoogle Scholar
  7. CIPEL (1995) Rapports sur les études et recherches entreprises dans le bassin lémanique. Technical report. Commission internationale pour la protection des eaux du Léman contre la pollution, Lausanne, SuisseGoogle Scholar
  8. De Angelis DL, Goldstein RA, O’Neill RV (1975) A model for trophic interaction. Ecology 56:881–892CrossRefGoogle Scholar
  9. Ellner, SP, Seifu, Y, Smith, RH (2002) Fitting population dynamic models to time series data by gradient matching. Ecology 83:2256–2270Google Scholar
  10. Froda S, Colavita G (2005) Estimating predator–prey systems via ordinary differential equations with closed orbits. Aust N Z J Stat 42: 235–254CrossRefGoogle Scholar
  11. Froda S, Nkurunziza S (2007) Prediction of cyclic predator–prey populations modeled by perturbed ODE. J Math Biol 54:407–451PubMedCrossRefGoogle Scholar
  12. Fuller W (1987) Measurement error models. Wiley, New YorkCrossRefGoogle Scholar
  13. Gasull A, Kooij RE, Torregrosa J (1997) Limit cycles in the Holling-Tanner model. Publ Mat 41:149–167Google Scholar
  14. Getz WM (1984) Population dynamics: a unified approach. J Theor Biol 108:623–643CrossRefGoogle Scholar
  15. Ginzburg L (1998) Assuming reproduction to be a function of consumption raises some doubts about some popular predator–prey models. Ecology 67:325–327Google Scholar
  16. Hanski I, Hansson L, Henttonen H (1991) Specialist predators, generalist predators and the microtine rodent cycle. J Anim Ecol 60:353–367CrossRefGoogle Scholar
  17. Hanski I, Korpimäki E (1995) Microtine rodent dynamics in northern Europe: parametrized models for the predator–prey interaction. Ecology 76:840–850CrossRefGoogle Scholar
  18. Hanski I, Henttonen H, Korpimäki E, Oksanen L, Turchin P (2001) Small-rodent dynamics and predation. Ecology 82:1505–1520Google Scholar
  19. Holling CS (1959) The components of predation as revealed by a study of small mammal predation on the European pine sawfly. Can Entomol 91:293–320CrossRefGoogle Scholar
  20. Holling CS (1965) The functional response of predators to prey density and its role in mimicry and population regulation. Mem Entomol Soc Can 47:3–86Google Scholar
  21. Huffaker CB, Shea KP, Herman SG (1963) Experimental studies on predation: complex dispersion and levels of food in an acarine predator–prey interaction. Hilgardia 34:303–330Google Scholar
  22. Jost C, Arditi R (2001) From pattern to process: identifying predator–prey models from time-series data. Popul Ecol 43:229–243CrossRefGoogle Scholar
  23. Jost C, Arino O, Arditi R (1999) About deterministic extinction in ratio-dependent predator–prey models. Bull Math Biol 61:19–32CrossRefGoogle Scholar
  24. Jost C, Ellner SP (2000) Testing for predator dependence in predator–prey dynamics: a non-parametric approach. Proc R Soc Lond B 267:1611–1620CrossRefGoogle Scholar
  25. Jost C, Devulder G, Vucetich JA, Peterson RO, Arditi R (2005) The wolves of Isle Royale display scale-invariant satiation and ratio-dependent predation on moose. J Anim Ecol 74:809–816CrossRefGoogle Scholar
  26. Kolmogorov AN (1936) Sulla teoria di Volterra della lotta per l’esistenza. G dell’Inst Italiano per l’Attuari 7:74–80Google Scholar
  27. Kot M (2001) Elements of mathematical biology. Cambridge University Press, CambridgeGoogle Scholar
  28. Lehmann EL (1975) Nonparametrics: statistical methods based on ranks. Holden Day, San FranciscoGoogle Scholar
  29. Leslie PH (1948) Some further notes on the use of matrices in population mathematics. Biometrika 35:213–245Google Scholar
  30. Lotka AJ (1925) Elements of physical biology. Williams and Wilkins, BaltimoreGoogle Scholar
  31. May RM (1973) Stability and complexity in model ecosystems. Princeton University Press, PrincetonGoogle Scholar
  32. O’Donoghue M, Boutin S, Krebs CT, Zuleta G, Murray DL, Hofer EJ (1998) Functional responses of coyotes and lynx to the snowshoe hare cycle. Popul Ecol 43:229–243Google Scholar
  33. Rosenzweig ML, MacArthur RH (1963) Graphical representation and stability conditions of predator–prey interactions. Am Nat 97:29–223CrossRefGoogle Scholar
  34. Sáez E, Gonzales-Oliveres E (1999) Dynamics of a predator–prey model. SIAM J Appl Math 59:1867–1878CrossRefGoogle Scholar
  35. Skalski GT, Gilliam JF (2001) Functional responses with predator interference: viable alternatives to the Holling type II model. Ecology 82:3083–3092CrossRefGoogle Scholar
  36. Tanner JT (1975) The stability and the intrinsic growth rates of prey and predator populations. Ecology 56:855–867CrossRefGoogle Scholar
  37. Veilleux BG (1976) The analysis of a predatory interaction between Didinium and Paramecium. Master’s thesis, University of Alberta, EdmontonGoogle Scholar
  38. Veilleux BG (1979) An analysis of a predatory interaction between Didinium and Paramecium. J Anim Ecol 48:787–803CrossRefGoogle Scholar
  39. Volterra V (1931) Leçons sur la théorie mathématique de la lutte pour la vie. Gauthiers-VillarsGoogle Scholar
  40. Wollkind DJ, Collins JB, Logan J (1988) Metastability in a temperature-dependent model system for predator–prey mite outbreak interactions on fruit trees. Bull Math Biol 50:379–409Google Scholar
  41. Zahedi A (2008) Statistical inference for some predator–prey models. Master’s thesis, Department of Mathematics, Université du Québec à MontréalGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité du Québec à MontréalMontréalCanada

Personalised recommendations