Skip to main content

Parametric Analysis of a Predator–prey System Stabilized by a Top Predator

Journal of Mathematical Biology volume 53, pages 305–335 (2006)Cite this article

Abstract

We present a complete parametric analysis of a predator–prey system influenced by a top predator. We study ecosystems with abundant nutrient supply for the prey where the prey multiplication can be considered as proportional to its density. The main questions we examine are the following: (1) Can the top predator stabilize such a system at low densities of prey? (2) What possible dynamic behaviors can occur? (3) Under which conditions can the top predation result in the system stabilization? We use a system of two nonlinear ordinary differential equations with the density of the top predator as a parameter. The model is investigated with methods of qualitative theory of ODEs and the theory of bifurcations. The existence of 12 qualitatively different types of dynamics and complex structure of the parametric space are demonstrated. Our studies of phase portraits and parametric diagrams show that a top predator can be an important factor leading to stabilization of the predator-prey system with abundant nutrient supply. Although the model here is applied to the plankton communities with fish (or carnivorous zooplankton) as the top trophic level, the general form of the equations allows applications of our results to other ecological systems.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

43,34 €

Price includes VAT (Finland)
Tax calculation will be finalised during checkout.

References

  1. Arnold V.I. (1983) Geometrical Methods of Ordinary Differential Equations. Springer, Berlin New York

    MATH  Google Scholar 

  2. Bazykin, A.D. Nonlinear Dynamics of Interacting Populations. In: Chua, L.O. (ed.) Series on Nonlinear Science, Series A, vol. 11. World Scientific (1998)

  3. Berryman A.A. (1981) Population Systems: a General Introduction. Plenum, New York

    Google Scholar 

  4. Boer M.P., Kooi B.W., Kooijman S.A. (2001) Multiple attractors and boundary crises in a tri-trophic food chain. Math. Biosci. 169, 109–28

    MATH  Article  MathSciNet  Google Scholar 

  5. Bollens S.M., Frost B.W. (1989) Predator induced diel vertical migration in a marine planktonic copepod. J. Plankton Res. 11, 1047–1065

    Article  Google Scholar 

  6. Caswell H., Neubert M.G. (1998) Chaos and closure terms in plankton food chain models. J. Plankton Res. 20, 1837–1845

    Article  Google Scholar 

  7. Doveri F., Scheffer M., Rinaldi S., Muratori S., Kuznetsov Y.A. (1993) Seasonality and chaos in a plankton-fish model. Theor. Popul. Biol. 43, 159–183

    MATH  Article  Google Scholar 

  8. Edwards A.M., Brindley J. (1996) Zooplankton mortality and the dynamical behavior of plankton population models. Bull. Math. Biol. 61, 303–339

    Article  Google Scholar 

  9. Edwards A.M., Yool A. (2000) The role of higher predation in plankton population models. J. Plankton Res. 22, 1085–1112

    Article  Google Scholar 

  10. Eisenberg J.N., Maszle D.R. (1995) The structural stability of a three-species food chain model. J. Theor. Biol. 176, 501–510

    Article  Google Scholar 

  11. Evans G.T., Parslow J.S. (1985) A model of annual plankton cycles. Biol. Oceanogr. 3, 327–347

    Google Scholar 

  12. Guckenheimer J., Holmes P. (1983) Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields. Springer, Berlin Heidelberg New York

    Google Scholar 

  13. Han B.P., Straskraba M. (1998) Modeling patterns of zooplankton diel vertical migration. J. Plankton Res. 20, 1463–1487

    Article  Google Scholar 

  14. Hassard B., Kazarinoff N., Wan Y.-H. (1981) Theory and Application of Hopf Bifurcation. Cambridge University Press, London

    Google Scholar 

  15. Hastings A., Powell T. (1991) Chaos in the three species food chain. Ecology 72, 896–903

    Article  Google Scholar 

  16. James A., Pitchford J.W., Brindley J. (2003) The relationship between plankton blooms, the hatching of fish larvae, and recruitment. Ecol. Model. 160, 77–90

    Article  Google Scholar 

  17. Kohlmeier C., Ebenhoh W. (1995) The stabilization role of cannibalism in a predator–prey system. Bull. Math. Biol. 57, 401–411

    MATH  Google Scholar 

  18. Kuznetsov Y.A. (1995) Elements of Applied Bifurcation Theory. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  19. Malchow H. (1994) Non-equilibrium structures in plankton dynamics. Ecol. Model. 75, 123–134

    Article  Google Scholar 

  20. Marsden J., McCracken M. (1976) Hopf Bifurcation and its Applications. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  21. May R.M. (1974) Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton

    Google Scholar 

  22. Medvinsky A.B., Petrovskii S.V., Tikhonova I.A., Malchow H., Li B.L. (2002) Spatiotemporal complexity of plankton and fish dynamics. SIAM Rev. 44, 311–370

    MATH  Article  MathSciNet  Google Scholar 

  23. Mills E.L., Forney J.L. (1983) Impact of Daphnia pulex of predation by young yellow perch Perca flavescens in Oneida Lake, New York. Trans. Am. Fish. Soc. 110, 479–488

    Article  Google Scholar 

  24. Morozov A.Y. (2003). Mathematical Modelling of Ecological Factors Affecting Stability and Pattern Formation in Marine Plankton Communities. PhD Dissertation. SIO, Moscow

    Google Scholar 

  25. Morozov A.Y., Nezlin N.P., Petrovskii S.V. (2005) Invasion of a top predator into epipelagic ecosystem can bring a paradoxical top-down trophic control. Biol. Invasions 7, 845–861

    Article  Google Scholar 

  26. Murray J.D. (1989) Mathematical Biology. Springer, Berlin Heidelberg New york

    MATH  Google Scholar 

  27. Olson D.B., Cosner C., Cantrell R.S., Hastings A. (2005) Persistence of fish populations in time and space as a key to sustainable fisheries. Bull. Marine Sci. 76, 213–232

    Google Scholar 

  28. Pitchford J., Brindley J. (1998) Intratrophic predation in simple predator-prey models. Bull. Math. Biol. 60, 937–953

    MATH  Article  Google Scholar 

  29. Rosenzweig M.L. (1971) Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171, 385–387

    Article  Google Scholar 

  30. Scheffer M. (1991) Fish and nutrients interplay determines algal biomass: a minimal model. Oikos 62, 271–282

    Article  Google Scholar 

  31. Scheffer M. (1998) Ecology of Shallow Lakes. Chapman& Hall, London

    Google Scholar 

  32. Scheffer M., Rinaldi S., Kuznetsov Y.A. (2000) Effects of fish on plankton dynamics: a theoretical analysis. Can J. Fish Aquat. Sci. 57, 1208–1219

    Article  Google Scholar 

  33. Steele J.H., Henderson E.W. (1992) The role of predation in plankton models. J. Plankton Res. 14, 157–172

    Article  Google Scholar 

  34. Truscott J.E., Brindley J. (1994) Equilibria, stability and excitability in a general class of plankton population models. Philos. Trans. R. Soc. London Ser. A 347, 703–718

    MATH  Article  Google Scholar 

  35. Van den Bosch F., de Roos A.M., Gabriel W. (1988) Cannibalism as a life boat mechanism. J. Math. Biol. 26, 619–633

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

  1. Ecological Complexity and Modeling Laboratory, Department of Botany and Plant Sciences, University of California, Riverside, CA, 92521-0124, USA

    Andrew Y. Morozov & Bai-Lian Li

Authors
  1. Andrew Y. Morozov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bai-Lian Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bai-Lian Li.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Morozov, A.Y., Li, BL. Parametric Analysis of a Predator–prey System Stabilized by a Top Predator. J. Math. Biol. 53, 305–335 (2006). https://doi.org/10.1007/s00285-006-0008-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-006-0008-z

Keywords

  • Predator–prey system
  • Top predator
  • Plankton models
  • Stabilization
  • Bifurcation analysis