Skip to main content
SpringerLink
Log in

Incorporating prey refuge in a prey–predator model with a Holling type I functional response: random dynamics and population outbreaks

  • Original Paper
  • Published:
Journal of Biological Physics Aims and scope Submit manuscript

Abstract

A prey–predator discrete-time model with a Holling type I functional response is investigated by incorporating a prey refuge. It is shown that a refuge does not always stabilize prey–predator interactions. A prey refuge in some cases produces even more chaotic, random-like dynamics than without a refuge and prey population outbreaks appear. Stability analysis was performed in order to investigate the local stability of fixed points as well as the several local bifurcations they undergo. Numerical simulations such as parametric basins of attraction, bifurcation diagrams, phase plots and largest Lyapunov exponent diagrams are executed in order to illustrate the complex dynamical behavior of the system.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Notes

  1. The shape of the Holling type I functional response is linear. So if the predators are spiders and the preys are biting flies, the number of flies killed by one spider is proportional to the flies’ density [4].

  2. The results of the stability analysis have been exhibited by using the software package Maxima 5.27.0 (http://maxima.sourceforge.net/).

  3. All numerical simulations were executed using the software package E&F Chaos [21].

  4. This low-resting metabolic rate may be due to the fact that they use hydrostatic pressure for extending their appendages [26].

  5. In addition, spiders can further reduce their metabolic rate below their already low levels when they experience periods of food limitation [27].

  6. Many spiders, including both web-building and wandering spiders, are sit-and-wait predators that spend very little time in active locomotion [28]. For example, for the wolf spider Pardosa amentata, the daily energy loss attributed to locomotion was estimated to be only 1% of the daily energy usage of spiders [29].

  7. The energetic costs of web production are relatively small because web building is often a short process and some spiders are able to recycle web proteins, which can substantially reduce the metabolic cost of silk production [29].

  8. For example for the wolf spider Pardosa lugubris, the females invest 26% in reproduction and males invest only 16% [30].

  9. Male spiders invest 81% of their ingested energy into respiration, while females invest 73% [30]. This difference in respiration rate between the sexes is likely related to the high energetic needs of males for agonistic encounters with competing males and for courting females [31].

  10. We use the same parameter values that were also used in Danca et al.’s [9] paper, so that our results are comparable with those of the basic model (1).

  11. Positive Lyapunov exponents correspond to diverging neighboring orbits, negative Lyapunov exponents correspond to converging orbits and zero Lyapunov exponents correspond to bifurcations that occur in the system [34].

References

  1. Verhulst, P.F.: Notice sur la loi que la population suit dans son accroissement. Corresp. Math. et Phys. 10, 113–121 (1838)

    Google Scholar 

  2. Lotka, A.J.: Elements of Physical Biology. Williams and Wilkins, Baltimore MD (1925)

    MATH  Google Scholar 

  3. Volterra, V.: Fluctuations in the abundance of a species considered mathematically. Nature 118(2972), 558–560 (1926)

    Article  ADS  Google Scholar 

  4. Holling, C.S.: The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. Can. Entomol. 91(5), 293–320 (1959)

    Article  Google Scholar 

  5. Holling, C.S.: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Entomol. Soc. Can. 45, 1–60 (1965)

    Article  Google Scholar 

  6. Rosenzweig, M.L., MacArthur, R.H.: Graphical representation and stability conditions of predator–prey interactions. Am. Nat. 97(895), 209–223 (1963)

    Article  Google Scholar 

  7. Maynard, S.J.: Mathematical Ideas in Biology. Cambridge University Press, Cambridge (1968)

    Google Scholar 

  8. Hadeler, K.P., Gerstmann, I.: The discrete Rosenzweig model. Math. Biosci. 98(1), 49–72 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  9. Danca, M., Codreanu, S., Bako, B.: Detailed analysis of a nonlinear prey–predator model. J. Biol. Phys. 23(1), 11–20 (1997)

    Article  Google Scholar 

  10. Liu, X., Xiao, D.: Complex dynamic behaviors of a discrete-time predator–prey system. Chaos, Solitons Fractals 32(1), 80–94 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Maynard, S.J.: Models in Ecology. Cambridge University Press, London (1974)

    MATH  Google Scholar 

  12. Hassel, M.P.: The Dynamics of Arthropod Predator–Prey Systems. Princeton University Press, Princeton (1978)

    Google Scholar 

  13. Taylor, R.J.: Predation. Chapman & Hall, New York (1984)

    Book  Google Scholar 

  14. Pollock, S.T.: Ecology, vol. 10. Dorling Kindersley, London, Eyewitness Science (1993)

  15. Last, J.M.: A Dictionary of Epidemiology. Oxford University Press (2000)

  16. Ryan, K.J., Ray, C.G.: Sherris Medical Microbiology: An Introduction to Infectious Diseases, 4th edn. McGraw-Hill Medical (2003)

  17. Mandell, G.L., Bennett, J.E., Dolin, R.: Mandell, Douglas, and Bennett’s Principles and Practice of Infectious Diseases, 7th edn. Churchill Livingstone (2009)

  18. Oldstone, M.B.A.: Viruses, Plagues, & History, 1st edn. Oxford University Press, USA (1998)

    Google Scholar 

  19. Lehrer, S.: Anopheles mosquito transmission of brain tumor. Med. Hypotheses 74(1), 167–168 (2010)

    Article  Google Scholar 

  20. Jury, E.I.: Inners and Stability of Dynamic Systems. Wiley, New York (1974)

    MATH  Google Scholar 

  21. Diks, C., Hommes, C., Panchenko, V., van der Weide, R.: E&F Chaos: a user-friendly software package for nonlinear economic dynamics. Comput. Econ. 32(1–2), 221–244 (2008)

    Article  MATH  Google Scholar 

  22. Edgar, W.D.: Prey and predators of wolf spider Lycosa lugubris. J. Zool. Lond. 159, 405–411 (1969)

    Article  Google Scholar 

  23. Nyffeler, M., Breene, R.G.: Evidence of low daily food consumption by wolf spiders in meadowland and comparison with other cursorial hunters. J. Appl. Entomol. 110, 73–81 (1990)

    Article  Google Scholar 

  24. Ruppert, E.E., Fox, R.S., Barnes, R.B.: Invertebrate Zoology: A Functional Evolutionary Approach, 7th edn. Brooks Cole Thomson, Belmont, CA (2004)

    Google Scholar 

  25. Anderson, J.F.: Metabolic rates of spiders. Comput. Biochem. Phys. 33, 51–72 (1970)

    Article  Google Scholar 

  26. Anderson, J.F., Prestwich, K.N.: The fluid pressure pumps of spiders (Chelicerata, Araneae). Z. Morph. Tiere. 81, 257–277 (1975)

    Article  Google Scholar 

  27. Anderson, J.F.: Responses to starvation in the spiders Lycosa lenta Hentz and Filistate hibernalis (Hentz). Ecology 55, 576–585 (1974)

    Article  Google Scholar 

  28. Prestwich, K.N.: The energetics of web-building in spiders. Comput. Biochem. Phys. A 57, 321–326 (1977)

    Article  Google Scholar 

  29. Ford, M.J.: Energy costs of the predation strategy of the web-spinning spider Lepthyphantes zimmermanni Bertkau (Linyphiidae). Oecologia. 28, 341–349 (1977)

    Article  Google Scholar 

  30. Edgar, W.D.: Aspects of the ecological energetic of the wolf spider Pardosa (Lycosa) lugubris (Walckenaer). Oecologia 7, 136–154 (1971)

    Article  Google Scholar 

  31. Kotiaho, J.S., Alatalo, R.V., Mappes, J., Nielsen, M.G., Parri, S., Rivero, A.: Energetic costs of size and sexual signaling in a wolf spider. Proc. R. Soc. B 265, 2203–2209 (1998)

    Article  Google Scholar 

  32. Anderson, J.F., Prestwich, K.N.: Respiratory gas exchange in spiders. Physiol. Zool. 55, 72–90 (1982)

    Google Scholar 

  33. Blanchard, P., Devaney, L.R., Hall, R.G.: Differential Equations. Thomson Brooks/Cole (2006)

  34. Shone, R.: Economic Dynamics: Phase Diagrams and Their Economic Application, 2nd edn. Cambridge University Press, New York (2002)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Economics, University of Thessaly, 43 Korai str., 38333, Volos, Greece

    Amalia Gkana & Loukas Zachilas

Authors
  1. Amalia Gkana
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Loukas Zachilas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Loukas Zachilas.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gkana, A., Zachilas, L. Incorporating prey refuge in a prey–predator model with a Holling type I functional response: random dynamics and population outbreaks. J Biol Phys 39, 587–606 (2013). https://doi.org/10.1007/s10867-013-9319-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10867-013-9319-7

Keywords

Search

Navigation