Skip to main content
SpringerLink
Log in

Mathematical Study About a Predator–Prey Model with Anti-predator Behavior

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

Species use their defense mechanisms to fight, kill or escape from predators. Each species shows its unique defense mechanism to avoid the predation. In this paper, we propose and analyze a three species predator–prey system with group defense mechanism. Conditions for boundedness and positivity of equilibrium points have been discussed. We establish conditions for stability of positive equilibrium points and periodic solutions via Hopf-bifurcation. The predator–prey system is numerically studied with the help of phase portrait, time evolution and bifurcation diagrams. Lyapunov exponent and sensitivity analysis ensure the chaotic behaviour of the model system. Period doubling and period halving bifurcations show dynamical complexities of the food chain system. This study suggests that the repercussion of better group defense by the intermediate predator is able to produce chaos in food chain models.

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
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Tenar, J.S.: Muskoxen. Queens Printer, Ottawa (1965)

    Google Scholar 

  2. Holmes, J.C., Bethel, W.M.: Modification of intermediate host behaviour by parasite. Zool. J. Linn. Soc. 51, 123–149 (1972)

    Google Scholar 

  3. Edmunds, M.: Defense in Animals: A Survey of Anti-predator Defenses. Longman, Harlow (1974)

    Google Scholar 

  4. Olive, C.W.: Foraging specialization in orb-weaving spiders. Ecology 61, 1133–1144 (1980)

    Article  Google Scholar 

  5. Pearlman, Y., Tsurim, T.: Daring, risk assessment and body condition interactions in steppe buzzards Buteo buteo vulpinus. Avian Biol. 39, 226–228 (2008)

    Article  Google Scholar 

  6. Swaffer, S.M., O’Brien, W.J.: Spines of Daphnia lumholtzi create feeding difficulties for juvenile bluegill sunfish (Lepomis macrochirus). J. Plankton Res. 18, 1055–1061 (1996)

    Article  Google Scholar 

  7. Freedman, H.I.: Deterministic Mathematical Model in Population Ecology. Monographs and Textbooks in Pure and Applied Mathematics, vol. 57. Marcel Dekkar, New York (1980)

    MATH  Google Scholar 

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

    Article  Google Scholar 

  9. Andrews, J.F.: A mathematical model for continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol. Bioeng. 10, 707–723 (2012)

    Article  Google Scholar 

  10. Collings, J.B.: The effects of the functional response on the bifurcation behavior of a mite predator–prey interaction model. J. Math. Biol. 36, 149–168 (1997)

    Article  MathSciNet  Google Scholar 

  11. Ruan, S., Xiao, D.: Global analysis in a predator–prey system with non-monotonic functional response. SIAM J. Appl. Math. 61, 1445–1472 (2001)

    Article  Google Scholar 

  12. Tayor, R.J.: Predation. Chapman and Hall, New York (1984)

    Google Scholar 

  13. Liznarova, A., Pekar, A.: Dangerous prey is associated with a type 4 functional response in spiders. Anim. Behav. 85, 1183–1190 (2013)

    Article  Google Scholar 

  14. Sokol, W., Howell, J.A.: Kinetics of phenol oxidation by washed cell. Biotechnol. Bioeng. 23, 2039–2049 (1980)

    Article  Google Scholar 

  15. Lotka, A.J.: Elements of Physical Biology. Williamsand Wilkins, Baltimore (1924)

    MATH  Google Scholar 

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

    Article  Google Scholar 

  17. Hasting, A., Powell, T.: Chaos in a three-species food chain. Ecology 72, 896–903 (1991)

    Article  Google Scholar 

  18. Huang, J., Xiao, D.: Analyses of bifurcations and stability in a predator–prey system with Holling type-IV functional response. Acta Math. Appl. Sin. Engl. Ser. 20, 167–178 (2004)

    Article  MathSciNet  Google Scholar 

  19. Pal, R., Basub, D., Banerjee, M.: Modelling of phytoplankton allelopathy with Monod–Haldane type functional response—a mathematical study. BioSystems 95, 243–253 (2009)

    Article  Google Scholar 

  20. Pei, Y.Z., De’Aguiar, M.M.M.: Complex dynamics of one-prey multi-predator system with defensive ability of prey and impulsive biological control on predators. Adv. Comput. Syst. 8, 483–495 (2005)

    Article  MathSciNet  Google Scholar 

  21. Upadhyay, R.K., Rai, V.: Chaotic population dynamics and biology of the top predator. Chaos Solitons Fractals 21, 1195–1204 (2004)

    Article  MathSciNet  Google Scholar 

  22. Freedman, H.I.: Hopf bifurcation in three species food chain models with group defense. Math. Biol. 111, 73–87 (1992)

    MathSciNet  MATH  Google Scholar 

  23. Wolkowicz, G.S.K.: Bifurcation analysis of a predator–prey system involving group defense. SIAM J. Appl. Math. 48, 592–606 (1988)

    Article  MathSciNet  Google Scholar 

  24. Xiao, D., Zhu, H.: Multiple focus and Hopf bifurcations in a predator–prey system with non-monotonic functional response. SIAM J. Appl. Math. 66, 802–819 (2006)

    Article  MathSciNet  Google Scholar 

  25. Raw, S.N., Mishra, P.: Modeling and analysis of inhibitory effect in plankton-fish model: application to the hypertrophic Swarzedzkie Lake in Western Poland. Nonlinear Anal. Real World Appl. 46, 465–492 (2019)

    Article  MathSciNet  Google Scholar 

  26. Zhu, H., Campbell, S.A., Wolkowicz, G.S.K.: Bifurcation analysis of a predator–prey system with non-monotonic functional response. SIAM J. Appl. Math. 63, 636–682 (2006)

    Article  Google Scholar 

  27. Freedman, H.I., Hongshun, Q.: Interaction leading to persistence in predator–prey systems with group defense. Bull. Math. Biol. 50, 517–530 (1988)

    Article  MathSciNet  Google Scholar 

  28. Roy, S., Chattopadhyay, J.: Towards a resolution of the paradox of the plankton: a brief overview of the proposed mechanisms. Ecol. Complex. 4, 26–33 (2007)

    Article  Google Scholar 

  29. Li, S.: Complex dynamical behaviors in a predator–prey system with generalized group defense and impulsive control strategy. Discrete Dyn. Nat. Soc. 2013, 881–892 (2013)

    MathSciNet  Google Scholar 

  30. Raw, S.N., Mishra, P., Kumar, R., Thakur, S.: Complex behavior of prey–predator system exhibiting group defense: a mathematical modeling study. Chaos Solitons Fractals 100, 74–90 (2017)

    Article  MathSciNet  Google Scholar 

  31. Mishra, P., Raw, S.N., Tiwari, B.: Study of a Leslie–Gower predator–prey model with prey defense and mutual interference of predators. Chaos Solitons Fractals 120, 1–16 (2019)

    Article  MathSciNet  Google Scholar 

  32. Klebanoff, A., Hastings, A.: Chaos in three species food chains. J. Math. Biol. 32, 427–451 (1994)

    Article  MathSciNet  Google Scholar 

  33. Upadhyay, R.K., Raw, S.N., Rai, V.: Restoration and recovery of damaged eco-epidemiological systems: application to the Salton Sea, California USA. Math. Biol. 242, 172–187 (2013)

    MathSciNet  MATH  Google Scholar 

  34. Li, M., Chen, B., Ye, H.: A bio-economic differential algebraic predator–prey model with nonlinear prey harvesting. Appl. Math. Model. 42, 17–28 (2017)

    Article  MathSciNet  Google Scholar 

  35. Gakhad, S., Naji, R.K.: Order and chaos in a food web consisting of a predator and two independent preys. Commun. Nonlinear Sci. Numer. Simul. 10, 105–120 (2005)

    Article  MathSciNet  Google Scholar 

  36. Bairagi, N.: Age-structured predator–prey model with habitat complexity: oscillations and control. Dyn. Syst. Int. J. 27(4), 475–499 (2012)

    Article  MathSciNet  Google Scholar 

  37. Li, S., Xiong, Z., Wang, X.: The study of a predator–prey system with group defense and impulsive control strategy. Appl. Math. Model. 34, 2546–2561 (2014)

    Article  MathSciNet  Google Scholar 

  38. Banerjee, M., Venturino, E.: A phytoplankton-toxic phytoplankton–zooplankton model. Ecol. Complex. 8(3), 239–248 (2011)

    Article  Google Scholar 

  39. Pal, N., Samanta, S., Rana, S.: The impact of constant immigration on a tri-trophic food chain model. Int. J. Appl. Comput. Math. 3(4), 3615–3644 (2017)

    Article  MathSciNet  Google Scholar 

  40. Birkhoff, G., Rota, G.C.: Ordinary Differential Equations. Ginn, Boston (1982)

    MATH  Google Scholar 

Download references

Acknowledgements

This work is supported to the corresponding author (SNR) by Science and Engineering Research Board (SERB), Department of Science and Technology, New Delhi, with a Grant (File No. ECR/2017/000141), under Early Career Research (ECR) Award.

Author information

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Raipur, Raipur, CG, 492010, India

    S. N. Raw, P. Mishra & B. Tiwari

Authors
  1. S. N. Raw
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Mishra
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Tiwari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. N. Raw.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Raw, S.N., Mishra, P. & Tiwari, B. Mathematical Study About a Predator–Prey Model with Anti-predator Behavior. Int. J. Appl. Comput. Math 6, 68 (2020). https://doi.org/10.1007/s40819-020-00822-5

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s40819-020-00822-5

Keywords

Mathematics Subject Classification

Search

Navigation