Skip to main content
SpringerLink
Log in

Dynamics of Predator–Prey Model Based on Fear Effect with Bifurcation Analysis and Chaos Control

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

Recent studies on terrestrial vertebrates have unveiled a correlation between fear of predators and a reduction in anti-predator defenses, leading to a significant decline in prey reproduction. Given the critical importance of prey population levels, we have introduced a fear factor into a discrete-time predator–prey model to investigate its influence on the complex and dynamic behavior of the system This research primarily focuses on assessing the local stability of both trivial and boundary equilibrium points and exploring the criteria for Neimark–Sacker and period-doubling bifurcations. Moreover, we have applied chaos control techniques to identify the stability region and corroborated our theoretical findings through experimental data. Unlike previous research, our study incorporates chaos control methodology and experimental data validation. Consequently, it contributes significantly to the comprehension of predator–prey interactions by incorporating the influence of fear on prey reproduction. Furthermore, this research introduces novelty and reliability for future investigations by integrating chaos control techniques and validating the findings using experimental data.

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

References

  1. Lotka, A.J.: Elements of Physical Biology. Williamsand Wilkins, New York (1925)

    Google Scholar 

  2. Volterra, V.: Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Societá anonima tipografica Leonardo da Vinci (1927)

  3. Creel, S.D., Christianson, D.: Relationships between direct predation and risk effects. Trends Ecol. Evol. 23(4), 194–201 (2008)

    Google Scholar 

  4. Wang, X., Zanette, L., Zou, X.: Modelling the fear effect in predator-prey interactions. J. Math. Biol. 73(5), 1179–1204 (2016)

    MathSciNet  Google Scholar 

  5. Cresswell, W.: Predation in bird populations. J. Ornithol. 152(1), 251–263 (2011)

    Google Scholar 

  6. Abramsky, Z., Rosenzweig, M.L., Subach, A.: The costs of apprehensive foraging. Ecology 83(5), 1330–1340 (2002)

    Google Scholar 

  7. Elgar, M.A.: Predator vigilance and group size in mammals and birds: a critical review of the empirical evidence. Biol. Rev. 64(1), 13–33 (1989)

    Google Scholar 

  8. Preisser, E.L., Bolnick, D.I.: The many faces of fear: comparing the pathways and impacts of nonconsumptive predator effects on prey populations. PLoS ONE 3(6), 24–65 (2008)

    Google Scholar 

  9. Creel, S., Christianson, D., Liley, S., Winnie, J.A.: Predation risk affects reproductive physiology and demography of elk. Science 315(5814), 960–960 (2007)

    Google Scholar 

  10. Hik, D.S.: Does risk of predation influence population dynamics? Evidence from cyclic decline of snowshoe hares. Wildl. Res. 22(1), 115–129 (1995)

    Google Scholar 

  11. Pierce, B.M., Bowyer, R.T., Bleich, V.C.: Habitat selection by mule deer: forage benefits or risk of predation. J. Wildl. Manag. 68(3), 533–541 (2004)

    Google Scholar 

  12. Partridge, B.L., Johansson, J., Kalish, J.: The structure of schools of giant bluefin tuna in Cape Cod Bay. Environ. Biol. Fishes 9, 253–262 (1983)

    Google Scholar 

  13. Zanette, L.Y., White, A.F., Allen, M.C., Clinchy, M.: Perceived predation risk reduces the number of offspring songbirds produce per year. Science 334(6061), 1398–1401 (2011)

    Google Scholar 

  14. Sasmal, S.K., Takeuchi, Y.: Dynamics of a predator-prey system with fear and group defense. J. Math. Anal. 481(1), 123–471 (2020)

    MathSciNet  Google Scholar 

  15. Panday, P., Pal, N., Samanta, S., Chattopadhyay, J.: Stability and bifurcation analysis of a three-species food chain model with fear. Int. J. Bifurcat. Chaos. 28(1), (2018)

  16. Liu, J., Lv, P., Liu, B., Zhang, T.: Dynamics of a predator–prey model with fear effect and time delay. Complexity. 2021 (2021)

  17. Mahapatra, G.S., Santra, P.K., Bonyah, E.: Dynamics on effect of prey refuge proportional to predator in discrete-time prey-predator model. Complexity 2021, 1–12 (2021)

    Google Scholar 

  18. Hossain, M., Pal, S., Tiwari, P., Pal, N.: Bifurcations, chaos, and multistability in a nonautonomous predator-prey model with fear. Chaos 31(12), 123–134 (2021)

    MathSciNet  Google Scholar 

  19. Santra, P.K., Mahapatra, G.S.: Dynamical study of discrete-time prey-predator model with constant prey refuge under imprecise biological parameters. J. Biol. Syst. 28(03), 681–699 (2020)

    MathSciNet  Google Scholar 

  20. Barman, D., Roy, J., Alrabaiah, H., Panja, P., Mondal, S.P., Alam, S.: Impact of predator incited fear and prey refuge in a fractional order prey predator model. Chaos Solitons Fractals 142, 110–420 (2021)

    MathSciNet  Google Scholar 

  21. Roy, J., Barman, D., Alam, S.: Role of fear in a predator-prey system with ratio-dependent functional response in deterministic and stochastic environment. Biosystems 197, 104–176 (2020)

    Google Scholar 

  22. Duarte, J., Januário, C., Martins, N., Sardanyés, J.: Chaos and crises in a model for cooperative hunting: a symbolic dynamics approach. Chaos 19(4), 43–102 (2009)

    Google Scholar 

  23. Pal, S., Pal, N., Samanta, S., Chattopadhyay, J.: Effect of hunting cooperation and fear in a predator-prey model. Ecol. Complex. 39, 100–770 (2019)

    Google Scholar 

  24. Arditi, R., Perrin, N., Saïah, H.: Functional responses and heterogeneities: an experimental test with cladocerans. Oikos 60(1), 69–75 (1991)

    Google Scholar 

  25. Zhang, H., Cai, Y., Fu, S., Wang, W.: Impact of the fear effect in a prey-predator model incorporating a prey refuge. Appl. Math. Comput. 356, 328–337 (2019)

    MathSciNet  Google Scholar 

  26. Sih, A.: Optimal behaviour: can foragers balance two conflicting demands. Science 210(4473), 1041–1043 (1980)

    Google Scholar 

  27. Gilliam, J.F., Fraser, D.F.: Habitat selection under predation hazard: test of a model with foraging minnows. Ecology 68(6), 1856–1862 (1987)

    Google Scholar 

  28. Lima, S.L., Dill, L.M.: Behavioural decisions made under the risk of predation: a review and prospectus. Can. J. Zool. 68(4), 619–640 (1990)

    Google Scholar 

  29. Santra, P.K., Mahapatra, G.S., Phaijoo, G.R.: Bifurcation analysis and chaos control of discrete prey-predator model incorporating novel prey-refuge concept. Comput. Math. Methods 3(6), 1–17 (2021)

    MathSciNet  Google Scholar 

  30. Wang, J., Cai, Y., Fu, S., Wang, W.: The effect of the fear factor on the dynamics of a predator-prey model incorporating the prey refuge. Chaos 29(8), 83–109 (2019)

    MathSciNet  Google Scholar 

  31. Hossain, M., Pal, S. Samanta, S., Chattopadhyay, J.: Fear induced stabilization in an intraguild predation model. Int. J. Bifurc. Chaos 30(4), (2020)

  32. Qiao, T., Cai, Y., Fu, S., Wang, W.: Stability and Hopf bifurcation in a predator-prey model with the cost of anti-predator behaviors. Int. J. Bifurc. Chaos 29(13), 1950185 (2019)

    MathSciNet  Google Scholar 

  33. Sasmal, S.K.: Population dynamics with multiple Allee effects induced by fear factors–a mathematical study on prey-predator interactions. Appl. Math. Model. 64, 1–14 (2018)

    MathSciNet  Google Scholar 

  34. Upadhyay, R.K., Mishra, S.: Population dynamic consequences of fearful prey in a spatiotemporal predator-prey system. Math. Biosci. Eng. 16(1), 338–372 (2018)

    MathSciNet  Google Scholar 

  35. Allen, M.C., Clinchy, M., Zanette, L.Y.: Fear of predators in free-living wildlife reduces population growth over generations. Proc. Natl. Acad. Sci. USA 119(7), (2022)

  36. Agarwal, R.P., El-Sayed, A.M.A., Salman, S.M.: Fractional-order Chaos system: discretization, bifurcation and chaos. Adv. Differ. Equ. 2013(1), 320 (2013)

    Google Scholar 

  37. Yuan, L.G., Kuang, J.H.: Stability and a numerical solution of fractional-order Brusselator chemical reaction system. J. Fract. Calc. Appl. 8(1), 38–47 (2017)

    MathSciNet  Google Scholar 

  38. Pal, S., Pal, N., Chattopadhyay, J.: Hunting cooperation in a discrete-time predator-prey system. Int. J. Bifurc. Chaos 28(07), 1850083 (2018)

    MathSciNet  Google Scholar 

  39. He, Z., Lai, X.: Bifurcation and chaotic behaviour of a discrete-time predator-prey system. Nonlinear Anal. Real World Appl. 12, 403–417 (2011)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  41. Li, B., He, Z.: Bifurcations and chaos in a two-dimensional discrete Hindmarsh-Rose model. Nonlinear Dyn. 76(1), 697–715 (2014)

    MathSciNet  Google Scholar 

  42. Chen, Q., Teng, Z., Hu, Z.: Bifurcation and control for a discrete-time prey-predator model with Holling-IV functional response. Int. J. Appl. Math. Comput. Sci. 23(2), 247–261 (2013)

    MathSciNet  Google Scholar 

  43. Cheng, L., Cao, H.: Bifurcation analysis of a discrete-time ratio-dependent predator-prey model with Allee effect. Commun. Nonlinear Sci. Numer. Simul. 38, 288–302 (2016)

    MathSciNet  Google Scholar 

  44. Ishaque, W., Din, Q., Taj, M.: Mutual interference and its effects on searching efficiency between predator-prey interaction with bifurcation analysis and chaos control. J. Vib. Control 29(5–6), 1031–1045 (2023)

    MathSciNet  Google Scholar 

  45. Din, Q., Ishaque, W., Iqbal, M.A., Saeed, U.: Modification of Nicholson-Bailey model under refuge effects with stability, bifurcation, and chaos control. J. Vib. Control 28(23–24), 3524–3538 (2022)

    MathSciNet  Google Scholar 

  46. Jing, Z., Yang, J.: Bifurcation and chaos in discrete-time predator-prey system. Chaos Solitons Fractals 27(1), 259–277 (2006)

    MathSciNet  Google Scholar 

  47. Marotto, F.R.: Snap-back repellers imply chaos in \(R^{n}\). J. Math. Anal. 63(1), 199–223 (1978)

    MathSciNet  Google Scholar 

  48. Marotto, F.R.: On redefining a snap-back repeller. Chaos Solitons Fractals 25(1), 25–28 (2005)

    MathSciNet  Google Scholar 

  49. Chen, G., Fang, J., Hong, Y., Qin, H.: Controlling Hopf bifurcations: discrete-time systems. Discrete Dyn. Nat. Soc. 5, 29–33 (2000)

    Google Scholar 

  50. Chen, G., Yu, X.: On time-delayed feedback control of chaotic systems. IEEE Trans. Circuits Syst. 46(6), 767–772 (1999)

    MathSciNet  Google Scholar 

  51. Wen, G.L., Xu, D.L., Xie, J.H.: Controlling Hopf bifurcations of discrete-time systems in resonance. Chaos Solitons Fractals 23(5), 1865–1877 (2005)

    MathSciNet  Google Scholar 

  52. Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64(11), 1196–1199 (1990)

    MathSciNet  Google Scholar 

  53. Romeiras, F.J., Grebogi, C., Ott, E., Dayawansa, W.P.: Controlling chaotic dynamical systems. Physica D 58(1–4), 165–192 (1992)

    MathSciNet  Google Scholar 

  54. Ogata, K.: Modern Control Engineering, 2nd edn. Prentice-Hall, New Jersey (2010)

    Google Scholar 

  55. Hi, X., Li, C., Pan, X., Peng, M.: Impulsive control and Hopf bifurcation of a three-dimensional chaotic system. J. Vib. Control 20(9), 1361–1368 (2014)

    MathSciNet  Google Scholar 

  56. Ghandchi-Tehrani, M., Wilmshurst, L.I., Elliott, S.J.: Bifurcation control of a Duffing oscillator using pole placement. J. Vib. Control 21(14), 2838–2851 (2015)

    Google Scholar 

  57. Mobayen, S., Baleanu, D., Tchier, F.: Second-order fast terminal sliding mode control design based on LMI for a class of non-linear uncertain systems and its application to chaotic systems. J. Vib. Control 23(18), 2912–2925 (2017)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank the main editor and anonymous referees for their valuable comments and suggestions leading to improvement of this paper. This research work was funded by Higher Education Commission (HEC) Pakistan under NRPU Project No. 20–16985/NRPU/R &D/HEC/2021.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Poonch Rawalakot, Rawalakot, 12350, Pakistan

    Waqas Ishaque & Qamar Din

  2. Department of Mathematics, University of Sargodha, Sargodha, Pakistan

    Khuram Ali Khan

  3. Department of Maths and Computer Science, Faculty of Science, University of Kinshasa, Kinshasa, Democratic Republic of the Congo

    Rostin Matendo Mabela

Authors
  1. Waqas Ishaque
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qamar Din
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Khuram Ali Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Rostin Matendo Mabela
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors reviewed the manuscript.

Corresponding author

Correspondence to Waqas Ishaque.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ishaque, W., Din, Q., Khan, K.A. et al. Dynamics of Predator–Prey Model Based on Fear Effect with Bifurcation Analysis and Chaos Control. Qual. Theory Dyn. Syst. 23, 26 (2024). https://doi.org/10.1007/s12346-023-00878-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-023-00878-w

Keywords

Mathematics Subject Classification

Search

Navigation