Skip to main content
SpringerLink
Log in

Non-smooth dynamics emerging from predator-driven discontinuous prey dispersal

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In ecology, the refuge protection of the prey plays a significant role in the dynamics of the interactions between prey and predator. In this paper, we investigate the dynamics of a non-smooth prey–predator mathematical model characterized by density-dependent intermittent refuge protection of the prey. The model assumes the population density of the predator as an index for the prey to decide on when to avail or discontinue refuge protection, representing the level of apprehension of the prey by the predators. We apply Filippov’s regularization approach to study the model and obtain the sliding segment of the system. We obtain the criterion for the existence of the regular or virtual equilibria, boundary equilibrium, tangent points, and pseudo-equilibria of the Filippov system. The conditions for the visibility (or invisibility) of the tangent points are derived. We investigate the regular or virtual equilibrium bifurcation, boundary-node bifurcation and pseudo-saddle-node bifurcation. Further, we examine the effects of dispersal delay on the Filippov system associated with prey vigilance in identifying the predator population density. We observe that the hysteresis in the Filippov system produces stable limit cycles around the predator population density threshold in some bounded region in the phase plane. Moreover, we find that the level of apprehension and vigilance of the prey play a significant role in their refuge-dispersion dynamics.

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

Availability of data and material

Not applicable.

Code availability

Not applicable.

References

  1. Amo, L., López, P., Martín, J.: Refuge use: a conflict between avoiding predation and losing mass in lizards. Physiol. Behav. 90(2–3), 334–343 (2007)

    Article  Google Scholar 

  2. Beauchamp, G.: What can vigilance tell us about fear? Animal Sent. 2(15), 1 (2017)

    Google Scholar 

  3. Beauchamp, G.: External body temperature and vigilance to a lesser extent track variation in predation risk in domestic fowls. BMC Zool. 4(1), 1–8 (2019)

    Article  Google Scholar 

  4. Belgrad, B.A., Griffen, B.D.: Predator-prey interactions mediated by prey personality and predator hunting mode. Proc. R. Soc. B Biol. Sci. 283(1828), 20160408 (2016)

    Article  Google Scholar 

  5. Berezovskaya, F.S., Song, B., Castillo-Chavez, C.: Role of prey dispersal and refuges on predator-prey dynamics. SIAM J. Appl. Math. 70(6), 1821–1839 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bhattacharyya, J., Roelke, D.L., Pal, S., Banerjee, S.: Sliding mode dynamics on a prey-predator system with intermittent harvesting policy. Nonlinear Dyn. 98(2), 1299–1314 (2019)

    Article  Google Scholar 

  7. Bhattacharyya, J., Roelke, D.L., Walton, J.R., Banerjee, S.: Using \(yy\) supermales to destabilize invasive fish populations. Theor. Popul. Biol. 134, 1–14 (2020)

    Article  MATH  Google Scholar 

  8. Blake, C.A., Andersson, M.L., Hulthén, K., Nilsson, P.A., Brönmark, C.: Conspecific boldness and predator species determine predation-risk consequences of prey personality. Behav. Ecol. Sociobiol. 72(8), 133 (2018)

    Article  Google Scholar 

  9. Boukal, D.S., et al.: Lyapunov functions for Lotka–Volterra predator-prey models with optimal foraging behavior. J. Math. Biol. 39(6), 493–517 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brown, G.E., Rive, A.C., Ferrari, M.C., Chivers, D.P.: The dynamic nature of antipredator behavior: prey fish integrate threat-sensitive antipredator responses within background levels of predation risk. Behav. Ecol. Sociobiol. 61(1), 9–16 (2006)

    Article  Google Scholar 

  11. Carthey, A.J., Bucknall, M.P., Wierucka, K., Banks, P.B.: Novel predators emit novel cues: a mechanism for prey naivety towards alien predators. Sci. Rep. 7(1), 1–9 (2017)

    Article  Google Scholar 

  12. Chattopadhyay, J., Bairagi, N., Sarkar, R.: A predator-prey model with some cover on prey species. Nonlinear Phenom. Complex Syst. Minsk 3(4), 407–420 (2000)

    MathSciNet  Google Scholar 

  13. Chen, L., Chen, F., Chen, L.: Qualitative analysis of a predator-prey model with holling type ii functional response incorporating a constant prey refuge. Nonlinear Anal. Real World Appl. 11(1), 246–252 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chittka, L., Skorupski, P., Raine, N.E.: Speed-accuracy tradeoffs in animal decision making. Trends Ecol. Evolut. 24(7), 400–407 (2009)

    Article  Google Scholar 

  15. Choh, Y., Ignacio, M., Sabelis, M.W., Janssen, A.: Predator-prey role reversals, juvenile experience and adult antipredator behaviour. Sci. Rep. 2, 728 (2012)

    Article  Google Scholar 

  16. Cooper, W.E., Jr., Perez-Mellado, V.: Historical influence of predation pressure on escape by podarcis lizards in the Balearic islands. Biol. J. Lin. Soc. 107(2), 254–268 (2012)

    Article  Google Scholar 

  17. Donelan, S.C., Grabowski, J.H., Trussell, G.C.: Refuge quality impacts the strength of nonconsumptive effects on prey. Ecology 98(2), 403–411 (2017)

    Article  Google Scholar 

  18. Dowling, L.M., Godin, J.G.J.: Refuge use in a killifish: influence of body size and nutritional state. Can. J. Zool. 80(4), 782–788 (2002)

    Article  Google Scholar 

  19. Drakunov, S.V., Utkin, V.I.: Sliding mode control in dynamic systems. Int. J. Control 55(4), 1029–1037 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fardell, L.L., Pavey, C.R., Dickman, C.R.: Fear and stressing in predator-prey ecology: considering the twin stressors of predators and people on mammals. PeerJ 8, e9104 (2020)

    Article  Google Scholar 

  21. Feyten, L.E., Brown, G.E.: Ecological uncertainty influences vigilance as a marker of fear. Animal Sent. 2(15), 7 (2018)

    Google Scholar 

  22. Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides: Control Systems, vol. 18. Springer, Berlin (2013)

    Google Scholar 

  23. Filippova, T.: A note on the evolution property of the assembly of viable solutions to a differential inclusion. Comput. Math. Appl. 25(2), 115–121 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hauzy, C., Gauduchon, M., Hulot, F.D., Loreau, M.: Density-dependent dispersal and relative dispersal affect the stability of predator-prey metacommunities. J. Theor. Biol. 266(3), 458–469 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jana, D., Ray, S.: Impact of physical and behavioral prey refuge on the stability and bifurcation of Gause type Filippov prey-predator system. Model. Earth Syst. Environ. 2(1), 24 (2016)

    Article  Google Scholar 

  26. Jeffrey, M.R.: Dynamics at a switching intersection: hierarchy, isonomy, and multiple sliding. SIAM J. Appl. Dyn. Syst. 13(3), 1082–1105 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ji, L., Wu, C.: Qualitative analysis of a predator-prey model with constant-rate prey harvesting incorporating a constant prey refuge. Nonlinear Anal. Real World Appl. 11(4), 2285–2295 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kar, T.K.: Modelling and analysis of a harvested prey-predator system incorporating a prey refuge. J. Comput. Appl. Math. 185(1), 19–33 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kavaliers, M., Choleris, E.: Antipredator responses and defensive behavior: ecological and ethological approaches for the neurosciences. Neurosci. Biobehav. Rev. 25(7–8), 577–586 (2001)

    Article  Google Scholar 

  30. Křivan, V.: Behavioral refuges and predator-prey coexistence. J. Theor. Biol. 339, 112–121 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Manarul Haque, M., Sarwardi, S.: Dynamics of a harvested prey-predator model with prey refuge dependent on both species. Int. J. Bifurc. Chaos 28(12), 1830040 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Martin, R.A., Hammerschlag, N.: Marine predator-prey contests: ambush and speed versus vigilance and agility. Mar. Biol. Res. 8(1), 90–94 (2012)

    Article  Google Scholar 

  33. Mirza, R.S., Ferrari, M.C., Kiesecker, J.M., Chivers, D.P.: Responses of American toad tadpoles to predation cues: behavioural response thresholds, threat-sensitivity and acquired predation recognition. Behaviour 143(7), 877–889 (2006)

    Article  Google Scholar 

  34. Perko, L.: Differential Equations and Dynamical Systems, vol. 7. Springer, Berlin (2013)

    MATH  Google Scholar 

  35. Reaney, L.T.: Foraging and mating opportunities influence refuge use in the fiddler crab, uca mjoebergi. Anim. Behav. 73(4), 711–716 (2007)

    Article  Google Scholar 

  36. Ruxton, G.: Short term refuge use and stability of predator-prey models. Theor. Popul. Biol. 47(1), 1–17 (1995)

    Article  MATH  Google Scholar 

  37. Sih, A.: Prey refuges and predator-prey stability. Theor. Popul. Biol. 31(1), 1–12 (1987)

    Article  MathSciNet  Google Scholar 

  38. Sih, A., Cote, J., Evans, M., Fogarty, S., Pruitt, J.: Ecological implications of behavioural syndromes. Ecol. Lett. 15(3), 278–289 (2012)

    Article  Google Scholar 

  39. Tang, G., Tang, S., Cheke, R.A.: Global analysis of a holling type ii predator-prey model with a constant prey refuge. Nonlinear Dyn. 76(1), 635–647 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. Utkin, V.I.: Sliding Modes in Control and Optimization. Springer, Berlin (2013)

    Google Scholar 

  41. Welp, T., Rushen, J., Kramer, D., Festa-Bianchet, M., De Passille, A.: Vigilance as a measure of fear in dairy cattle. Appl. Anim. Behav. Sci. 87(1–2), 1–13 (2004)

    Article  Google Scholar 

  42. Zhou, Y., Sun, W., Song, Y., Zheng, Z., Lu, J., Chen, S.: Hopf bifurcation analysis of a predator-prey model with holling-ii type functional response and a prey refuge. Nonlinear Dyn. 97(2), 1439–1450 (2019)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

JB is supported by the Grants from Science and Engineering Research Board (SERB), Govt. of India (File No. TAR/2018/000283).

Funding

SERB, India (File No. TAR/2018/000283).

Author information

Authors and Affiliations

  1. Department Mathematics, Karimpur Pannadevi College, Nadia, West Bengal, 741152, India

    Joydeb Bhattacharyya

  2. Agricultural and Ecological Research Unit, Indian Statistical Institute, Kolkata, West Bengal, 700108, India

    Joydev Chattopadhyay

Authors
  1. Joydeb Bhattacharyya
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joydev Chattopadhyay
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joydeb Bhattacharyya.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethics approval

Not applicable.

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

Bhattacharyya, J., Chattopadhyay, J. Non-smooth dynamics emerging from predator-driven discontinuous prey dispersal. Nonlinear Dyn 106, 3647–3668 (2021). https://doi.org/10.1007/s11071-021-06963-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-06963-6

Keywords

Mathematics Subject Classification

Search

Navigation