Skip to main content
SpringerLink
Log in

Dynamics of a Prey–Predator System with Herd Behaviour in Both and Strong Allee Effect in Prey

  • COMPLEX SYSTEMS BIOPHYSICS
  • Published:
Biophysics Aims and scope Submit manuscript

Abstract

This paper mainly deals with the prey−predator dynamics where both the prey and predator exhibit herd behavior. Positivity, boundedness, some extinction criteria, stability of possible equilibrium points are discussed with some global behavior, in an innovative approach. Numerical simulations are represented to clarify the analytical discussions. The biological implications of analytical and numerical findings are discussed separately for the purpose of maintaining ecological balance in nature. At the end, the possibilities of future work related to this topic are represented.

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.

Similar content being viewed by others

REFERENCES

  1. H. Malchow, S. Petrovskii, and E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology (CRC, Boca Raton, 2008).

    MATH  Google Scholar 

  2. J. D. Murray, Mathematical Biology (Springer-Verlag, New York, 1989).

    Book  MATH  Google Scholar 

  3. V. Volterra and U. D’Ancona, in VII Congr. Int. Acqui. et de Peche (Paris, 1931), pp. 1–14.

  4. A. J. Lotka, Elements of Mathematical Biology (Dover, New York, 1956).

    MATH  Google Scholar 

  5. C. S. Holling, Mem. Entomol. Soc. Can. 45, 3 (1965).

    Google Scholar 

  6. J. T. Tanner, Ecology 56, 855 (1975).

    Article  Google Scholar 

  7. P. A. Braza, SIAM J. Appl. Math. 63 (3), 889 (2003).

    Article  MathSciNet  Google Scholar 

  8. S. B. Hsu, T. W. Hwang, and Y. Kuang, J. Math. Biol. 42, 490 (2001).

    Article  Google Scholar 

  9. S. B. Hsu, T. W. Hwang, and Y. Kuang, Math. Biosci. 181, 55 (2003).

    Article  MathSciNet  Google Scholar 

  10. P. H. Leslie and J. C. Gower, Biometrika 47, 219 (1960).

    Article  MathSciNet  Google Scholar 

  11. H. R. Akcakaya, Ecol. Monogr. 62, 119 (1992).

    Article  Google Scholar 

  12. P. A. Abrams, Ecology 75 (6), 1842 (2003).

    Article  Google Scholar 

  13. P. A. Abrams and L. R. Ginzburg, Trends Ecol. Evol. 15, 337 (2000).

    Article  Google Scholar 

  14. T. Matsuoka and H. Seno, Math. Model. Nat. Phenom. 3 (4), 131 (2008).

    Article  MathSciNet  Google Scholar 

  15. H. I. Freedman and G. Wolkowitz, Bull. Math. Biol. 48, 493 (1986).

    Article  MathSciNet  Google Scholar 

  16. A. Maiti, P. Sen, D. Manna, and G. P. Samanta, Nonlinear Dyn. Syst. Theory 16 (1), 86 (2016).

    MathSciNet  Google Scholar 

  17. A. Maiti, R. Paul, and S. Alam, J. Stat. Math. Eng. 2 (1), 1 (2016).

    Google Scholar 

  18. A. Maiti, R. Paul, and S. Alam, Int. J. Res. Eng. Technol. 5 (9), 233 (2016).

    Google Scholar 

  19. A. Maiti, R. Paul, and S. Alam, Int. J. Res. Publ. Eng. Technol. 2 (9), 22 (2016).

    Google Scholar 

  20. L. Berec, E. Angulo, and F. Courchamp, Trends Ecol. Evol. 22, 185 (2006).

    Article  Google Scholar 

  21. T. H. Clutton-Brock, D. Gaynor, G. M. Mcllrath, et al., J. Anim. Ecol. 68, 672 (1999).

    Article  Google Scholar 

  22. F. Courchamp, T. Clutton-Brock, and B. Grenfell, Trends Ecol. Evol. 14, 405 (1999).

    Article  Google Scholar 

  23. F. Courchamp, L. Berec, and J. Gascoigne, Allee Effects in Ecology and Conservation (Oxford Univ. Press, Oxford, 2008).

    Book  Google Scholar 

  24. M. S. Mooring, T. A. Fitzpatrick, T. T. Nishihira, and D. D. Reisig, J. Wildl. Manag. 68, 519 (2004).

    Article  Google Scholar 

  25. D. J. Rinella, M. S. Wipfli, C. A. Stricker, et al., Canad. J. Fish. Aquat. Sci. 69, 73 (2012).

    Article  Google Scholar 

  26. P. A. Stephens, W. J. Sutherland, and R. P. Freckleton, Oikos 87, 185 (1999).

    Article  Google Scholar 

  27. M. Kot, Elements of Mathematical Biology (Cambridge Univ. Press, Cambridge, 2001).

    Google Scholar 

  28. G. A. K. van Voorn, L. Hemerik, M. P. Boer, and B. W. Kooi, Math. Biosci. 209, 451 (2007).

    Article  MathSciNet  Google Scholar 

  29. M. H. Wang and M. Kot, Math. Biosci. 171, 83 (2001).

    Article  MathSciNet  Google Scholar 

  30. J. Wang, J. Shi, and J. Wei, J. Math. Biol. 62, 291 (2011).

    Article  MathSciNet  Google Scholar 

  31. C. W. Clark, Mathematical Bioeconomic: The Optimal Management of Renewable Resources (Wiley, New York, 1990).

    MATH  Google Scholar 

  32. C. W. Clark, The Worldwide Crisis in Fisheries: Economic Models and Human Behavior (Cambridge Univ. Press, Cambridge, 2007).

    Book  Google Scholar 

  33. M. Liermann and R. Hilborn, Fish Fish. 2, 33 (2001).

    Article  Google Scholar 

  34. G. Wang, X. G. Liang, and F. Z. Wang, Ecol. Model. 124, 183 (1999).

    Article  Google Scholar 

  35. A. D. Bazykin, F. S. Berezovskaya, A. S. Isaev, and R. G. Khlebopros, J. Theor. Biol. 186, 267 (1997).

    Article  Google Scholar 

  36. E. D. Conway and J. A. Smoller, SIAM J. Appl. Math. 46, 630 (1986).

    Article  MathSciNet  Google Scholar 

  37. J. D. Flores, J. Mena-Lorca, B. Gonźalez-Yaňez, and E. Gonźalez-Olivares, in Proc. Int. Symp. on Mathematical and Computational Biology, ed. by R. Mondaini and R. Dilao (E-Papers Servios Editoriais, 2007), pp. 219–232.

  38. E. Gonźalez-Olivares and A. Rojas-Palma, Bull. Math. Biol. 73, 1378 (2011).

    Article  MathSciNet  Google Scholar 

  39. P. Aguirre, E. Gonźalez-Olivares, and E. Saez, Nonlinear Anal. RWA 10, 1401 (2009).

    Article  Google Scholar 

  40. P. Agiurre, E.Gonźalez-Olivares, and E. Saez, SIAM J. Appl. Math. 69, 1244 (2009).

    Article  MathSciNet  Google Scholar 

  41. C. S. Holling, Can. Entomol. 91, 385 (1959).

    Article  Google Scholar 

  42. A. Hastings and T. Powell, Ecology 72, 896 (1991).

    Article  Google Scholar 

  43. S. Gakkhar and A. Shing, J. Math. Anal. Appl. 385, 423 (2012).

    Article  MathSciNet  Google Scholar 

  44. T. V. Ton and A. Yagi, Commun. Stochast. Anal. 2, 371 (2011).

    Google Scholar 

  45. D. Xiao, W. Li, and M. Han, J. Math. Anal. Appl. 324, 14 (2006).

    Article  MathSciNet  Google Scholar 

  46. B. Mukhopadhyay and R. Bhattacharyya, Nonlinear Anal. Model. Control 16 (1), 77 (2011).

    Article  MathSciNet  Google Scholar 

  47. C. S. Holling, Can. Entomol. 91, 293 (1959).

    Article  Google Scholar 

  48. A. Maiti and G. P. Samanta, J. Math. Ed. Sci. Tech. 36, 65 (2005).

    Article  Google Scholar 

  49. S. Ruan and D. Xiao, SIAM J. Appl. Math. 61, 1445 (2001).

    Article  Google Scholar 

  50. J. D. Murray, Mathematical Biology (Springer-Verlag, New York, 1993).

    Book  MATH  Google Scholar 

  51. C. Cosner, D. L. DeAngelis, J. S. Ault, and D. E. Olson, Theor. Pop. Biol. 56, 65 (1999).

    Article  Google Scholar 

  52. J. Chattopadhyay, S. Chatterjee, and E. Venturino, J. Theor. Biol. 253, 289 (2008).

    Article  Google Scholar 

  53. V. Ajraldi, M. Pittavino, and E. Venturino, Nonlinear Anal. RWA 12, 2319 (2011).

    Article  Google Scholar 

  54. A. P. Braza, Nonlinear Anal. RWA 13, 1837 (2012).

    Article  Google Scholar 

  55. S. P. Bera, A. Maiti, and G. P. Samanta, World J. Model. Simulation 11, 3 (2015).

    Google Scholar 

  56. S. P. Bera, A. Maiti, and G. P. Samanta, Nonlinear Anal. Model. Control 21 (3), 345 (2016).

    Article  MathSciNet  Google Scholar 

  57. D. K. Arrowsmith and C. M. Place, Dynamical Systems: Differential Equations, Maps, and Chaotic Behaviour (Chapman & Hall, London, 1992).

    Book  MATH  Google Scholar 

  58. B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Application of Hopf-Bifurcation (Cambridge Univ. Press, Cambridge, 1981).

    MATH  Google Scholar 

  59. M. R. Garvie, Bull. Math. Biol. 69 (3), 931 (2007).

    Article  MathSciNet  Google Scholar 

  60. L. N. Guin and P. K. Mandal, Appl. Math. Model. 38 (17–18), 4417 (2014).

  61. L. N. Guin and P. K. Mandal, Int. J. Biomath. 7 (5), 1450047 (2014).

    Article  MathSciNet  Google Scholar 

  62. L. N. Guin, Appl. Math. Comput. 226, 320 (2014).

    MathSciNet  Google Scholar 

  63. L. N. Guin, M. Haque, and P. K. Mandal, Appl. Math. Model. 36 (5), 1825 (2012).

    Article  MathSciNet  Google Scholar 

  64. L. N. Guin, S. Chakravarty, and P. K. Mandal, Nonlinear Anal. Model. Control 20 (4), 509 (2015).

    Article  MathSciNet  Google Scholar 

  65. L. N. Guin and H. Baek, Math. Comput. Simulation 146, 100 (2018).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology-Puducherry, 609609, Karaikal, India

    S. Biswas & G. S. Mahapatra

  2. Chandrahati Dilip Kumar High School (H.S.), Chandrahati, 712504, West Bengal, India

    D. Pal

  3. Department of Mathematics, Indian Institute of Engineering Science and Technology, 711103, Shibpur, Howrah, India

    G. P. Samanta

Authors
  1. S. Biswas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Pal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. S. Mahapatra
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. G. P. Samanta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to S. Biswas, D. Pal, G. S. Mahapatra or G. P. Samanta.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Biswas, S., Pal, D., Mahapatra, G.S. et al. Dynamics of a Prey–Predator System with Herd Behaviour in Both and Strong Allee Effect in Prey. BIOPHYSICS 65, 826–835 (2020). https://doi.org/10.1134/S0006350920050036

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0006350920050036

Keywords:

Search

Navigation