Advertisement

Computational dynamics of the Nicholson-Bailey models

  • Sk. Sarif Hassan
  • Divya Ahluwalia
  • Ravi Kiran MaddaliEmail author
  • Monika Manglik
Regular Article
  • 29 Downloads

Abstract.

In population dynamics, the Nicholson-Bailey model describes the host-parasitoid system which has been well studied since 1930 with a consideration that the parameters are all positive real numbers. In this article, the dynamics of the Nicholson-Bailey model \(x_{n+1} = x_{n} (e^{r(1-\frac{x_{n}}{\kappa}) - ay_{n}})\) and \(y_{n+1} = x_{n} (1-e^{-ay_{n}})\) is reinvestigated computationally where all the parameters are considered as real numbers. The model has all sorts of dynamical behavior such as chaotic, periodic and locally stable/unstable equilibriums. In addition, the dynamics of the scaled Nicholson-Bailey \(x_{n+1}=(x_{n} + \alpha) (e^{r(1-\frac{(x_{n}+\alpha)}{\kappa})-a(y_{n}+\beta)})\), \( y_{n+1}=(x_{n} +\alpha) (1-e^{-a(y_{n}+\beta)})\) where \(\alpha\) and \(\beta\) are scaling factors and of the noisy model \( x_{n+1}=x_{n} (e^{r(1-\frac{x_{n}}{\kappa})-ay_{n}})+\nu_{1}\), \( y_{n+1}=x_{n} (1-e^{-ay_{n}})+\nu_{2}\), where \((\nu_{1}, \nu_{2})\) is uniformly distributed noise over the interval (0,1), are also reconnoitered computationally.

References

  1. 1.
    M.P. Hassell, The Dynamics of Arthropod Predator-Prey Systems (Princeton University Press, 1978)Google Scholar
  2. 2.
    M.P. Hassell, H.N. Comins, R.M. Mayt, Nature 353, 255 (1991)ADSCrossRefGoogle Scholar
  3. 3.
    M. Mangel, B.D. Roitberg, Theor. Populat. Biol. 42, 308 (1992)CrossRefGoogle Scholar
  4. 4.
    A.A. Berryman, Ecology 73, 1530 (1992)CrossRefGoogle Scholar
  5. 5.
    A.J. Nicholson, V.A. Bailey, The Balance of Animal Populations: Part I, in Proceedings of the Zoological Society of London, Vol. 105, No. 3 (Blackwell Publishing, Ltd., Oxford, UK, 1935) pp. 551--598Google Scholar
  6. 6.
    R.M. May, J. Anim. Ecol. 47, 833 (1978)CrossRefGoogle Scholar
  7. 7.
    M.P. Hassell, R.M. May, J. Anim. Ecol. 42, 693 (1973)CrossRefGoogle Scholar
  8. 8.
    M.N. Qureshi, A.Q. Khan, Q. Din, Adv. Differ. Equ. 2014, 62 (2014)CrossRefGoogle Scholar
  9. 9.
    Z. Zhou, X. Zou, Appl. Math. Lett. 16, 165 (2003)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    X. Liu, Appl. Math. Model. 34, 2477 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    J.R. Beddington, C.A. Free, J.H. Lawton, Nature 225, 58 (1975)ADSCrossRefGoogle Scholar
  12. 12.
    H.N. Agiza, E.M. Elabbasy, E.K. Metwally, A.A Elsadany, Nonlinear Anal. Real World Appl. 10, 116 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    A.D. Taylor, Am. Nat. 132, 417 (1988)CrossRefGoogle Scholar
  14. 14.
    Q. Din, Adv. Differ. Equ. 2013, 95 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Q. Din, Chaos, Solitons Fractals 59, 119 (2014)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    C. Huffaker, Hilgardia 27, 343 (1958)CrossRefGoogle Scholar
  17. 17.
    A.J. Lotka, Elements of Physical Biology (Williams and Wilkins, Baltimore, Md, 1925)Google Scholar
  18. 18.
    E. McCauley, W.G. Wilson, A.M. de Roos, Am. Nat. 142, 412 (1993)CrossRefGoogle Scholar
  19. 19.
    A. Bergman, B. Gligorijevic, Eur. Phys. J. Plus 130, 203 (2015)CrossRefGoogle Scholar
  20. 20.
    S. Ruan, D. Xiao, SIAM J. Appl. Math. 61, 1445 (2001)CrossRefGoogle Scholar
  21. 21.
    S. Agarwal, M. Fan, Appl. Anal. 81, 801 (2002)MathSciNetCrossRefGoogle Scholar
  22. 22.
    R. Asheghi, J. Biol. Dyn. 8, 161 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Physica D 126, 285 (1985)ADSCrossRefGoogle Scholar
  24. 24.
    B.E. Kendall, Chaos, Solitons Fractals 12, 321 (2001)ADSCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsPingla Thana MahavidyalayaWest BengalIndia
  2. 2.Department of Mathematics, School of EngineeringUniversity of Petroleum and Energy StudiesDehradunIndia

Personalised recommendations