Advertisement

Bifurcation and mixed tracking of the discrete fractional LPA model

  • Ibiyinka Fuwape
  • Samuel Ogunjo
Article

Abstract

The memory effect in fractional calculus has made it a more realistic approach for the practical modeling of real life phenomena ranging from population, electric circuits, etc. In this paper, a discrete fractional order model for the Larva–Pupa–Adult (LPA) beetle population dynamics is proposed. The bifurcation of the proposed model showed novel dynamics. The mixed tracking control of the discrete fractional order LPA model was also considered to adjust individually and independently the population of each of larva, pupa or adult.

Keywords

Active control Fractional order calculus LPA model Flour beetles Chaos control Synchronization 

References

  1. 1.
    May RM (1974) Biological populations with non-overlapping generations: stable points, stable cycles and chaos. Science 186:645CrossRefGoogle Scholar
  2. 2.
    Edelstein-Keshet L (1988) Mathematical models in biology. Random House, New YorkzbMATHGoogle Scholar
  3. 3.
    Danca M, Codreanu S, Bakó B (1997) Detailed analysis of a nonlinear prey-predator model. J Biol Phys 23(1):11.  https://doi.org/10.1023/A:1004918920121 CrossRefGoogle Scholar
  4. 4.
    Shukla MK, Sharma BB (2017) Investigation of chaos in fractional order generalized hyperchaotic Henon map. AEU Int J Electron Commun 78:265.  https://doi.org/10.1016/j.aeue.2017.05.009. http://linkinghub.elsevier.com/retrieve/pii/S1434841117302273
  5. 5.
    Wu GC, Baleanu D, Zeng SD (2014) Discrete chaos in fractional sine and standard maps. Phys Lett A 378(5–6):484.  https://doi.org/10.1016/j.physleta.2013.12.010. http://linkinghub.elsevier.com/retrieve/pii/S0375960113011092
  6. 6.
    Wu GC, Baleanu D (2014) Discrete fractional logistic map and its chaos. Nonlinear Dyn 75(1–2):283.  https://doi.org/10.1007/s11071-013-1065-7 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Atici FM, Eloe PW (2009) Initial value problems in discrete fractional calculus. Proc Am Math Soc 137:981MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Abdeljawad T (2011) On Riemann and Caputo fractional differences. Comput Math Appl 62(3):1602.  https://doi.org/10.1016/j.camwa.2011.03.036. http://linkinghub.elsevier.com/retrieve/pii/S089812211100188X
  9. 9.
    Boccaletti S (2000) The control of chaos: theory and applications. Phys Rep 329(3).  https://doi.org/10.1016/S0370-1573(99)00096-4. http://linkinghub.elsevier.com/retrieve/pii/S0370157399000964
  10. 10.
    Boccaletti S, Kurths J, Osipov G, Valladares D, Zhou C (2002) The synchronization of chaotic systems. Phys Rep 366(1–2):1 .  https://doi.org/10.1016/S0370-1573(02)00137-0. http://linkinghub.elsevier.com/retrieve/pii/S0370157302001370
  11. 11.
    Ojo KS, Njah AN, Ogunjo ST (2013) Comparison of backstepping and modified active control in projective synchronization of chaos in an extended Bonhoffer van der Pol oscillator. Pramana 80(5):825. http://link.springer.com/article/10.1007/s12043-013-0526-3
  12. 12.
    Ogunjo ST, Ojo KS, Fuwape IA (2017) Comparison of three different synchronization scheme for fractional chaotic systems. In: Azar AT, Vaidyanathan S, Ouannas A (eds) Fractional order control synchronization chaotic syst. Stud Comput Intell. Springer: Berlin. https://link.springer.com/chapter/10.1007/978-3-319-50249-6_16
  13. 13.
    Ojo K, Ogunjo ST, Williams O (2013) Mixed tracking and projective synchronization of 5D hyperchaotic system using active control. Cybern Phys 2(1):31Google Scholar
  14. 14.
    Cushing J (2003) Cycle chains and the LPA model. J Differ Equ Appl 9(7):655.  https://doi.org/10.1080/1023619021000042216. http://www.tandfonline.com/doi/abs/10.1080/1023619021000042216
  15. 15.
    Kuang Y, Cushing JM (1996) Global stability in a nonlinear difference-delay equation model of flour beetle population growth. J Differ Equ Appl 2(1):31.  https://doi.org/10.1080/10236199608808040 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ogunjo ST, Fuwape IA, Olufemi OI (2013) Chaotic dynamics in a population of Tribolium. FUTA J Res Sci 9(2):186Google Scholar
  17. 17.
    Chen F, Luo X, Zhou Y (2010) Existence results for nonlinear fractional difference equation. Adv Differ Equ 2011(1):713201.  https://doi.org/10.1155/2011/713201 MathSciNetzbMATHGoogle Scholar
  18. 18.
    Pecora LM, Carroll TL (1990) Synchronization in chaotic systems. Phys Rev Lett 64(8):821.  https://doi.org/10.1103/PhysRevLett.64.821 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Federal University of TechnologyAkureNigeria
  2. 2.Michael and Cecilia Ibru UniversityUghelliNigeria

Personalised recommendations